Spectral near field data of LED systems for optical simulations

This work presents, summarizes and validates a fast, accurate and general measurement and modeling technique to obtain spectral near field data of LED systems in order to improve the optical design process of modern high quality LED systems. It requires only a minimum of goniophotometric near field measurements as well as no time-consuming angularly resolved spectral measurements. The procedure is named physically motivated basis spectra (PMBS) as its main assumption is that each piece of angularly and spatially varying spectral information can be described as the weighted sum of its physical basis spectra such as the individual semiconductors or a phosphor. Based on detailed spectral information regarding the goniophotometric measurement setup, the spectral model is obtained by solving a simple system of linear equations using the obtained near field measurements. The complete process is validated and applied to different state-of-the-art LED systems. The obtained results can be used directly in state-of-the-art ray tracers.


INTRODUCTION
The sole requirement of the first commercially used artificial light sources -gas lights -was to provide light. However, starting with the development of incandescent lamps, artificial light sources began to compete in economic aspects such as production costs, lifetime and energy efficiency. Both technologies improved and soon artificial light sources became common and the field of general lighting started to grow. During the 1960s and 1970s, in the shadows of the established light sources, another light source technology was invented [1]. This was the starting point of the development of light emitting diodes (LEDs) [2], [3].
The basis of the LED breakthrough originated in 1994. It was the invention of the first high brightness blue LED [4] by Nakamura et. al.
as it enabled white LED based lighting. The LED technology rapidly improved in the following years and quickly started to be competitive in terms of lifetime and energy efficiency. Nowadays, LEDs have become one of the most important light sources in lighting technology.
LED lighting applications range from medical [5], [6], horticulture [7] and automotive lighting [8] over several general lighting applications such as street lighting, work place lighting as well as shop and home lighting [9]. It is expected that the importance of LEDs will further increase in the next few years [3], [10].
However, not only the direct efficiency of the light source itself helped LEDs to become such a success story. Their small sizes and comparable low operation temperatures also enabled the possibility of placing INTRODUCTION cost-efficient plastic optics near to the LED source [11]. Examples are light guides, free-form lenses or metal coated free-form reflectors.
A feasible optical design of these free-form elements ensures a high optical system efficiency, specific photometric quantities and a compact system size. However, the design of these compact systems requires the optical designer to include the near field radiation properties of each light source in the system [12], [13]. This means the light source model has to describe the direction of the light rays coming from the source and their starting points on the spatially extended source. This information is essential for designing specific, efficient and robust high quality optical systems.
The measurement technique which provides this near field information is called near field goniophotometry [14], [15]. The measurement result obtained is called ray file. It can be used in commercial ray tracing software [16], [17]. This software enables to trace light rays according to the rules of geometrical optics and thus to assess an optical design. Therefore, near field goniophotometry, ray files and ray tracing have become important optical measurement and simulation tools which enable the design of efficient high quality lighting systems considering the near field of LEDs [12], [18].
However, in modern lighting technology, "lighting quality" refers to a permanently increasing number of different aspects, which also depends on the specific application. Besides photometric quantities and efficiency this may include visual and aesthetic properties such as uniformity, color uniformity and color rendering or non-visual effects of light on human beings called human centric lighting [2], [3], [19].
Examples are the influence of the color temperature on the day and night rhythm [20], [21], the concentration during work or the learning process of school children [22].
All these additional quality aspects are directly related to the spectrum. Different material systems and LED chip designs provide different INTRODUCTION spectral power distributions. These can be combined with different phosphors, which convert spectral parts of the light from the LED chip into other spectral parts. This enables the design of specific spectral power distributions. These distributions can be optimized in terms of several different quality aspects such as efficiency, chromaticity, color rendering or other color appearance metrics as well as non-visual aspects [23]- [26]. At the same time, the desired uniformity requirements have led to the need for efficient mixing optics for different LEDs to ensure a high lighting quality in all the above mentioned aspects [27]- [29].
However, the optical design of color mixing optics as well as the simulation of complete spectral distributions in an optical system within a ray tracer is a challenging task. The simulations are not as precise as photometric simulations and require a comparably large amount of computation time [30]. Furthermore, in contrast to conventional incandescent or fluorescent lamps, the required modern LED multi-chip systems as well as plain white LEDs can show strong spectral variations in both the angular and the spatial dimension. These variations can also differ for individual LEDs due to manufacturing tolerances [16], [31]. Thus, the optical design requires a accurate LED models which precisely describe the complete spectral radiation in each direction from each point of the physical light sources. These models can be referred to as hyperspectral LED models.
Since conventional near field goniophotometry does not directly provide spectrally resolved near field information, there is a lack of suitable hyperspectral LED light source models for the ray tracing simulations. Nevertheless, if the optical design is based on wrong assumptions regarding the light source properties, the best design and the most precise and time-consuming simulation will still lead to an unsatisfactory result. This leads to a need for precise hyperspectral LED models in terms of the angular and the spatial dimension.

AIMS OF THE THESIS
The main aim of this thesis is to propose, develop and validate a measurement and modeling technique to obtain hyperspectral LED models which can be used in conventional non-sequential ray tracing software. In contrast to the existing state of the art, this method has to combine a suitable measurement time with acceptable spectral deviations in the context of lighting technology. This also includes the required measurement systems, as well as the model assumptions and the deduced limitations so that the method can be applied to state-ofthe-art LED systems. The models have to be capable of simulating a spectrally varying distribution, which may be based on an angularly and spatially varying LED source and a spectrally sensitive optical

STRUCTURE OF THE THESIS
The thesis develops, applies and validates one specific method to obtain hyperspectral LED models. The fundamentals, which are required in the thesis, are briefly summarized in chapter 2 which mainly focuses on state-of-the-art near field goniophotometry and some aspects of ray tracing and optical design as well as light source models. Furthermore, spectral properties of LEDs as well as selected colorimetric terms and concepts are briefly explained.

CHAPTER 2 FUNDAMENTALS
The aim of this chapter is to summarize the necessary fundamentals regarding the spectral characteristics of LEDs, photometry and colorimetry. Furthermore, ray tracing and its typical light sources as well as near field goniophotometry are briefly introduced. A more complete and general overview of the fundamentals is given in the cited literature.

SPECTRAL PROPERTIES OF LEDS
A Light Emitting Diode (LED) is an optoelectronic semiconductor converting electrical energy into incoherent radiation. A part of the electrical energy is also converted into thermal energy. The emitted radiant power distribution in watt per frequency or wavelength λ is called spectral power distribution S(λ) or in short spectrum. The spectrum depends on the specific material system of the LED as well as its thermal and electrical operation conditions. The wavelength at which the highest radiation occurs is called peak wavelength. In this thesis, the peak wavelength is the only characteristic wavelength associated with an LED spectrum. Examples of LED spectra and their peak wavelengths are given in Fig. 2

.1. It is common in lighting
technology to visualize all quantities as a function of the wavelength in vacuum rather than the frequency.
The spectral width of LED spectra prohibit the direct generation of white light using solely one semiconductor chip. However, as the gen-eration of white light is an important aspect in lighting technology, two possibilities for creating white LED based light have been developed.
One is the additive mixture of different LED spectra with spatially separated LED chips, for instance red, green and blue. The color is tuned by adapting the power of the individual chips. However, the color mixing of the spatially separated chips is a challenging task.
White light can also be created by combining typically a blue or ultraviolet LED chip with one or several conversion phosphors. The phosphor partly absorbs the high-energy LED radiation and re-emits radiation with longer wavelengths. Due to the additional scattering properties of the phosphor, the resulting radiation is partly mixed. The thickness of the phosphor layer can be used to tune the light color. A thicker layer results in a yellower or warm white lighting. Figure 2.2 (a) shows white LED spectra, which are based on a RGB LED system, and uniformity. This results from an angular dependent "LED radiation" to "phosphor radiation" ratio caused by deviating phosphor conversion lengths. An example of this phenomenon is discussed in detail in section 8.1. More information on LEDs, LED systems, and conversion phosphors is provided in [36], [37].  [38], [39].

RADIOMETRY, PHOTOMETRY AND COLORIMETRY
Radiometry is the science of measuring the optical part of electromagnetic radiation ranging from ultraviolet to infrared [40]. Photometry can be considered an area of radiometry, which focuses solely on the visible part of the spectrum. Another important difference between them is the application of specific actinic action spectra. Actinic action spectra are weighting functions, which are used to weight the measured spectral power distribution according to photobiological or photochemical effects. In radiometry, there is no specific weighting function. In photometry, the v(λ) function, which is used to model the sensitivity of the human eye, is the most important weighting function.
If an actinic action spectrum or any other spectral weighting function is applied, the spectrally integrated value changes. In terms of the v(λ) function, the resulting quantities have specific names such as luminous power, light intensity, illuminance or luminance to distinguish them from their radiometric counterparts such as radiometric power, radiant intensity, irradiance or radiance. In this thesis, all arbitrary weighted and spectrally integrated values obtained with any function different from the v(λ) function are marked with the index "pseudo" to distinguish them from the unweighted radiometric and photometric values. More information on the basis quantities of radiometry and photometry can be found in [40], [41].
Two other important actinic action spectra are the color matching functions x(λ) and z(λ), which are related to human color perception.
Similarly to the v(λ) function, they were defined by the Commission Internationale de l'Eclairage (CIE). The color matching functions x(λ), z(λ) and the y(λ) = v(λ) function are the basis of the CIE 1931 color space. The integration of a spectrum S(λ), which is weighted with a color matching function, results in a tristimulus value X CIE , Y CIE , or Z CIE : The tristimulus values serve as an important starting point for the science of color measurement called colorimetry. An important light color representation used in this thesis is the CIE 1976 uniform chromaticity scale diagram or in short CIE u v' diagram. The coordinates are the so called chromaticity coordinates u and v', which are calculated according to In contrast to a similar diagram, the CIE 1931 xy diagram, the euclidean distance ∆u v between two points u 1 , v 1 and u 2 , v 2 in the CIE u v' diagram is approximately proportional to the perceived difference of their light color stimuli. The CIE thus recommends evaluating white light color differences [42], [43] and also saturated light color differences with this diagram [43]. The color matching functions and the CIE u v' diagram are shown in Fig. 2.3.
Not all colorimetric values are based on the tristimulus values. The color rendering index (CRI) and its suggested successors such as TM- , which are all used to rate the capability of a light spectrum to render object colors, require the complete spectrum S(λ) because they are based on different spectral weighting functions. A comprehensive overview on this topic is provided in [44]. . More information on colorimetry and more advanced concepts such as color appearance models are provided in [45], [46].

RAY TRACING
The precise propagation of each light source's electromagnetic radiation can be described with Maxwell's equations. However, solving these differential equations is both computationally expensive and unnecessary in the typical optical design process of lighting technology. In lighting technology, the spatial dimensions of the optically relevant geometries are usually much larger than the wavelengths of the usually incoherent radiation and effects as interference or diffraction can be neglected without affecting the precision [12]. In these cases, the mathematical assumption λ → 0 is valid and leads to geometrical optics [47]. Then the light can be approximated in the form of rays, which are perpendicular to the wave front and only the rules of geometrical optics such as Snell's law of refraction, the law of reflection or the Fresnel equations are considered if a ray encounters a material interface [48].
A ray tracer is a software tool, which facilitates the application of the rules of geometrical optics 1 on up to several billion rays within a typically three dimensional opto-mechanical setup. Besides the geometries, all optical material properties such as refractive indices, dispersion relations or surface parameters are usually part of the calculations.
Ray tracers are typically used to design optical elements such as lenses, reflectors or total internal reflection optics in such a way that they provide a specific light distribution or illumination in combination with a specific light source or to predict the visual perception of an 1 Ray tracers may also support physical optics calculations.
illumination scenario. Therefore, light distributions can be evaluated regarding their radiometric, photometric or colorimetric properties. Furthermore, system analyses can be performed, for instance to visualize specific ray paths called sequences within the opto-mechanical setup. A typical ray tracing setup consists of a source, which serves as ray origin, the opto-mechanical setup and sensors, which evaluate the tracing results. An example is shown in Fig. 2.4. There is sequential and non-sequential ray tracing. In sequential ray tracers, the sequence of the optically active geometries is predefined. It is faster and typically used to design imaging optics. In contrast, non-sequential ray tracing has no predefined tracing sequence. It is typically used to design non-imaging optics or to simulate effects such as stray light. There are also cases in which both approaches are combined. An example is the design of modern head-up-displays [49].
In direct non-sequential ray tracing, rays start at a defined light source and are evaluated at defined sensors. In inverse ray tracing, the rays start at the sensor and propagate to the sources. Inverse ray tracing can be used if only a small number of rays from the source reach the sensor. In both cases, the rays are traced through the optical system until they are absorbed or until they have no more interactions with the optical system. In this thesis, the ray tracer OptisWorks is used to evaluate the light source models.
Several ray tracers, for example OptisWorks, use a Monte Carlo approach. This means that each ray is either completely reflected, completely transmitted or completely absorbed with a specific probability, which is defined by the rules of geometrical optics. This also means that if the same ray is traced twice, its sequence throughout the optical system may change. This has the consequence that the number of rays is an important precision criterion of the simulation. The statistical error of a Monte Carlo ray tracer is indirectly proportional to the square root of the number of rays used.
Typically, the optical design process consists of several steps such as the initial design and the design optimization [12]. The initial design provides a first geometry, which is based on simplified assumptions. It is followed by the design optimization, which aims to optimize the initial geometry by considering more realistic conditions. An important aspect in each step of the optical design is to choose an appropriate light source model. There are several types of light source models in ray tracers, which will be introduced below. More information regarding the optical design process and optical design is provided in [12], [48].  it is thus only described by its light intensity distribution curve. The error associated with this approximation is usually neglected, if the distance between the source and the observer or the optical element is beyond the photometric limiting distance. More information on this distance is provided in [50]. Although it is the least precise model, it is often chosen because it enables the use of advanced mathematical optical design techniques such as the supporting quadric method [12], [51] or tailoring [51] during the initial design step [12]. The point source model may be based on a solely mathematical model, a goniophotometric measurement or an indirect camera measurement [52].
Geometrical light source models are a possibility for modeling extended light sources [12]. They extend from simple models, for instance an extended plane, to complex physical CAD models with defined optical parameters of the different materials. A schematic example is provided in Fig. 2.5 (b). Often, the core of these models is another light source model or a mathematical description for instance a Lambertian emitting plane. Radiance or luminance maps may be used as source models as well [53], [54]. They are created by ray tracing of another light source model. These models speed up the ray tracing process and allow both direct and inverse ray tracing.
A ray file as shown in Fig. 2.5 (c) models an extended light source as well. It represents the near field of the source as a list of vectors. Each vector represents a ray with a three dimensional starting point and a normalized direction [16]. The luminous or (pseudo-)radiometric flux is either provided by the density of the rays with constant amplitudes or an individual amplitude for each ray. There are several different ray file formats for the individual ray tracing softwares. Some formats allow the inclusion of additional information like the wavelength or the polarization. In 2013, the TM-25 format was defined by the Illuminating Engineering Society (IES) as a general standard for the ray file format [17], [55]. Ray files are typically used in the optical design process if the extension of the source can no longer be neglected.
However, they allow only direct ray tracing. Ray files are typically provided on the home page of the light source manufacturers. They are usually based on near field goniophotometric measurements [14], [15].

NEAR FIELD GONIOPHOTOMETRY
Ray files are usually based on a set of luminance or pseudo-radiance images, which are measured goniometrically [56]. In this measurement technique, the luminance images are obtained from different angular positions. These measurements are performed with a so called near field goniophotometer 2 , as shown in Fig. 2.6 (a). A typical near field goniophotometer consists of at least two moving axles, an imaging luminance measurement device (ILMD 3 ) -also referred to as a luminance or color camera -, a photometer and sometimes also a spectroradiometer.
In the ILMD, an objective (lens) images the luminance of the device under test on a spatially resolved sensor, for instance a CCD (charge-coupled device) or CMOS (complementary metal-oxidesemiconductor). An additional optical transmission filter adapts the spectral sensitivity of the system to the v(λ) function. The ILMD 2 Strictly speaking, it is more often used as near field gonio(pseudo-)radiometer in this thesis. Nevertheless, in accordance with the commonly used terminology, the terms goniophotometry and goniophotometric are used in these cases as well. 3 Strictly speaking, it is more often used as imaging (pseudo-)radiometric measurement device in this thesis. Nevertheless, in accordance with the commonly used terminology the abbreviation ILMD is used in these cases as well.
NEAR FIELD GONIOPHOTOMETRY provides the luminance and pseudo-radiance images and thus the near field information. The spectral information is lost due to the optical integration on the sensor.
One or several filter wheels within the ILMD may allow the replacement of the v(λ) matching filter by another filter to adapt the spectral sensitivity of the system to other actinic action spectra such as x(λ) and z(λ). To reduce the complexity in the manufacturing process, the x(λ) adaption filter is sometimes represented by two filters. They are named x short (λ) and x long (λ) in this thesis. A schematic ILMD with a filter wheel is visualized in Fig. 2.6 (b). These systems can be used to apply selected actinic action spectra directly on the measured spectra and thus to obtain a ray file set, which represents the selected quantity.
However, these ray files are not suited to perform spectral simulations, which will be discussed in more detail in section 3.2. The photometer as well as the spectroradiometer provide no spatial resolution. Although they are mounted on a near field goniophotometer, the source models obtained solely with these sensors can be considered as far field models. The photometer's system sensitivity is also adapted to the v(λ) function. In near field goniophotometry the photometer, which also provides no spectrally resolved information, is used to obtain the absolute luminous flux of ray files, which are based on a v(λ) ILMD-based goniophotometric measurement.
The spectroradiometer is the only sensor which provides spectrally resolved information. There are array and scanning spectroradiometers. Although a scanning spectroradiometer offers a higher spectral resolution and a larger stray light suppression, which leads to a higher dynamic range and therefore a smaller measurement uncertainty [59], its higher measurement time is unfavorable in combination with a goniometer. This is why the faster but less precise array spectroradiometers are typically used for goniometric measurements. The measurement time of an angularly resolved goniometric spectral measurement is still very high compared to a similarly resolved ILMDbased measurement due to the difference of the integration times ultimately leading to a start-stop-motion of the goniometric spectral measurement [59].
In this thesis, the "RIGO801 near field goniophotometer" from the TechnoTeam Bildverarbeitung GmbH was used to perform the near field measurements. It is equipped with a photometer from the Czibula & Grundmann GmbH, a "LMK5 color" camera from TechnoTeam and a "specobs 1211" spectroradiometer from JETI Technische Instrumente GmbH. The camera has an integrated filter wheel, which is equipped with v(λ), x short (λ), x long (λ) andz(λ) matching functions as well as an infrared and a glass filter, which has no significant spectrally selective absorption.

CHAPTER 3 SPECTRAL RAY FILES
In this chapter, the concept of the 6-dimensional data set "spectral ray file" is briefly described and motivated. Subsequently, ideal properties of measurement techniques to obtain spectral ray files are deduced and used to assess the briefly and solely qualitatively described state of the art.

PLENOPTIC FUNCTION
The 7-dimensional plenoptic function P(X, Y, Z, φ, ϑ, λ, t) is the most general light source description in the context of the ray optic assumption if polarization is neglected [60]. Each ray is described by its 3-dimensional starting point (X, Y, Z), the angular ray direction (φ, ϑ) and the wavelength λ. In the lighting industry, temporal variations t are usually neglected as the lighting distribution is assumed to be static or at least quasi-static, which reduces the plenoptic function to the 6-dimensional non-temporal plenoptic function R(X, Y, Z, φ, ϑ, λ).
If there is no spectral information available, the plenoptic function is reduced to the 5-dimensional usually photometric or pseudo-radiometric non-spectral-non-temporal plenoptic function M(X, Y, Z, φ, ϑ).
A ray file is a discrete representation of the plenoptic function. Each directly measured ray file represents the 5-dimensional non-spectralnon-temporal plenoptic function because the spectral information is lost due to the optical integration on the ILMD (Imaging Luminance Measurement Device) as described in section 2.2 and 2.4.
In contrast, a spectral ray file represents one additional dimension, which is the wavelength as shown in Fig. 3.1 so that each ray has a specific starting point, angular direction and wavelength. Before the measurement properties and state-of-the-art techniques to create spectral ray files are discussed in section 3.3 and 3.4, section 3.2 explains the advantages of spectral ray files over a conventional ray file in commercial ray tracing software.

APPLICATION OF SPECTRAL RAY FILES
According to the assumption of geometrical optics, the wavelength is unimportant during ray tracing and only the geometric relations of rays and surfaces as well as material parameters are relevant. The wavelength is only required to calculate photometric quantities, which depend on actinic action spectra. Examples are light and object colors, color rendering, blue light hazard or circadian effects. Thus, the wavelength does not seem to be necessary for the tracing process itself.
However, this assumption is wrong because the wavelength does have an indirect influence on the ray tracing process.
If a ray hits a material interface, the ray tracer needs to assign the physical parameters of the materials such as absorption coefficient and the index of refraction. These parameters depend on the wavelength.
For instance, in the visible part of the spectrum the index of refraction of classical optical materials is usually higher for smaller wavelengths.
According to Snell's Law, this leads to different angles of refraction for different wavelengths, which is referred to as chromatic aberration.
The image of the prism shown in Fig The same can happen if different spectral absorptions occur in the system, which is for instance typical in rear lights.
This wavelength dependency is also the reason why it is not feasible to obtain any photometric quantity "prior" to the ray tracing. The effect of the wavelength dependent material parameters changes the results, as a function of angular or spatial dimension on the sensors of the ray tracer as it has in Fig. 3.2. Therefore, the design process of a complex optical system requires the inclusion of the wavelength dependent material parameters and the spectral information of the source.
In the case of conventional light sources such as tungsten halogen lamps, a global spectrum can be added to the ray file because the spectrum does not change as a function of spatial or angular dimension.
The spectrum serves as a global probability density function which statistically assigns a wavelength to each ray. The optical design process of complex systems thus only requires modeling the wavelength dependence of the material parameters. However, in the case of LEDs there might be spectral variations as a function of angular or spatial dimension which can be caused by the spatial separation or by different intensity distributions of the individual LED chips and conversion phosphors. Then, attaching a global spectrum leads to errors [61]. To minimize these errors and achieve a realistic light source model, the conventional 5-dimensional ray file has to be replaced by a 6-dimensional spectral ray file. To sum up, to achieve precise spectral and colorimetric simulation results in commercial ray tracing software, realistic light source models incorporating the complete spectral information of the source are required.

RAY FILE MEASUREMENTS
The aim of computer-aided optical design is to accelerate the development process of optical components and optical systems. The spectral ray files should allow colorimetric and spectral simulations with realistic predictions within established non-sequential simulation software.
Further, the measurement effort has to be reasonable with respect to PROPERTIES OF SPECTRAL RAY FILE MEASUREMENTS the cost and time saving aspect of the computer aided lighting process. On the one hand it makes no sense to create a "perfect ray file" because there are other model assumptions in the ray tracer, such as manufacturing tolerances of the CAD models, which create a bias as well and because the required storage and measurement effort would increase significantly. On the other hand a fast and cheap measurement technique which does not provide useful results will not be used in any application. Finding the right trade-off between these aspects is crucial for each measurement task.

Spectral deviation
The spectral deviation between two spectra may be described as residual sum of squares (RSS) or as chromaticity distance ∆u v . While the RSS is not prone to metamerism and suited for comparative analyses, its non-interpretable values are not suited for assessing absolute deviations. In these cases, the interpretable chromaticity distance ∆u v is used. Typically, both figures of merit are highly correlated. Naturally, the tolerable spectral deviation of a spectral ray file always depends on the precise task. A typical one would be ensuring the color uniformity of an illumination. This is challenging because the human eye is a very sensitive detector when it comes to color discrimination. Nevertheless, the spectral deviation of a spectral ray file has to ensure at least the usability in such applications. The exact color discrimination threshold in u v depends on the light color and the viewing conditions. Thus it varies from individual to individual and therefore also from study to study. In the case of white illumination, which has tough uniformity requirements, it typically lies in the range around 4 × 10 −3 < ∆u v < 1.6 × 10 −2 in practical applications [62].

Memory requirements and interface
The generated data format has to be capable of being integrated in existing non-sequential ray tracing software. Even if a measurement technique creates adequate results in a short amount of time, the results have to retain their properties in the established software without a complex adaption of the simulation software. Otherwise the usability decreases. The memory requirements should also remain reasonable because ray files are often provided on the home page of the LED manufacturers for download.
This may become critical in the case of a 6-dimensional quantity. In the best case a standardized format as TM-25 [17], [55] can be used directly.

Measurement system
As mentioned above the measurement effort has to be reasonable. Measuring the 5-dimensional quantity ray file with a moving scan typically requires between several minutes and a few hours. The measurement time depends on the measurement resolution, scanning range and the integration times of the sensors.
As these measurement times are currently tolerated, the measurement time of a spectral ray file should be in the same order of magnitude. In particular, a start-stop-motion of the goniometer should be avoided because it heavily increases the measurement time. Another aspect that should be considered is the potential need for additional nonconventional measurement equipment such as a hyperspectral camera.
It is a structural and financial advantage, if a method only requires established measurement equipment.

CREATE SPECTRAL RAY FILES
The most obvious and simple method to create spectral ray files would be an individual ILMD-based measurement of each LED source in the system followed by their combination. However, this approach has several problems [61]. ILMD-based measurements are required to be goniometric in order to include angularly resolved information.
The spectral information of an ILMD-based measurement is lost due to the optical integration of the spectrum on the ILMD sensor. In combination with a goniometer it provides the 5-dimensional M(X, Y, Z, φ, ϑ).
However, it is possible to weight the spectrum by using different optical transmission filters prior to the integration on the chip. Each filter optically weights the spectrum differently and therefore generates different spectrally integrated pseudo-radiometric information.
The number of required ILMD-based measurements with different optical filters varies in the state-of-art methods. On the other hand, the spectral measurements do not generate spatially resolved information because the spectroradiometer images one "pixel", but spectrally resolved information. In combination with the goniometric system the spectral information may be obtained as a function of the angle.
Nevertheless, only the combination of both the ILMD and the spectroradiometer provides enough information in terms of spatial, angular and spectral resolution to create a spectral ray file, which represents R(X, Y, Z, φ, ϑ, λ), if no additional measurement equipment, such as hyperspectral cameras, are used.
Rykowksy summed up the four different possibilities for combining the spectral measurements with the ILMD-based measurements with respect to the required measurement effort [63]. Table 3.1 shows proposed state-of-the-art methods and their assignment to these four options. The columns distinguish the measurement methods between the need of angularly resolved spectral measurements and one spectral measurement. Note that only one state-of-the-art method avoids angularly resolved spectral measurements. The rows distinguish the measurement methods between the required number of ILMD-based measurements with different optical transmission filters, which is either equal to one or larger.

SPECTRAL ADJUSTING
One of the first methods to create spectral ray files was published by Radiant Vision in 2011 [63]. This method generally requires angularly resolved spectral measurements and ILMD-based measurements with at least three optical filters to obtain spatially resolved colorimetric information at the same angular positions. For each selected starting location (ILMD pixel) the associated far field spectrum is adjusted to match the measured colorimetric information. The adjustments are based on an iterative algorithm such that "as much features of the original spectral power distribution as possible remain" [63]. A more precise description of the algorithm is not provided as it is the intellectual property of Radiant Vision. The algorithm proceeds until the spectrum matches the measured color within a user defined tolerance.
The authors suggest a colorimetric tolerance of u v < 0.003. The resulting data format provides an adjusted spectrum at each location and angle of radiation. This format cannot be used in a ray tracer directly because one ray requires one wavelength and not a complete spectrum.
Therefore, a specific wavelength is assigned to each ray based on the provided spectrum of the starting point and the ray direction to allow the ray tracing. The memory requirements of the 6-dimensional raw data prior to the wavelength assignment are rather high.
The spectral adjustment can be prone to metamerism as the spectral adjustment is not unique since different spectra can produce the same color stimulus. This problem intensifies if the number of spectral sources increases and the term "spectral feature" becomes harder to define. The largest disadvantage, however, is the measurement time because angularly resolved spectral measurements are recommended to obtain good results. Nevertheless, this method can produce angularly and spectrally resolved spectral ray files by using only conventional measurement techniques.

POLYCHROMATIC RAY DATA
The Opsira GmbH published another method to create spectral ray files [64]. It combines between seven and ten ILMD-based measurements with optical bandpass filters, which are evenly spaced across the visible spectrum, to create the same amount of different ray files.
This ray file set serves as the basis for the ray tracing simulation [61], [65]. An angularly resolved spectral measurement allows the absolute radiometric weighting of each ray file by integrating the measured spectra in the defined spectral intervals and over the angular dimension. The authors note that strong spectral variations along the angular measurements may require an individual weighting of each ILMD picture prior to the ray generation in order to improve the absolute weighting [61]. This requires the angular positions of the ILMD-based measurements to match those of the spectral measurements.
The spectral deviations as well as the memory requirements and the measurement time depend on the number of filters used to create the ray file set. However, the number of ILMD-based measurements is always comparably high and the required angularly resolved spectral measurements result in a start-stop-motion of the goniometer. Both facts lead to a rather high measurement time. Nevertheless, the obtained set of ray files can be used directly in ray tracing software and produces angularly and spatially resolved spectral information. Although the whole concept relies only on conventional measurement equipment, it is the most general concept so far as it works for all kinds of stable light sources and not just all kinds of LEDs.

PRINCIPAL COMPONENT ANALYSIS
Jacobs et. al. proposed a method to create spectral ray files which contain brightness variations as a function of angular and spatial dimension and spectral variations as a function of angular dimension [66]. This approach is the only method requiring just one ILMDbased measurement but still requiring angularly resolved spectral measurements.
The angularly resolved spectral measurements are used to deduce a set of principal components whose weighted sum is capable of re- A large disadvantage is that the spectral variation is only described as function of the angular dimension. Therefore, this approach is more suited to describe spatially uniform sources such as OLEDs (organic light emitting diodes) but lacks precision in the case of spatially varying spectral information.

BLUE/YELLOW APPROACH
The LED manufacturer Osram has proposed the only measurement technique which requires a single spectral measurement in addition to two ILMD-based measurements. This approach is mainly used to describe phosphor converted white LEDs and is typically referred to as the "Blue/Yellow" approach or the "Blue/Yellow" ray files [16], [67]. The main idea of this method is to separate a global measured spectrum into two parts, which represent the physical sources blue LED and yellow phosphor. ILMD-based measurements with a blue optical filter and a yellow optical filter are performed to create two ray files. The spectrum is separated at a specific wavelength into a blue and yellow part as well. Each ray file is assigned to its sharp spectral part describing its relative change as a function of angular and spatial dimension. The relative weighting between the ray files as well as the absolute weighting with respect to the radiometric flux is considered by a global weighting factor for each ray file. Although the procedure to obtain these two factors is not described in [16], [67], it is fair to assume that these weightings are based on the global measured spectrum and the overall flux of the LED. A similar method is described in subsection 4.2.2.
The main advantage of this approach is the short measurement time because there is only a need for two ILMD-based measurements and no angularly resolved spectral measurement. As the angularly and spatially resolved spectral information is provided by two ray files, it is directly available in the ray tracer and has only small memory requirements. In addition it is compatible to the standardized ray file format IES TM-25.
However, one has to accept the introduction of a systematic error if this method is used. Although the spectrum consists of two spectral sources, which are the physical LED and phosphor spectrum, there is a difference between the physical spectra and the assigned spectral parts.
The size of the error depends on the chosen separation wavelength, the characteristics of the source, the global spectrum and also on the actual filters used to create the ray files. If the filters transmit only a spectral range, which represents one spectral source directly without overlap to another source, the resulting ray files have the correct relative distribution.

HYPERSPECTRAL MEASUREMENTS
Lee et. al proposed a measurement technique which combines angularly resolved spectral measurements with one hyperspectral image [68]. First, all measurements are used to deduce sharply separated spectral parts. This is done based on a correlation analysis of the spectral change in the near-and far-field. Each spectral band is treated as an individual source.
The hyperspectral image provides the information regarding the 2dimensional starting points. The angularly resolved spectral measurements serve as a data base of the ray directions for each spectral band.
The 2-dimensional starting points on an emitting plane as well as the directions are determined by a Monte Carlo approach for each source.
A ray tracer is used to compare simulation results of the model which consists of the sum of all sources with additional photometric and colorimetric mid-field measurement data. The mid-field is defined according to [69]. It is the region between the near field (vector region) and the far field (Fraunhaufer region) in which the light pattern varies from a distance to another.
In general, the third dimension of the starting point is neither the same for different spectral bands nor provided by the geometrical 2-dimensional hyperspectral image. Therefore, the location of the emitting plane of each spectral band is also an optimization value.
This is for instance necessary in the case of phosphor converted white LEDs because the phosphor usually 3-dimensionally encapsulates the blue LED. The starting points, ray directions and emitting plane locations are altered until the normalized cross-correlation (NCC) of the simulation result and the mid-field measurement data is above a user defined threshold. The authors recommend a NCC above 0.99.
This method does not require a conventional ILMD-based goniophotometric measurement. However, although it uses hyperspectral imaging, angularly resolved spectral measurements are still required and thus lead to a high measurement time. In addition, the mid-field region has to be measured and the separation into sharp spectral bands differs from the true physical basis spectra as described in subsection 3.4.4.

PHYSICAL LED MODELS
The approach discussed in this subsection will only be mentioned peripherally as its main purpose is not the generation of spectral ray files but the analysis and improvement of LED sources and phosphor parameters [70]- [72]. The idea is to create a complete geometrical and optical physical model of the LED and, if present, the phosphor within the ray tracer.
The modeling starts with the creation of a precise LED-CAD model, which includes for instance the electrode structures and encapsulation geometries. The modeling also requires the inclusion of additional measurement data such as for instance the blue LED emission spectrum and the phosphor absorption and re-emission spectra in order to correctly predict the wavelength conversion of the blue light in the phosphor matrix. While a semiconductor is optically described as Lambertian surface radiator, the scattering in the phosphor is modeled using Mie theory and requires additional parameters that have to be measured [73] or modeled [71]. In the case of [71], unknown intrinsic phosphor parameters within the phosphor matrix are altered until simulation results in ray tracing software are in agreement with additional photometric, colorimetric and/or spectral measurement data.
Compared to the spectral ray files described from subsection 3.4.1 to 3.4.5, the resulting models are not just descriptive with respect to the angularly and spatially resolved spectral information, but offer a deeper understanding of the LED source itself. However, the individual modeling effort and required data amount and therefore the overall creation time is larger than in the other methods. Nevertheless, if the complex models exist they can be used in a similar way to spectral ray files [74].

SUMMARY
Several state-of-the-art methods exist for the creation of spectral ray files. Most of them focus on the angular color uniformity of phosphor converted white LED. The fastest amongst them is the Blue/Yellow approach because it requires no angularly resolved spectral measure-ments. The two ILMD-based measurements provide a high angular and spatial resolution. The sharp spectral separation is its main limitation, which is also true in the case of the hyperspectral measurements.
The PCA approach can also be used to obtain a spectral ray file of a solely angularly varying source. In contrast to the Blue/Yellow approach, it is not limited to LEDs and was successfully applied on a white OLED. It requires only one ILMD-based measurement, angularly resolved spectral measurements at a comparable low resolution and the assumption that the source has no spatially spectral variations.
To obtain a spectral ray file of a spatially and angularly varying source either the spectral adjusting or the polychromatic ray data can be used. However, both methods require time-consuming angularly resolved spectral measurements at the positions of the ILMD-based measurements. Although the spectral adjusting is usually faster than the polychromatic ray data because it utilizes only three ILMD-based measurements, it is also prone to metamerism and cannot be used for complex LED systems such as an RGBW LED. The only remaining measurement method besides setting up a physical model, is thus the polychromatic ray data. However, a spectral ray file of a Red/White LED system required eight ILMD-based measurements in addition to the angularly resolved spectral measurements [61]. As with the spectral adjusting, the measurement time can be reduced by decreas-

DIRECT GENERALIZATION OF THE BLUE/YELLOW APPROACH
The plenoptic function R(X, Y, Z, φ, ϑ, λ) describes the radiation of a source in terms of starting point (X, Y, Z), direction (φ, ϑ) and wavelength λ if temporal variations and polarization can be neglected. It is therefore the most complete description of a radiation source if the ray optic assumption is used. Although the state-of-the-art methods Taking a closer look, the Blue/Yellow approach from subsection 3.4.4 combines most principles. Its main drawback is that it is only described for phosphor converted white LEDs. Therefore, it has to be generalized.
The idea of the Blue/Yellow approach is to model each physical spectral source individually. The overall plenoptic function R(X, Y, Z, φ, ϑ, λ) is composed of the sum of two individual spectrally separated plenoptic functions such that in which R SB (X, Y, Z, φ, ϑ, λ) is the plenoptic function of the blue part, which models the LED, and R SY (X, Y, Z, φ, ϑ, λ) the plenoptic function of the yellow part, which models the phosphor. It is assumed that each individual plenoptic function can be divided into a nonspectral blue part M A,SB (X, Y, Z, φ, ϑ) and a non-spectral yellow part M A,SY (X, Y, Z, φ, ϑ) as well as an individual globally separated spectrum of the blue part S SB (λ) and the yellow part S SY (λ): The overall spectrum S(λ) can be written as where λ Cut is the separation wavelength. The spectral separation is therefore a sharp spectral separation. The principle is visualized in Fig. 4  If the weightings are absolute weightings, the combination of the plenoptic functions and spectra provides the correct radiometric flux. Then the absolute plenoptic functions M A,SB (X, Y, Z, φ, ϑ) and The absolute M A,SB (X, Y, Z, φ, ϑ) and M A,SY (X, Y, Z, φ, ϑ) provide the weighting as a function of spatial and angular dimension.
The direct generalization for n max different sharply separated spectral sources S S,n (λ) would be and λ Cut,0 = 0 nm. The variable λ Cut,n is the separation wavelength between the n th and n th +1 sharp spectral band. Each n th spectral band has to be measured individually with a suitable filter to obtain M S,n (X, Y, Z, φ, ϑ, λ). Afterwards, the plenoptic function R(X, Y, Z, φ, ϑ, λ) is the weighted sum of these ILMD-based goniophotometric measurements and the sharply separated spectra as described in Eq. 4.8.
The deduction of the weighting factors W S,n is similar to those described in subsection 4.2.2 and is therefore not described in detail at this point.
Although Eq. 4.6 to Eq. 4.8 are general, their combination is not physically valid for each LED spectrum and the associated error can become large for certain LED spectra. The problem occurs if a sharp spectral distribution S S,n (λ) changes as a function of angular and spatial dimension and therefore the association of the usually pseudo-radiometric plenoptic function M n (X, Y, Z, φ, ϑ) to a sharply separated spectrum S S,n (λ) becomes erroneous. Sometimes there is a full spectral overlap as in the case of the combination of phosphor spectra and red or green LED, which is typical to enhance the color rendering of a luminaire.
Then the generalization provided by Eq. 4.6 to Eq. 4.8 is not valid due to the non valid assumption of constant sharp separated basis spectra.
The problem of spectral overlap is visualized in Fig. 4.2, which shows three measured spectra at different angular positions from an LED system, which consists of phosphor converted white LEDs and additional red LEDs. The sharp spectral separation at λ Cut,1 is reasonable because the relative spectra S S,1 (λ) and the left hand side of the S S,2 (λ) are similar. However, the sharp spectral separation into a red and a yellow part at λ Cut,2 results in a different relative spectral distribution  To avoid this problem, a valid and more physical generalization of the Blue/Yellow approach is necessary. This generalization aims to avoid the less precise and arbitrary sharp spectral separation and leads to the physically motivated basis spectra approach.

PHYSICALLY MOTIVATED BASIS SPECTRA (PMBS)
To show the differences between the arbitrary and physical generalization of the Blue/Yellow approach, the physically relations of a single phosphor converted white LED will be formulated in equations similar to those in section 4.1. The physical realization of Eq. 4.3 uses the physical spectra of each source and describes the overall spectrum S(λ) as the sum of the blue LED spectrum S Blue (λ) and the yellow phosphor spectrum S Phosphor (λ).
The main difference between Eq. 4.3 and Eq. 4.9 is that the spectra S Blue (λ) and S Phosphor (λ) in Eq. 4.9 are the physical spectra of the individual sources. Then the assumption that the relative spectrum of each individual physical radiation source is constant over the spatial and angular dimension becomes valid. This means it is not erroneous to assign a plenoptic function A n (X, Y, Z, φ, ϑ) to the physical basis spectrum S n (λ) for the blue LED and the phosphor such that This is visualized in Fig. 4 is the direct result of an ILMD-based measurement and thus a usually pseudo-radiometric quantity of all sources. The radiometric distribution A n (X, Y, Z, φ, ϑ) in contrast is always assigned to precisely one individual source and not necessarily proportional to any direct measurement result.
The generalization for LED systems with n max different spectral sources S n (λ) would be similar to Eq. 4.6 and Eq. 4.8: This is one of the most important equations in this chapter. It means that although the plenoptic function R(X, Y, Z, φ, ϑ, λ) varies spectrally as a function of the angular and spatial dimension, its underlying relative spectral distributions are constant. R(X, Y, Z, φ, ϑ, λ) can therefore be described by the weighted sum of physical basis spectra, which occur in the system. All spatial and angular variations are described by the relative radiometric A n (X, Y, Z, φ, ϑ), which change as a function of angular and spatial dimension.
However, one problem remains. As previously mentioned, a direct ILMD-based measurement of the relative radiometric distributions A n (X, Y, Z, φ, ϑ) is neither assumed nor generally possible. To be more precise, there is no optical filter which can measure only the relative change of one basis spectrum, if that basis spectrum is fully overlapped by other basis spectra. The relative radiometric distributions A n (X, Y, Z, φ, ϑ) have to be obtained using the combination of different pseudo-radiometric ILMD-based measurements. The procedure is explained in the next subsection.

42
The plenoptic function R(X, Y, Z, φ, ϑ, λ) is described as the weighted sum of the n max physical basis spectra S n (λ), which exists in the DUT:

SPECTRAL RECONSTRUCTION MATRIX
The absolute M A,n (X, Y, Z, φ, ϑ) obtained with the ILMD-based measurements depends on the effective spectral sensitivity s τ,eff,n (λ) of the ILMD-system. According to Eq. 4.13 s τ,eff,n (λ) is the product of the transmission function of the actual optical filter τ n (λ) used to perform the measurement, the optional transmission function of a neutral density filter τ ND,n (λ) and the spectral system sensitivity function s Sys (λ). The spectral system transmission function usually remains constant and includes the sensitivity of the optoelectronic sensor and the transmission of the lens optics The absolute measurement signal M A (X, Y, Z, φ, ϑ) of an ILMD-based measurement with the effective spectral sensitivity s τ,eff (λ) is given by (4.14) It is assumed that besides the measured signal M A (X, Y, Z, φ, ϑ) the relative basis spectra S n (λ) as well as all spectral transmissions and sensitivities and therefore s τ,eff (λ) are known. This means that only the n max A n (X, Y, Z, φ, ϑ) are unknown variables in Eq. 4.14. If m different optical filters are used, they produce m different effective spectral sensitivities s τ,eff,m (λ) and thus m spectrally weighted pseudo-radiometric measurement values M A,m (X, Y, Z, φ, ϑ), which are given by These equations are a system of linear equations, which has a unique solution if the number of different optical filter measurements equals the number of physical basis spectra m = n max . Equation 4.15 can be rewritten in its matrix form as The matrix M Sτ summarizes all available information about the effective spectral sensitivity of the measurement system and the physical basis spectra of the DUT. Equation 4.16 therefore relates the absolute pseudo-radiometric measurement values M A,n (X, Y, Z, φ, ϑ) to the unknown radiometric amplitudes A n (X, Y, Z, φ, ϑ), which will be assigned to the n max physical basis spectra. The radiometric distributions A n (X, Y, Z, φ, ϑ) can be reconstructed by inverting the matrix M Sτ . The reconstruction equation is given as The matrix M −1 Sτ will be called the spectral reconstruction matrix. The reconstructed distributions can be used to calculate the plenoptic function and thus to create the spectral ray file. The general procedure is summed up in the following subsection.

RELATIVE AND ABSOLUTE WEIGHTINGS
It is important to note that Eq. 4.14 til 4.17 are only valid if absolute values are used. However, in this context the M n (X, Y, Z, φ, ϑ) are based on measured ray files. They are therefore provided as unweighted distributions. According to Eq. 4.5 the absolute distributions are given as . . .
where the diagonal matrix of the weightings can be written as the product of the relative weighting matrix W R and an absolute scal- The separation of the absolute and relative weightings makes sense for two reasons. Firstly, the absolute scaling factor can be based on the photometer, which provides absolute luminous flux values with a smaller uncertainty than the ILMD. Secondly, the mathematical importance of both weightings differs. Only the relatives weightings are essential for PMBS. If Eq. 4.18 and 4.19 are inserted into Eq. 4.17 and the matrix multiplications are carried out the result is given by This shows that the relative weightings are necessary to perform the reconstruction because they effectively change the complete reconstruction matrix. If the relative weightings are ignored, the spectral reconstruction does not provide the correct radiometric distributions In contrast, the absolute scaling factor W A can be factored out in Eq. 4.20 and therefore also remains as a global scaling factor, if the distributions A n (X, Y, Z, φ, ϑ) from Eq. 4.20 are inserted into Eq. 4.12. W A is chosen so that it relates the integral of the plenoptic function to an absolute radiometric or photometric value.
The determination of the relative weighting factors W R,n is mandatory and described in the following. Two methods to obtain the correct relative weightings W R,n will be discussed. The first method uses measured values obtained directly from the goniophotometric ILMDbased measurements. The availability of these values depends on the measurements. The second method relates the relative distribution M n (X, Y, Z, φ, ϑ) to a spectral measurement in the known geometrical setup and deduces the weightings by using Eq. 4.16. Finally, the absolute weighting W A can be applied to ensure the correct radiometric flux such that Furthermore, it should be mentioned that in contrast to the weightings of the measurement values, an arbitrary non-zero weighting of the basis spectra does not change the result in Eq. 4.12. The basis spectra weightings change the reconstructed A n (X, Y, Z, φ, ϑ) but also the initial weightings of S n (λ) and therefore not their product in the final Eq. 4.12. Mathematical details are provided in the Appendix A.1.

METHOD 1: CAMERA LUMINOUS FLUX
While the luminous flux of the ray file typically refers to the angular integrated photometer measurements, some near field goniophotometers sum up all preprocessed weighted pseudo-radiance images of the ILMD-based measurement. These values can be used directly as weightings for the corresponding ray file and therefore provide the weighting factor. 1 If these values are not available, the weighting can be estimated based on a spectral measurement using the second method. 1 In the case of the RIGO801 used in this thesis, this value is provided as "camera luminous flux" in the measurement results. It is corrected by a calibration factor to generate a luminous flux under the assumption that the device under test has a spectral distribution according to standard illuminant A. Each filter has its own calibration factor. The weighting of the camera luminous flux has to be negated by dividing the camera luminous flux by the individual calibration factor of the filter.

METHOD 2: INVERSE RECONSTRUCTION
The second and more general option to obtain the weighting is to use the geometrical information of the spectral measurement and the measurement ray files used to obtain M n (X, Y, Z, φ, ϑ). Typically, the spectrum is measured with the goniometer in the same setup as all ILMD-based measurements at a known position (φ, ϑ, r). The variable r is the measurement distance. The measured spectrum S M (λ, φ, ϑ, r) at the known position is a linear combination of the basis spectra S n (λ) according to The amplitudes A n (φ, ϑ, r) ∈ (0, ∞) for the known position can be estimated using an optimization algorithm, in which the amplitudes A n (φ, ϑ, r) are the optimization variables by minimizing the residual sum of squares RSS (A n (φ, ϑ, r)) defined as (4.23) The amplitudes A n (φ, ϑ, r) can be used to estimate the expected measurement values M A,n (φ, ϑ, r) at the known position (φ, ϑ, r) using Eq. 4.16. The relative weighting of the measurement can then be obtained by ray tracing of the relative measurement ray files towards the known position (φ, ϑ, r). which ensures that near field information is used. The diameter of the sensor element is equal to the diameter of the optical probe of the spectroradiometer. Each measurement ray file is traced individually to that sensor to obtain the ray number M n (φ, ϑ, r) at position (φ, ϑ, r). Finally, the weighting W R,n can be calculated by (4.24) in which N Ray,n is the number of rays in the n th measured ray file. Note that 4.24 requires that all rays in the ray file have a constant amplitude.
To sum up, the relative weightings W R,n are determined such that a spectral reconstruction at a specific position (φ, ϑ, r) is capable of reconstructing the measured spectrum at the same position. It is based on the measured ray files and the spectral reconstruction matrix.
Basically, the PMBS method is used inversely.
The spectral measurement should contain spectral data of all basis spectra if this method is used. Typically, the main radiance direction at φ = 0°and ϑ = 0°is a suitable position for the spectral measurement. If the spectral measurement is an absolute measurement, the relative weightings already include the correct absolute weighting and thus lead to W A ≈ 1. If the spectral measurement only provides a relative distribution S M (λ), an additional absolute weighting has to be calculated.

SPECIAL CASE: NORMALIZATION
The relative weightings can be replaced by a normalization of the spectral reconstruction matrix such that M R,Sτ is defined as, and if two conditions are fulfilled: 1. The basis spectra deduction is based on a spectral measurement of the complete spectrum S(λ).
2. All spectral sources follow the same relative distribution (for instance Lambertian) and have the same number of rays.
The first condition will be fulfilled in most cases because the spectral measurement is performed in the same goniometric setup as the ILMD-based measurements. The second one is the limiting condition but is for instance fulfilled in the case of RGB LEDs. Note that it is not fulfilled in the case of RGBW because the phosphor will have a different relative distribution.
It can be shown that this normalization equals the second method described above. The deduction of the amplitudes using the optimization in Eq. 4.23 results in A 1 = 1 = · · · = A n max because of the first condition. Then Eq. 4. 16 gives for all expected measurement values M A,n at the known position. The second condition provides that the ray tracing towards each position would result in the same M 1 = · · · = M n max . It is thus a constant for all measurement values and can be neglected such that the relative weightings of the measurements would be for all measurements. If these weightings are inserted into Eq. 4.21 and then into Eq. 4.16, the result can be written as (4.30) By multiplying the inverse of the weighting matrix from the left hand side the equation changes to (note that W A is a scalar) (4.31) Since the weighting matrix is a diagonal matrix, the inverse is a diagonal matrix, in which each element is the inverted element of the original matrix. Thus it can be rewritten according to Eq. 4.26 This is a convenient result because it simplifies the procedure in some cases such as RGB LEDs.

ABSOLUTE WEIGHTING
The absolute weighting W A can be based on the absolute photometric measurement, which is always performed during an ILMD-based goniophotometric measurement by using the photometer. The angularly integrated photometer values provide the luminous flux of the device under test. Another option is to use additional measurement equipment. Afterwards, the spectral ray files are weighted such that their luminous flux matches that of the absolute photometric measurement.

LIMITATIONS
The most limiting assumption of the whole process is the constant basis spectra assumption, which implies that the workflow does not cover nonlinear effects such as quenching caused by phosphor saturation due to a high radiant flux of the LED [75] or phosphor self absorption.
These effects cause a change of the relative phosphor spectrum as a function of angular or spatial dimension.
In the case of phosphor converted white LEDs without additional spectral sources, the influence of such effects can be estimated by obtaining angularly resolved colorimetric measurements 2 as shown in  Chromatic effects resulting from a primary optic cannot be modeled with the PMBS approach. Although it is common in near field photometry, it shall be mentioned that all interrelated measurements require the steady state condition with respect to electrical and thermal operation conditions.

WORKFLOW OF PMBS
Based on section 4.2, a general workflow for PMBS can be concluded.
The workflow is visualized in Fig. 4.6 and serves as an outline for the next chapters.
The process starts with the determination of the individual semiconductor and phosphor basis spectra using a spectral measurement. The determination of the n max basis spectra is discussed in detail in chapter 5. Subsequently, the goniophotometric measurement with n max different suitable filters are performed. Chapter 6 will focus on a precise definition of the term "suitable" in this context. Then the obtained results in the form of ray files and the spectral sensitivities of the measurement system and the basis spectra are combined to form the spectral reconstruction matrix M −1 Sτ . Subsequently, the ray files are used to solve the system of linear equations by applying the reconstruction equation Eq. 4.17. While this step is mathematically unproblematic as described above, the practical application on ray files is not trivial because a ray file is a randomized statistically discrete set of vectors. Therefore, chapter 7 will focus on the practical application of the reconstruction equation Eq. 4.17.

PMBS: MODELING THE BASIS SPECTRA
In a best case scenario, the semiconductor and phosphor basis spectra, which are necessary to set up the reconstruction matrix of the PMBS method, would already be available due to prior knowledge of the system. Then this chapter might be skipped. However, more often than not they are not available.
Therefore, this chapter describes an algorithm, which estimates the basis spectra based on predefined phenomenological LED models and one spectral measurement. While this "initial modeling" performs well in most cases, complex spectral systems can be optimized further if an additional assumption and the relations from chapter 4 are used.

PHENOMENOLOGICAL BASIS SPECTRA OF LEDS
Each spectral model aims to describe the spectral power distribution of an LED semiconductor as accurately as possible. The difference between a phenomenological model and a physical model is its foundation. A phenomenological modeled LED spectrum describes the correct spectral power distribution with a solely mathematical distribution. A physically modeled LED spectrum describes the correct spectral power distribution based on physical parameters.
Before the basis spectra estimation process is introduced in the next section, some possible LED basis spectra functions will be discussed.
There are plenty of different phenomenological LED models in the literature [23], [76]- [80]. Reifergerste et. al. summed up and evaluated some of them [81]. Table 5.1 shows the models which were considered in this thesis. Each model has a parameter which represents the peak wavelength λ C of the LED and an initial amplitude p. In the case of PMBS and according to section 4.2.2 and appendix A.1, the initial amplitude is unimportant as long as p ∈ (0, ∞). The remaining parameters σ n describe the width and skewness of each model function. Their number differs between the model functions.
models. In the case of symmetric models, such as for example a Gaussian or a second order Lorentzian distribution, it is assumed that the distribution is symmetric towards the peak wavelength λ C . This is not true for real LED spectra and therefore the models lack accuracy [81]. But it is advantageous that the models are very simple and that the parameter σ 1 can be interpreted directly as the width of the spectral power distribution. The Pearson VII distribution is also symmetric and in the case of σ 2 = 2 identical to the second order Lorentzian (σ 1 differs only by a factor).
The asymmetric models are in general more precise in describing the spectral power distribution than the symmetric models, but they are also more complicated. The skewness is described by at least two parameters σ n . The phenomenological Logistic power peak function and the physically modeled Boltzmann based distribution (which is also the only one which models the spectrum as a function of frequency ν) have the advantage of already having been extended to describe thermal and in the case of the Logistic power peak also current dependencies of the spectra [81], [82]. The Logistic power peak and the Asymmetric logistic peak require only one more parameter than the symmetric models, which is an advantage in the case of numerical optimization techniques.
In contrast, the Boltzmann based model, which is shown in the last row of table 5.1, requires five more parameters than the Logistic power peak. Three of them are needed to fit the Gaussian distribution (the second term), which is added to the Boltzmann based first term to adjust the model to measurement data. In practice the second term is obtained in a second fitting process after the result of the first fit (first term) is subtracted from the measurement data [82]. Therefore, this additional Gaussian slightly reduces the physical background and adds a phenomenological part to that model.
In the case of PMBS, the parameters of the models should be characteristic and assignable to a specific feature of the spectral power distributions. Most spectra in

INITIAL MODELING
This section is divided into three subsections. The first describes the modeling process of a single spectrum, the second the slightly more complex case of overlapping LED spectra and the third the most complex case which is LED spectra and a phosphor.

SINGLE LED SPECTRUM
The aim of the initial modeling is to estimate the model parameters x = {p, λ C , σ 1 ...σ n } for each spectral source. The most straightforward approach is to measure each spectral source in the system individually and fit a parametrized model. An optimal fit is obtained by finding the minimal residual sum of squares RSS, which is defined as by directly using an optimization algorithm.  provided chromaticity distance u v helps to assess the physiological impact of the fitting error because the residual sum of squares has no physiological meaning. Note that a difference in u v alone is prone to metamerism and can theoretically give a value u v = 0 although the fit shows clear differences. The analysis of both figures of merit (RSS and u v ) leads to the conclusion that apart from the Gaussian, all tested model spectra might be used to model individually measured LED spectra (at least those without a phosphor).  However, as discussed at the beginning of section 3.4, an individual measurement can be problematic in practical applications and may even affect the shape of the obtained spectral power distribution. Therefore an estimation has to be possible if only one measured spectrum is available.

OVERLAPPING LED SPECTRA
Due to the spectral overlap the modeling gets more complicated if all spectra are estimated from one measured spectrum, which contains the sum of all basis spectra. Besides the measurement spectrum S M (λ), the proposed algorithm requires prior knowledge of the system, which is the number of different LED spectra n max . This knowledge is typically directly available. The optimization problem is the estimation of n max different parameter sets ..x n max } using the already defined Eq. 5.1 and Eq. 5.2 from the last subsection.  As long as the spectral overlap of the individual spectral sources is small, the optimization of the sum will provide results as good as those from an individual fit. But the result may get worse as consequence of a spectral overlap as in the case of Pearson VII. If the spectral overlap becomes larger, it is helpful to use models with less optimization parameters and easily definable constraints as for instance the Lorentzian function. Nevertheless, the comparison shows that in simple cases without a phosphor the basis spectra obtained are good enough to proceed with PMBS directly.

SPECTRAL SOURCES WITH PHOSPHOR
In case of the presence of a phosphor the optimization becomes more complicated for two reasons: • The spectral overlap between a phosphor and a LED spectrum can be very large (as in the case of a green or red LED and a yellow phosphor, such as in RGBW systems).
• There is no general phenomenological parameterized model function, which can describe the spectral power distribution of a phosphor.
Again the following approach requires prior knowledge of the system, which is the number of basis spectra n max and the occurrence of the phosphor as well as the measured spectrum S M (λ). Due to the occurrence of the phosphor the number of LED basis spectra is reduced to n max − 1. Consequently, the following approach assumes that the phosphor spectrum can be described by one distribution. The modeling procedure visualized in Fig. 5.2 consists of a four step process: Step 1 is similar to the optimization without a phosphor. The measured spectrum S M (λ) is used to estimate n max LED basis spectra by using Eq. 5.1 and Eq. 5.2, although it is known that there are only n max − 1 LED spectra. Then the obtained model parameters are used to identify the model spectrum having the least probability of being an LED spectrum. Hence a characteristic model parameter is required.
In the case of a Lorentzian this would be σ 1 as it describes the width of the model spectrum. The broadest spectrum is assumed to be the phosphor.
Step 2 directly follows the decision which assigns the individual models to the LED and phosphor. The remainder phosphor spectrum S R,P (λ), which is defined as serves as the basis of the phosphor. A smoothed spline is used to describe the phosphor. The smooth spline is used to represent the general distribution of the phosphor because it models the only known general attributes, which are smoothness and continuity, with the highest precision. The smoothing parameter p smooth is an optimization variable. However, as p smooth = 1 would equal a cubic spline, it needs a user defined upper constraint to maintain the smoothness. The result is a first estimate of the phosphor spectrum S P (λ). Step 3 is similar to step 2 but focuses on the LED instead of the phosphor. The phosphor spectrum S P (λ) is used to recalculate the n max − 1 LED spectra, which are based on the remainder spectrum S R,LED (λ) defined as S R,LED (λ) and the former n max − 1 LED spectra are then used in the optimization Eq. 5.1 and Eq. 5.2. The result is a set of n max − 1 improved LED models S n (λ, x).
Step 4 is the iterative improvement of the model spectra by alternately repeating step 2 and step 3. The results of step 3 are used to revise S R,P (λ) and the spline S P (λ) and subsequently, the results of step 2 can again be used to revise the LED basis spectra.
Step 4 can be repeated until it converges or until a predefined number is reached as indicated by Fig. 5.2. Finally, the n max basis spectra are given by the n max − 1 LED basis spectra and S P (λ) = S n max (λ).
The approach is similar to obtaining the Boltzmann based model described in [82] which models a Gaussian distribution that is also based on a remainder spectrum in a second optimization. This is also the reason why the Boltzmann based model is not used in this thesis apart from its high number of model parameters. Modeling the phosphor would lead to a more complex 3-step process and the additional Gaussian might disturb the phosphor spectrum. being an LED basis spectrum, which is S 2 (λ, x).
Step 4 Step 1: First estimation according to Eq. 5.1 and Eq. 5.2 based on the measured spectrum S M (λ). As the LED model spectrum S 2 (λ, x) is the broadest, it is assumed to be the that of the phosphor.
Step 2: Calculate remainder phosphor spectrum S R,P (λ) as the difference between the measured spectrum S M (λ) and S 1 (λ, x) + S 3 (λ, x) and fit the smoothed spline S P (λ).
Step 3: Re-estimation according to Eq. 5.1 and Eq. 5.2 based on the remainder spectrum S R,LED (λ) of the LEDs.
Step 4 (first part): Recalculate remainder phosphor spectrum S R,P (λ) and smoothed spline S P (λ) and proceed as shown in Fig. 5 The characteristic inhomogeneities of the phosphor remainder spectrum S R,P (λ) around the peak wavelength of the LED spectra are caused by the difference between the LED model spectra and the measured spectrum. They are similar to the differences observed in      The problem is the same as that which caused the deterioration of the Pearson VII model described in subsection 5.2.2. However, the effect is stronger because the spectral overlap of the green LED and the phosphor is much stronger.   Finally, these validations may be used as additional merit functions to optimize basis spectra, which were for instance obtained during the solely mathematical initial modeling described in the last section.

SPATIAL SEPARATION
This validation incorporates an additional physical attribute of LED systems, which is the spatial separation of colored LEDs (often also the phosphor as part of only the white/blue LED). Most often, the different LED colors lie next to each other. A green LED is green, a red LED is red and a white LED is white (or rather blue and yellow). The green, red and white LEDs do not overlap in the 3-dimensional spatial dimension.
Although there might be a remote phosphor, which overlaps with all LEDs, the single colored chips are both spatially separated and highly localized in the three dimensional spatial dimension. Those known attributes can be used.
A set of single pseudo-radiance images (or pseudo-irradiances based on measured ray files) M n (X, Y) is used to reconstruct the single radiance images (or irradiances) A n (X, Y) of the individual basis spectra using the mathematical relations described in chapter 4 by applying Eq. 4.17. The reconstructed A n (X, Y) can be used to quantify the "spatial separation" of the individual spectral sources. Often a 2dimensional spatial separation in the technical LED plane is sufficient.
A high spatial separation and localization consists of a few high values and as many zeros or dark areas as possible (the red LED does not lie on a white LED). To quantify the term "as many zeros or dark areas as possible", the figure of merit P sum,n has been developed. The mathematical details and an artificial example are provided in appendix A.2.
P sum,n becomes small if there are few high peaks and many values near zero. If there are artifacts (for instance a green LED spatially on top of a white LED) beside the high peaks, the values corresponding to the artifacts result in a higher P sum,n . As there are n max P sum,n (one for each basis spectrum) the final figure of merit is defined by A lower P sum indicates a better spatial separation and therefore a better set of basis spectra.  (b), which also leads to the higher P sum,2 compared to sub figure (a).
Note that the spatial resolution of the irradiances should be low to avoid influence of noise caused by high resolutions of the A n (X, Y).

PUNCTUAL SPECTRAL MEASUREMENTS
Another option to validate the basis spectra would be to perform additional spectral measurements at different angular positions (φ, ϑ) before or after the ILMD-based measurements. The ray files can be used to reconstruct the spectra using the PMBS method as described in chapter 4 using Eq. 4.12 and Eq. 4.17 at the measured angular positions (φ, ϑ). Then the reconstructed spectra can be compared to the spec- 1 Of course the two LED types green (5.6 (a), middle) and white (5.6 (a) right and left) are also spatially separated in the middle region. The irradiance images merge only because the plane of projection is above the LED plane. To see the separation clearly, it would be necessary to display the images directly in the LED plane. This is not done in Fig. 5.6 because each LED is as large as one pixel due to the large distance of the two outer regions compared to the small distance of the LEDs within the middle region. On closer inspection one can notice a slight spatial shift between A 2 (X, Y) and the other reconstructed images.
tral measurements at these positions (φ, ϑ) in terms of chromaticity distance u v or the residual sum of squares RSS. As described in section 4.2.2 the process may take into consideration that the spectral measurement occurred in the near field as the measurement distance and the diameter of the sensor of the spectroradiometer are known during the ray tracing. The ray tracing is necessary to compute the measurement amplitudes as input for PMBS.
However, this method should only be used if the ray files or rather the light intensity distribution at the spectral measurement position does not lie within a high gradient because then the uncertainty of the measurement amplitudes obtained is too large. Furthermore, a comparison to noisy spectral data (for instance measured at the dark areas) should be avoided.

COLOR MIXING LINE
In the case of phosphor-converted white LEDs there is a similar but slightly advanced option compared to the punctual spectral measure- This means that the alignment of the theoretical mixing line and some measured points also gives a hint of the precision of the basis spectra.
Nevertheless, due to metamerism there is a theoretical chance that incorrect basis spectra could still lead to a good alignment. Therefore, the punctual spectral measurements required to draw the true mixing line should also be used to reconstruct the spectrum at the measure-ment positions as described above. The color mixing line is a simple and intuitive representation but also limited to cases with only two basis spectra and therefore mainly to phosphor converted white LED.
The restrictions regarding the quality of the spectral measurements are identical to those of the punctual spectral measurements.

OPTIMIZATION OF BASIS SPECTRA
In the last three subsections additional figures of merit for the purpose of basis spectra validation have been developed. A logical consequence of their definitions is to use them directly within an optimization approach to further optimize the basis spectra. The spatial separation P sum in particular can be minimized directly because the reconstruction of the A n (X, Y) depends on the reconstruction matrix M −1 Sτ , which is a function of the parameters of the basis spectra 2 . As the spatial separation basically optimizes the reconstruction matrix M −1 Sτ it focuses on integrated spectral values but not on a spectrally resolved distribution. Therefore the RSS or the punctual spectral measurements are used as a constraint during the optimization to avoid a solution which would only optimize P sum without considering a measured spectral distribution. In this thesis, the constraint was defined as a tolerated maximum factor for the deterioration the RSS. As the results from the initial modeling provide reasonable starting points, local optimization techniques were used. 2 The relative weighting should be based on the camera fluxes if this method is used because otherwise the optimization is influenced by the deduction of the relative weighting factors, which also rely on the basis spectra in the second method.

MODELING THE UNCERTAINTY OF THE SPECTRAL RECONSTRUCTION
Similarly to the n max basis spectra, the different n max effective spectral sensitivities s τ,eff,n (λ) are required to set up the reconstruction matrix M Sτ , which is necessary to apply PMBS. According to Eq. 4.13, the effective spectral sensitivity s τ,eff,n (λ) of the ILMD-based goniophotometric measurement depends on the filter transmission τ n (λ), the neutral density filter transmission τ ND,n (λ) and the remaining system sensitivity s Sys (λ). In contrast to the basis spectra the determination of s τ,eff,n (λ) is straightforward. Each filter transmission τ n (λ) and τ ND,n (λ) as well as the sensitivity s Sys (λ) can be measured individually, as for instance described in [84]- [87].
The effective spectral sensitivities s τ,eff,n (λ) also influence the measurement values M n (X, Y, Z, φ, ϑ) which are required to reconstruct the radiometric amplitudes A n (X, Y, Z, φ, ϑ). Then, according to chapter 4 and Eq. 4.16, the only requirement of the effective spectral sensitivities is fulfilled if s τ,eff,1 (λ) = ... = s τ,eff,n max (λ). This means that at least in theory any arbitrary filter combination is sufficient for a spectral reconstruction with PMBS.
An example is shown in Fig. 6  The uncertainty η n from Eq. 6.1 has two different physical origins as shown in Eq. 6.2.
The first factor consists of uncertainties η τ,n regarding the effective spectral sensitivities s τ,eff,n (λ) and their interactions with the basis spectra. The uncertainty η τ,n depends on the individual filter used in the measurement and is added to the true value M t,n . The second term η N describes uncertainties which apply for all filter measurements independent of the specific filter function s τ,eff,n (λ) at each step of the goniophotometric measurements. Examples are noise from the sensor chip or the quantification errors caused by the ray file resolutions. Fur-Eq. 4.17. This error propagation can be estimated with the condition number of the reconstruction matrix M Sτ and according to Eq. 6.3 [88], In the following sections, empirical models for optical glass and common interference filters as they are provided by the manufacturers are extended to describe the physically caused measurement uncertainty η τ,n . The models allow the analysis of the influence of different effective spectral sensitivities s τ,eff,n (λ) on the reconstruction result of PMBS. This offers the possibility of choosing the optimal filter combination from a given filter set.

OPTICAL GLASS FILTERS
An optical glass filter is characterized by its internal transmission spectrum τ in (λ) given at a reference thickness d 0 . The ideal transmission function τ(λ) is the product of the resulting internal transmission τ in (λ) for a thickness d and the Fresnel losses τ Fr (γ, n r ) at the air/glass thermore, the uncertainties propagate while solving the reconstruction and glass/air interfaces, which depend on the angle of incidence γ and the refractive index n r . The ideal relation is given in Eq. 6.4 [89].
The uncertainties of the room temperature u(T) and temperature shifts u(T abs ) due to absorption as well as those of spatial roughness u(d), angle of incidence u(γ), the uncertainty of the transmission curve measurement u(τ in ) and its wavelength precision u(λ) [84] are added to the model as absolute values. They influence Fresnel losses as well as the internal transmissions. All uncertainties are assumed to be correlated and are therefore the same for all wavelengths. Note that all uncorrelated uncertainties of τ(λ) are averaged out due to the spectral integration on the ILMD sensor. The model of the transmission function therefore depends on the angle of incidence γ, the refractive index n r , the thickness of the glass d and the temperature coefficient dλ/dT. Hence the model is defined as d 0 cos δ (6.5) using δ = arcsin sin [γ + u(γ)] n r .
Multiple reflections in the Fresnel term τ Fr (γ + u(γ), n r ) are not modeled since the absolute difference between multiple reflections and a two boundary approach is small. By neglecting the effect in the model this small absolute difference is not assigned to τ Fr (γ + u(γ), n r ) 2 . For the same reason, dispersion effects and thermal expansion [86] are neglected as well. Also note that the linear temperature dependence in Eq. 6.5 is limited to long pass glass edge absorption filters [89].

INTERFERENCE FILTERS
The transmission spectrum τ(λ) of interference filters relies on con- - [92]. Decreasing transmission, the broadening of the transmission spectrum and a separation of s-and p-polarization can be neglected for the angle of incidence used in this thesis [91]. The model of the interference filters for small angles γ < 15 • can therefore be written as

INPUT UNCERTAINTIES
To take into account the influence of the uncertainty terms in Eq. 6.5 and 6.6, the uncertainties need to be modeled as well. In accordance with [93], a probability density function (PDF) is chosen or determined for each term. Table 6

FILTER SELECTION
As described above, the reconstruction result is influenced by the optical filters of the ILMD-based measurements. Due to the fact that these measurements are a time-consuming aspect of PMBS, it is not feasible to try out all different filters in real measurements. Therefore, the ideal set of n max optimal filters from all possible filter combinations has to be determined prior to the actual measurements. Given a set of m different filter functions τ(λ), one can determine the ideal subset of n max filters out of N comb possible filter combinations for each specific R(X, Y, Z, φ, ϑ, λ) consisting of n max basis spectra S n (λ). N comb can be calculated with the binomial coefficient: The following subsections introduce two methods to determine the optimal filters by considering the modeled uncertainties. Figure 6.3 shows the workflow of PMBS under consideration of both methods.
The first method uses Monte Carlo simulations to determine the ideal filter combination from a given filter set. The second -optional and less accurate, but faster -method which is based on factorial design, may be used prior to the actual Monte Carlo simulation to reduce the number of all possible combinations to save computation time.

MONTE CARLO BASED FILTER SELECTION
To determine the ideal filter combination a Monte Carlo simulation can be performed. In this simulation, random amplitudes A n of the basis spectra are generated to create a defined number of known random spectra S(λ). Furthermore, random uncertainty values are created by using inverse transform sampling of the predefined PDFs in table 6.1.
The uncertainties are sampled individually for each filter.
Subsequently, the measurements are simulated by solving Eq. 4.15 using the known random spectra amplitudes A n and the transmission profiles, which are influenced by the uncertainty terms according to Eq. 6.5 and 6.6. This results in a simulated measurement value M η,n = M n + η n for each filter for each random spectrum. Then the sensor noise is added according to Eq. 6.2. Finally the reconstruction equation Eq. 4.17 is performed to obtain A η,n = A n + ∆A n . These A η,n define the reconstructed spectrum S η (λ), which can be compared to the known random spectrum S(λ) with the residual sum of squares RSS according to One or several weighting functions k i (λ), for example the color matching functions, can be added to rate the RSS according to the desired application. The mean value of all random spectra RSS mean is used as a figure of merit for the specific filter combination. The mean value is chosen because it includes outliers stronger than the median, which would overestimate the filter performance. Note that choosing the maximum would base the selection on too noisy data. The filter combination minimizing RSS mean would be the preferred one for the goniophotometric measurements and the spectral reconstruction.
Since the absolute values of RSS mean do not allow an intuitive judgment of the colorimetric precision, the known random spectra and the reconstructed spectra can also be compared in terms of absolute colorimetric distance u v mean . As described in chapter 4, u v mean alone is not suitable to evaluate the performance of the reconstruction, if there are more than three basis spectra, due to metamerism.

FACTORIAL DESIGN BASED FILTER PRESELECTION
An individual Monte Carlo simulation for each possible filter according to Eqs. 4.14, 6.2, 6.5, and 6.  16 iterations for each interference filter according to Eqs. 6.5 and 6.6. If more detailed models which account for more factors than Eq. 6.5 and 6.6 are used the number of iterations can be reduced by using a fractional factorial design [96]. Subsequently, the standard deviation is calculated based on all iterations to represent η τ,n for each individual filter.
After estimating the uncertainty of each filter, the error propagation resulting from the solution of Eq. 4.17 has to be considered. Therefore, the second part of the preselection process estimates the mathematical error of each filter combination with a 3-level full factorial design [95]. The sensor signal noise η N is added to the results of the first simulation according to Eq. 6.2. This results in a minimal, mean and maximal value for each filter: Note that M mean,n represents the true value M t,n without any uncertainty. The test spectrum S(λ) is reconstructed for all 3 n max possible permutations for each filter combination by solving the reconstruction equation Eq. 4.17 to define S η (λ). Again, the number of iterations can be reduced by either using a two level factorial design, a fractional factorial design or both. As in the Monte Carlo simulation, the RSS mean between the original spectrum S(λ) and the 3 n max reconstructed spectra S η (λ) is used as figure of merit. After selecting a certain percentage of the N comb possible combinations which performed best in the preselection process, the full Monte Carlo simulation is performed to select the optimal combination from the reduced sub set.

VALIDATION
The Monte Carlo simulation has to determine the best or at least one of the best filter combinations such that only the required ILMD-based goniophotometric measurements are performed. The filter preselection has to determine the filter sets such that the most promising filter combinations are not excluded by the preselection. This section aims to validate these requirements.

VALIDATION OF FILTER SELECTION
To validate the filter selection, simulations are compared to spectral reconstructions, which are based on measurements with different filters. The filter selection is more important and more complex in the case of a higher number of overlapping basis spectra. Therefore, a RGBW LED system was used to verify the filter selection process. To minimize the influences of the basis spectra modeling, the basis spectra are based on individual spectral measurements of the red, green, blue and white LEDs. The basis spectra are displayed in Fig. 6.4 (a).  The Monte Carlo simulation also included the spectral sensitivity of the remaining system as well as the transmission function of a neutral density filter which had to be added to all ILMD-based measurements in order to reduce the signal on the CCD. Both are displayed in Fig. 6.4 (d).
The uncertainties are based on table 6.1 with η N = 0.4% and a large angular distribution. The neutral density filter transmission uncertainty was taken into account and modeled similarly to u(τ in ) and u(λ) but only once for all filters as the same neutral density filter was used in each measurement.
In addition to the twelve ILMD-based goniophotometric measurements, an angularly resolved spectral measurement was performed.
The angularly resolved spectral measurement is used to evaluate the performance of each filter combination. The evaluation used is similar to the punctual spectral measurements described in subsection 5.3.2.
Each spectrum is reconstructed at the known angular positions and compared to the measured spectrum in terms of the chromaticity distance u v and the RSS. The mean value of all positions RSS mean is used to rate each filter combination. The mean chromaticity distance u v mean is used to assess the absolute influence of the filter selection with respect to the requirements of the spectral ray files.   u v mean > 0.1. The last group consists of filter combinations which cannot reconstruct the spectra due to missing information such as filter combination B and C. Impossible chromaticity distance values such as u v mean > 0.6 result from physically impossible chromaticity coordinates, which are based on partly negative spectral distributions. The impossible area is marked in all figures. The filter selection succeeds in separating these three groups. The order within the groups is not monotone but particularly in the first group the differences are very small. Nevertheless, the filter selection identifies the optimal filter combination, which is filter combination A, from all 15 possibilities.  include this specific filter. There are two possible explanations for this behavior. The first one is neglected uncertainties, such as those of the CCD, whose spectral sensitivity function might affect the interference filter at 480 nm more than the alternative filters at 500 nm and 520 nm.
But it is more likely that the disparity is caused by deviations from the basis spectra model of the blue LED and the phosphor to the real basis spectra. This difference is stronger in the spectral region in which both spectra overlap. Since the transmission region of the 480 nm filter is located directly in this spectral region, the assumed reconstruction matrix might differ from the true matrix. The filter selection cannot address this circumstance, because it assumes that the basis spectra are correct. This issue will be discussed in more detail in section 6.4.
Nevertheless, the filter selection of the interference filters also identifies the best filter combinations from all possible combinations.

VALIDATION OF FILTER PRESELECTION
Similarly to the Monte Carlo simulation, the result of the preselection is an arrangement of all possible filter combinations according to their anticipated reconstruction success. To validate the performance of the preselection, the resulting arrangement is compared to the results of the Monte Carlo simulation. Therefore, simulations with four different test spectra as shown in  The uncertainty terms were modeled according to table 6.1, such that realistic values are used for this verification. The simulation included a constant system function, which described the sensitivity of a CCD sensor without an uncertainty and without any neutral density filter.
As the angular distribution varies strongly depending on the geometric extension, the angular uncertainty distribution was modeled twice as shown in Fig. 6.2 to present a large and small measurement object. The same is true for the sensor noise and the quantification error of the ray files η N . Therefore, the total number of filter selection scenarios adds up to 16.  However, one can for example save a factor 50 of computation time by simply simulating the best 2% from the preselection.

FILTER TECHNOLOGY COMPARISON
The validation shows that the filter selection as well as the preselection fulfill their requirements. But it also shows that there is no significant difference regarding the reconstruction performance of the two differ- If the cell color within a row does not change, the filter selection can be assumed to be stable. Table 6.3 shows that the filter selection based on the glass edge absorption filters is quite stable. Although there is a change of sequence within the second group and in the case of model 3 also a change between groups, the filter selection stays stable. Table 6.4 and Fig. 6.10 show that the filter selection which is based on the interference filters has more sequence changes between the groups and in one instance even fails to select the optimal filter combination.
For a fair comparison to the glass edge absorption filters, the blue cell colors have to be interpreted as either green or yellow. Nevertheless, it can be concluded that in the case of uncertain basis spectra the transmission region of the filters should remain large, which is a more typical attribute of glass edge absorption filters.

PMBS: CREATION OF HYPERSPECTRAL LED MODELS
As discussed in chapter 4, the generation of spectral ray files using PMBS requires solving a system of linear equations by applying the reconstruction equation. However, problems occur due to the resolution and precision requirements of spectral ray files and the discrete nature of the input data. This chapter introduces histogram based light source models to apply PMBS. It focuses on the relationship of artifacts and resolutions and introduces a physically motivated calculation concept using neighborhood relations to overcome the highly discrete nature of ray files, which finally leads to spectral ray files 1 .

CALCULATION CONCEPT
Theoretically, the general problem of obtaining spectral ray files is solved by an adequate basis spectra modeling to obtain S n (λ) as described in chapter 5 followed by an optimal filter selection to determine the optimal s τ,eff,n (λ) as described in chapter 6. Then it is only necessary to apply the reconstruction Eq. 4.17 on the 5-dimensional measured ray files. However, due to the discrete nature of ray files, the rays of different ray files will not be defined at the same positions (X, Y, Z) with the same directions (φ, ϑ). To this end the calculation concept as shown in Fig. 7.1 has been developed.
The calculation process starts with the transformation of the measured ray files into continuous models of their plenoptic function M n (X, Y, Z, φ, ϑ). Subsequently, Eq. 4.17 is applied to the measurement models to achieve the spectral models A n (X, Y, Z, φ, ϑ). Finally, the spectral models can be used to create spectral ray files.

HISTOGRAM BASED MEASUREMENT MODELS
The model creation and calculation time should not exceed the overall measurement and creation time of the ray files because the calculation should not become the bottleneck of PMBS. Additionally, it would be advantageous, if the spectral model could be used directly for inverse ray tracing. This would require the models to be competitive with existing models in terms of memory requirements and access time. To achieve these aims, concepts from similar ray file based light source models [53], [54], [99] have been adapted.

FROM RAY FILE TO MODEL
The transformation of the original ray files to the models can be divided into three main steps, which are visualized in Fig. 7.2. The corresponding transformations for the case of ray files with constant amplitudes are illustrated in Fig. 7.3.
Step 1 focuses solely on the spatial information (X, Y, Z). Firstly, all rays are projected onto an enveloping spatial hemisphere to reduce the spatial dimension to two 2 and to enable the possibility of describing them with spherical coordinates (φ S , ϑ S ) [99]. Subsequently, a grid defined by the resolution parameter N spatial is created and all rays are assigned to a specific bin to create a 2-dimensional histogram.  Fig. 7.2 (a).
The false color represents the measurement amplitudes, which are the number of rays. Step 2 defines different local regions within the spatial model as shown in Fig. 7.2 (b). The envelope is divided into N Voron regions such that each region covers approximately the same amount of ray starting points as shown on the upper part of the right hand side in Fig. 7.3. That way it is ensured that each region has approximately the same amount of rays. As suggested in [54], this thesis uses vector quantification (more specific Lloyds algorithm [100]) to distribute the points associated to the voronoi regions for this step.
Step 3 is similar to step 1 but focuses solely on the angular ray directions (φ, ϑ), by creating one angular distribution curve (ADC) for each voronoi region. The ADCs are described as histograms with N angular bins based on the directions of all rays in the voronoi region and can thus be interpreted as point source models. In total there are N Voron ADCs. The process is visualized in the right hand side of Fig. 7.3 and the final model in Fig. 7.2 (c).
In this thesis, the spatial hemispheres as well as all ADCs are divided into Cartesian coordinates with approximately constant solid angles as described in [99], [101], [102]. First, the hemisphere is divided into

FROM MODEL TO RAY FILE
Since a histogram M spatial (φ S , ϑ S ) is an estimation of a probability function, the normalized spatial histogram can be interpreted as the probability of occurrence of a random ray starting point (X, Y, Z) on the hemisphere. That ray starting point also defines the voronoi region Voron(φ S , ϑ S ) and thus its specific angular histogram. The angular histograms M angular (Voron(φ S , ϑ S ), φ, ϑ) can be normalized and interpreted similarly as the probability of occurrence of a random ray direction (φ, ϑ). Since there is an ADC for each voronoi region, M angular (Voron(φ S , ϑ S ), φ, ϑ) also depends on the starting point. Finally, the overall model value (or if normalized, the probability of occurrence) of a ray is given by the product of M spatial (φ S , ϑ S ) and M angular (Voron(φ S , ϑ S ), φ, ϑ) as described in Eq. 7.1.
The creation of a ray file from the model is basically the inverse of the model creation process. Firstly, randomized starting points are created.
Then the starting points are assigned to a specific ADC. Finally, the randomized ray directions are assigned to the starting points. As suggested in [54], the random variables are created using inverse transform sampling of the spatial histogram M spatial (φ S , ϑ S ) and the ADCs M angular (Voron(φ S , ϑ S ), φ, ϑ) respectively. Furthermore, it is necessary to randomly distribute the starting points within each bin.
This distribution has to take into account the original allocation process.
In the case of Cartesian coordinates with approximately constant solid angles as described above, the randomized values in φ/φ S direction can be distributed equally within the bin. In the polar direction the distribution is defined as sinusoidal because the solid angle (or surface area) is proportional to the sinus of its polar angle. This is important because an equal distribution leads to artifacts at low polar angles.

PRECISION ESTIMATION
The precision of the generated ray file is determined by three different resolution parameters N spatial , N angular and N Voron . However, it also depends on the number of rays in the original ray file and the measurement resolution during the goniophotometric measurement.
Furthermore, the precision of the spectral ray files cannot be estimated in general as there is no reference data. However, it is possible to estimate the precision of the quantification by comparing the original measured ray files to ray files, which were created based on the measurement models as shown in Fig. 7.4. As proposed in [68], [69] the similarity of two ray files is obtained by evaluating the normalized cross-correlation (NCC) values of their angular distributions at different distances r to cover near-, mid-and far field. The NCC of measurement model n at distance r is defined in Eq. 7.2.
The variable I n,r (φ, ϑ) is the achieved pseudo-intensity distribution 3 at distance r of a ray file, which was created with measurement model n, Each pseudo-intensity distribution is achieved by assigning the rays to equally distributed points on an enveloping hemisphere as described above. Unfortunately, I n,r (φ, ϑ) and J n,r (φ, ϑ) also require a resolution parameter. However, this resolution can be achieved by obtaining the NCC between the "same" measurement ray file. This requires the ray file to be created twice using the same measurement raw data. If the generation parameters are changed slightly (for instance the number of rays, or the start value in the case of RIGO801) the ray file is only statistically identical to the first. Then these ray files can be compared at different resolutions. If the resolution gets too high, the NCC will drop below the suggested value of NCC = 0.99 [68], [69].

HISTOGRAM BASED SPECTRAL MODELS
All resolutions N spatial , N Voron and N angular as well as all positions of the voronoi regions are the same for each measurement model. In that way the reconstruction Eq. 4.17 can be performed on the spatial histogram M spatial,n (φ S , ϑ S ) and M angular,n (Voron(φ S , ϑ S ), φ, ϑ) to create the histograms A spatial,n (φ S , ϑ S ) and A angular,n (Voron(φ S , ϑ S ), φ, ϑ) of all models. Finally, these histograms contain the amplitudes of the physical basis spectra and can be used to create a spectral ray file as described above.
at distance r of the original ray file, which was used to create mea-PMBS: CREATION OF HYPERSPECTRAL LED MODELS

NEGATIVE AND FALSE POSITIVE AMPLITUDES
A high NCC requires high resolution parameters but this also leads to a problem, namely the occurrence of negative reconstruction results in the spectral models. Figure 7.5 shows three measurement histograms M 1 (φ, ϑ) − M 3 (φ, ϑ) and one resulting spectral histogram A 1 (φ, ϑ) following the application of the reconstruction Eq. 4.17 at different resolutions. If the resolution increases, the ray number in the bins further decreases until statistically empty bins start to arise in the measurement models (white bins in Fig. 7.5). This is caused by the discrete nature of the original ray files. In that case, bins are statistically subtracted from empty bins. This results in negative values (yellow to red bins in Fig. 7.5) at these bins, which indicate a mismatch between the measurement models. Higher resolutions lead to more statistically empty bins and a stronger occurrence of negative amplitudes.
As each A spatial,n (φ S , ϑ S ) and A angular,n (Voron(φ S , ϑ S ), φ, ϑ) represents an amplitude of a physical basis spectrum, negative amplitudes A − n can be assumed to be solely reconstruction errors. However, as the negative amplitudes occur due to a mismatch of a minuend with its supposed subtrahend, the non-subtracted minuends lead to false positive amplitudes. Even if the non-physical negative amplitudes are ignored (or eliminated during the inverse transform sampling), the false positive amplitudes lead to visible artifacts in the spectral models and the created spectral ray files. Therefore, the amount of negative/false positive amplitudes Q n (A n ), which is the relation of negative amplitudes |A − n | to all amplitudes |A − n | + A + n , is defined as a quality metric for each spectral model A low value of Q n (A n ) indicates that there are only a few negative/false positive amplitudes and therefore only few artifacts in the spectral model. Unfortunately, the resolutions required to obtain a low Q n (A n ) contradict the resolution requirements and lead to a low NCC.
In the next subsection an algorithm which reduces Q n (A n ) without affecting the NCC is introduced.

NEAREST NEIGHBOR CALCULATION
The

CALCULATION PROCEDURE
Due to the improvement generated by the nearest neighbor calculation for models with high resolutions, the resolution can be optimized until a user defined NCC is reached. The auxiliary variable calculation success ζ is defined as the 1 ζ fraction of incorrectly calculated bins in the spectral models. If the calculation success is 100, statistically only 1/100 of all bins are incorrectly calculated. If it is one, all bins were incorrectly calculated. The amount of negative amplitudes Q n (ζ) of a spectral model directly depends on the calculation success and the matrix row of the reconstruction matrix used to create the spectral model. The relation is described in Eq. 7.5.
The variable α n is the sum of all negative matrix values and the vari- The final workflow is shown in Fig. 7 .4 is used to estimate the current artifacts Q n (A n ) = Q current,n of the spectral models. The initial calculation success ζ defines a Q n (ζ) = Q aim,n via Eq. 7.5. Then Q current,n and Q aim,n are compared. Algorithms such as bisection or false position are used [104] until Q current,n ≈ Q aim,n to adapt N spatial for the spatial histogram and N angular for all ADCs. These algorithms only require the assumption that the functional relation is one-dimensional and monotone and an allowed tolerance . Strictly speaking, is also an input parameter. However, if the precision of the false position or bisection algorithm is small, for instance below 1% of Q n (ζ), its influence can be neglected.
If Q current,n ≈ Q aim,n for the most critical spectral model, n max ray files are created from the measurement models to calculate their NCC values with the original ray files as described in the section above.
The NCC obtained is compared with the desired NCC. If the obtained NCC is smaller than the desired NCC, the remaining resolution parameter N Voron is optimized for the given calculation success ζ. If N Voron is too small, the NCC in the far field gets higher than in the near field because few ADCs with a high angular resolution are associated with large starting areas. If N Voron is too large, the NCC in the far field is smaller than in the near field for the same reason.
The model introduces a fuzziness between far field and near field or rather the angular distributions resolution and its associated area for a given calculation success ζ, which is adjusted with N Voron . If the desired NCC cannot be reached by adapting N Voron , the initial ζ is reduced. If ζ can remain high, the initial artifacts are small. However, if ζ becomes small, the spectral models show artifacts. The process stops when the desired NCC is obtained. Finally, the nearest neighbor calculation reduces the initial calculation artifacts and completes the creation of the hyperspectral LED models.

VALIDATION OF CALCULATION PROCEDURE
The concept in Fig. 7.7 works for every arbitrary LED spectrum which might also contain phosphor conversion. Nevertheless, each step is illustrated by the example of the spectral reconstruction of the RGB LED F50360 from Seoul Semiconductor [105]. The RGB LED is chosen because the spectrum and position of each spectral source are well separated and therefore well known. Thus, the RGB LED enables the possibility of easily detecting spectral reconstruction errors and offers a reasonable validation of the whole calculation process described in this chapter.
The basis spectra of the test RGB LED are shown in Fig. 7.9 (a). The standard functions x short (λ), x long (λ) and v(λ) and a neutral density filter as shown in Fig. 7.9 (b) were used to create these ray files. According to Eq. 4.16 and Eq. 4.17 the basis spectra S n (λ) and the effective spectral sensitivities s τ,eff,n (λ) define the reconstruction matrix M −1 It should be noted that the filters used are not optimal according to chapter 6, but they provide a strong spectral overlap and therefore allow the validation of the calculation procedure. The first matrix row shows that the first measurement with s τ,eff,1 (λ) = x short (λ) × τ ND (λ) covers approximately one spectral source because the first matrix value is nearly one and the remaining values are orders of magnitude smaller. The first row describes the reconstruction of the blue LED.
However, the reconstruction of the green and red LED requires a linear combination of all measured ray files, because neither s τ,eff,2 (λ) nor s τ,eff,3 (λ) measured only the individual green or red LED. The second filter s τ,eff,2 (λ) = x long (λ) × τ ND (λ) measured a linear combination of the green and red LED and the third filter s τ,eff,3 (λ) = v(λ) × τ ND (λ) measured a linear combination of all LEDs. This is also indicated by the second and third row of M −1 Sτ,RGB , because at least two values in each row are in the same order of magnitude.
Before the calculation procedure can start, the resolution and positions of the NCC calculation and the desired NCC, as well the Q final,n need to be defined. As the envelope of all measured ray files was set to 5 mm, the intensity distributions were evaluated to calculate the NCC (see subsection 7.2.3) at the distances 6, 7, 8, 9, 10, 20, 30, 40, 50 and 300 mm. Each of the equally distributed sampling points used to determine the intensity of the RGB LED covered approximately a solid angle of 0.0025 sr. At this resolution the NCC between the original ray files containing 10 6 rays and the original ray files containing 10 × 10 6 rays from the same measurement is above 0.996 at each distance. The difference can assumed to be noise in the smaller ray file resulting from the high resolution. As an NCC of above 0.99 indicates good agreement [68], [69], that resolution is suitable for the precision estimation of the created light source models. larger with increasing resolution parameters. If Q n (A n ) is as small as 1% or 10%, only the green LED is represented, but the irradiance is also blurred out due to the low angular and spatial resolution of the histograms, which is quantitatively verified by the NCC. However, if the resolutions are very high, spectral artifacts as high as 26% or 33%, which are caused by false positive amplitudes, start to occur. The model assigns parts of the blue and the red LED also to the green LED. The relation between this incomplete calculation and the NCC is also visualized in Fig. 7.11. The graphs show the achieved NCC displayed as 1 − NCC on a logarithmic scale at the tested distances for different calculation successes ζ with original ray files containing 10 6 rays in Fig. 7.11 (a) and 10 × 10 6 rays in Fig. 7.11 (b). The NCC with the same calculation success is always higher in Fig. 7.11 (b). This is to be expected, because higher model resolutions can be achieved, before the noise of the larger and therefore higher resolved original ray files reduces the calculation success. The graphs also verify that the calculation success ζ remains high for NCCs above 0.99 with typical original ray amounts, for instance 10 6 or 10 × 10 6 . According to [68], [69] an NCC > 0.99, or rather 1 − NCC < 1 × 10 −2 , indicates good conformity between model and measurement.    with far field measurements of the same system and the same LED source.

COMPARISON TO THE BLUE/YELLOW APPROACH
As described in section 3.4.4 and 4.1, the Blue/Yellow approach is a feasible, fast and therefore industrially used measurement method to create spectral ray files of phosphor converted white LEDs with two ILMD-based measurements. The main idea, which is the allocation of the physical sources to basis spectra and ray files, is identical to PMBS. While the Blue/Yellow approach is less complicated to apply, PMBS is a physically more valid approach and does not introduce a systematic error by virtue of its underlying assumptions. The required measurement amount of both methods is identical in the case of a phosphor converted white LED.
To quantitatively compare both methods, an LED with a strong angular color uniformity was chosen. Angularly resolved spectral measurements have been performed to assess the results.  The Blue/Yellow ray file of the LED was obtained from an industrial partner according to the state-of-the-art procedure 1 . The separation wavelength was set to 520 nm 2 . Figure 8.2 shows the sharp spectra and its theoretical color mixing line in addition to the measured chromaticities. The chromaticity shift is well described by the Blue/Yellow approach. There is only a small misalignment between the measured data and the theoretical color mixing line, which is caused by the 1 It is important to note that the same LED was used for both Blue/Yellow and PMBS. 2 It can be assumed that the separation wavelength was defined by the physical filters used to measure the blue and yellow ray files.
simplification of the sharp spectral separation as described in section 3.4.4 and 4.1.
The reconstructed spectra deviate stronger, the stronger the spectrum to be reconstructed deviates from the global spectrum used for the sharp spectral separation. As the global spectrum used for the separation is based on the sum of all measured spectra, the largest deviations occur at the main radiance direction ϑ = 0 • and the largest polar angle measured at ϑ = 65 • . They are displayed in Fig. 8.3 and not only show a characteristic discontinuity at the separation wavelength, but also differ stronger in their relation and their relative distribution in the region of the spectral overlap.   The basis spectra estimation of PMBS is also based on the global spectrum used for the sharp spectral separation. Figure 8.4 shows estimated basis spectra models and their theoretical color mixing lines.
The alignment of the color mixing lines compared to the angularly resolved spectral measurements is used to select the final basis spectra.
As the color mixing line of the basis spectrum Logistic power peak (green dashed-dotted line) shows the best alignment, it is chosen for PMBS. Furthermore, it should be noted that all PMBS basis spectra are more suited to describe the measurement values compared to the sharp spectral separation (solid line). The ILMD-based measurements have been performed with the standard optical filters for x short (λ) and x long (λ) and an additional neutral density filter. It is an advantage if the reconstruction works well with these filters because they are often available in ILMDs, which are able to measure colorimetric values.
Their suitability is to be expected because the system is simple and the filters separate the basis spectra well. The camera luminous fluxes are used to obtain the absolute weighting of the ray files. Finally, the reconstruction is performed.
The results of the PMBS simulation are visualized in Fig. 8.5. The comparison of reconstructed spectra to measured spectra in Fig. 8.5 shows that the deviations are small in the case of PMBS. The spectra differ more strongly in the spectral region of the basis spectra overlap but the overall differences are smaller than those of the Blue/Yellow spectra in Fig. 8.3.
The comparison of the reconstructed chromaticities to measurements in Fig. 8.5 and the comparison to Fig. 8.3 lead to the same conclusion.
PMBS is not only able to describe the relative chromaticity shift more precisely, but also differs less in terms of the absolute chromaticity differences, whose mean value is ∆u v mean ≈ 0.002. Note that these results are achieved although the obtained basis spectra still differ from the true basis spectra. The small intensity peak of the phosphor between 400 nm and 420 nm as well as the right hand side of the LED basis spectrum from the Logistic power peak can be assumed to be artifacts. However, as in the case of the Blue/Yellow approach, the combination of both weighted basis spectra lead to a cancellation of the individual basis spectra deviations and thus to a good representation of the true LED spectra.

RECONSTRUCTION OF TYPICAL SPECTRAL LED COMBINATIONS
Although PMBS performs well compared to the Blue/Yellow approach in the case of phosphor converted white LED, its main advantage is the generality for more complex LED systems. This section shows reconstruction results of two more complex but typical LED systems.
In both cases, the measurement object shown in Fig. 8.6 from [106] is used as an example. An aperture set creates a strong angular and spatial color variation in the near field, limits the solid angle and thus reduces the measurement time of this object [106]. This temperature controlled device was operated in two different colorimetric setups, namely Red/White and Red/Green/Blue/White. Both setups represent typical cases of modern lighting technology such as color tunable luminaries.
The following two subsections each focus on one specific example and can be divided into four different parts. The first part is the initial basis spectra estimation. The second part focuses on the measurements and a first precision estimation. The third part describes the basis spectra optimization and the last part assesses the final results.  However, it is hard to visualize artifacts and reconstructions in this plane. Therefore, the reconstructions and measured ray files are all shown in the aperture plane, which is also visualized in Fig. 8.6 (b) and (c).

RED/WHITE (RW)
The first setup combines white (W) and red (R) LEDs, which is a typical combination to enhance colorimetric parameters as for instance the color rendering index. The specific setup and angularly resolved spectral measurements are shown in Fig. 8  First, the initial modeling is used to obtain the three basis spectra from a spectrum measured at main radiance direction. The comparison of the obtained basis spectra to the individually measured spectra is shown in Fig. 8.9. The obtained spectra match the true basis spectra.
The chromaticity distances of the individual LEDs are smaller than u v = 0.001. Only the estimated phosphor deviates slightly which leads to a chromaticity distance of u v = 0.007.   The optimized basis spectra are shown in Fig. 8.11. As the initial modeling already achieved good results, the basis spectra optimization does not change the model spectra significantly. This can also be seen by comparing the spatial separation using the basis spectra from the initial modeling in Fig. 8.10 (b) to the spatial separation using the optimized basis spectra in Fig. 8.12. The spatial separation obtained with the optimized basis spectra shows a slight improvement in the case of the red LED model. The chromaticity distance of the optimized phosphor spectrum supports this impression as it reduces to u v = 0.004.
Finally, the spectral ray files are reconstructed using the calculation procedure described in section 7.4. Figure 8.13 shows spectral ray files  The mean chromaticity distance of the spectral reconstruction on the virtual spectroradiometer sphere is ∆u v mean ≈ 0.005 and therefore slightly higher than the median chromaticity distance ∆u v median ≈ 0.003. These values also indicate that the spectral reconstruction performs well, even if only standard optical filters are used. The performance in an actual ray tracer is evaluated in section 8.3.

RED/GREEN/BLUE/WHITE (RGBW)
The second case, shown in Fig. 8.14, combines white (W), blue (B), red (R) and green (G) LEDs. RGBW LEDs are typically used in modern adaptive and intelligent lighting systems. The angularly measured chromaticities shown in the CIE u v diagram include pure colors as well as mixtures. Therefore it is necessary to model all four basis spectra. Note that the blue LED chip of the white LED is assumed to be identical to the pure blue LED.
The initial basis spectra modeling was performed on a spectrum that had been measured at main radiance direction. Figure 8.15 shows the estimated basis spectra as well as a comparison to individually measured and modeled spectra. The chromaticity distance between the LED models is smaller than ∆u v ≈ 0.003. However, the phosphor basis spectrum shows a larger difference and a chromaticity distance as high as ∆u v ≈ 0.037. The first measurements are based only on standard filters for x short (λ), x long (λ), v(λ) and "Glass", which has no mentionable absorption. As in the case of Red/White and the phosphor converted LED, the filter selection was omitted such that in the first try only standard filters are used. The expected reconstruction performance is assessed using spatial separation, which is visualized in Fig. 8.16. It shows a weak performance. Instead of a spatial separation most LED models are located at the same positions. The LED in the lower left is assumed to be a green, red and blue at once. Furthermore, the spatial separation concludes that each LED has an additional phosphor.  The optimized measurements are performed with glass edge absorption filters with the cut off wavelengths 475, 610 and 695 nm and the standard filter for x short (λ). The obtained measurement ray files as normalized pseudo-irradiances in the aperture plane are shown in Fig. 8.17. Again spatial separation is used to validate the reconstruction and to rate the basis spectra. The spatial separation in Fig. 8.18 still shows slight artifacts. There is a negative green spectral part at the location of the white LED. But overall the result is an improvement compared to the reconstruction based on standard filters in Fig. 8.16.
The images show fewer LED locations, and reasonable combinations, as for instance a weak blue peak, which is located at a large phosphor peak and spatially separated colored LEDs.  The basis spectra have been optimized using spatial separation. They are shown in Fig. 8.19 (a)-(d). While the LED spectra show no significant change regarding the colorimetric distance, the phosphor spectrum has improved and the chromaticity distance decreased to ∆u v ≈ 0.021. The improvement is also verified by the spatial separation in Fig. 8.19 (e). In particular the green LED model shows fewer artifacts. However, there is still a negative spectral part from the red LED at the location of the white LED, which is an artifact. This artifact has to be considered in the calculation procedure according to chapter 7 by adapting Q final,n to the negative values of the spatial separation.
Otherwise the calculation converges slowly because it falsely assumes that all negative amplitudes are caused by the model resolutions. The spectral models obtained are shown in Fig. 8.19. As the blue LED model is part of both the blue and white LED, the irradiance in the aperture plane shows a good spatial separation and thus represents the chosen setup. The green LED model shows an artifact, as it associates 5 percent of its radiometric power to that of the white LED. For the remaining models this value is below 1.5 percent.
The mean chromaticity distance of the spectral reconstruction on the spectroradiometer sphere is ∆u v mean ≈ 0.012 and therefore higher than in previous examples. This value is mainly caused by comparably high deviations at high polar angles in the color mixing regions, which occur due to a decreased ray file resolution in these regions. Figure 8.20 shows a histogram of the chromaticity distances. Note that the median value, which ignores the outliers, provides a chromaticity distance of approximately ∆u v median ≈ 0.007. The performance of the ray files in a ray tracer will be evaluated in the next section.

SPECTRAL RAY TRACING
To validate the suitability of the obtained spectral ray files, a far field measurement of a wavelength dependent optical system including the measurement object was compared to ray tracing simulations in the commercial ray tracer OptisWorks from OPTIS. The optical system consists of one dispersion prism because it shows a strong wavelength dependence and because the simple geometry minimizes deviations between the optical system and the CAD geometry, which can disturb the assessment of the spectral ray files. The experimental setup containing the temperature control (TEC controller) and the source as well as the dispersion prism on a rotating stage is shown in Fig. 8.21 (a). A colorimetric intensity distribution of the refracted part of the light was goniometrically measured with an automotive goniometer and the colorimeter C3300 from LMT.
The simulation setup is shown in Fig. 8.21 (b). In both simulation and measurement the prism was aligned with the rotating plate.
As in the measurement, the photometric center of the intensity sensor in the simulation was defined at the center of the prism. The ray file editor from OptisWorks was used to assign the basis spectra to the reconstructed ray files from section 8.2. Finally, a direct spectral simulation with the combination of all spectral ray files was performed.
Each source was represented by 10 × 10 6 rays. A spectral intensity detector obtained the complete spectrum at each solid angle element.
As the prism was located in the near field of the source, the influence of the prism is different for each LED source.  Due to the deviations at the edges, the mean chromaticity distance is not suitable for quantifying the performance of the spectral ray files.
The highlighted regions R, R/W and W in Fig. 8.24 (a) are less influenced by the misalignment and thus used. The deviations in the red region are as high as the basis spectra model precision from the used Asymmetric logistic peak (see Table 5.2 in chapter 5). The deviations in the white and red/white regions are higher but not solely noise induced. Note that the absolute deviations are within the color discrimination thresholds of white light colors in applications according to [62]. The main part of these deviations occurs due to an offset of ∆v ≈ 0.009, which may originate from the phosphor basis spectra precision or the relative weighting between the phosphor spectrum and the blue LED or the uncertainty of the array spectroradiometer and the colorimeter used in the near or far field measurement. However, as the main deviation is induced by an offset, the relative deviations are expected to be smaller and enable for instance a uniformity analysis of the light distribution. However, the curvature of the distributions seems again more pronounced for the measurement. Also note that in the measurement the green distribution is located slightly above the blue distribution, which is not true for the simulation. This again indicates a misalignment between the far field measurement and the simulation or the simulation and the near field measurement. Furthermore, a small green color shadow can be noted at the lower right part of the red LED in the non-filtered simulation in Fig. 8.25 (b). This is a reconstruction artifact of PMBS, which occurred due to the wrongly assigned flux in the green LED model. Appendix B shows that -in contrast to the green shadow -the slight misalignment between measurement and simulation is not caused by the spectral ray files created with PMBS.

RED/GREEN/BLUE/WHITE (RGBW)
A quantitative comparison in the form of the chromaticity distance is shown in Fig. 8.26. Nevertheless, as the overall mean chromaticity distance is influenced by these deviations at the edges, it is not suited to validate the spectral ray files. A better quantitative validation is provided by the mean values of the six locations G, Y, R, B, C and W, which are less influenced by the misalignment. In that case, the highest mean chromaticity distance of a single spectral source occurs in the red region R and the white region W. However, both can be expected. As the basis spectra model used was a Logistic power peak, the deviations between the red LED basis spectrum and the true red LED spectrum are in this order of magnitude (see Table 5  Even so, the absolute deviations are again within the color discrimination thresholds in practical applications according to [62]. However, it should be noted that these thresholds depend on the specific viewing conditions and were determined for white light colors. Recent studies [107], [108] hint that the color discrimination threshold is higher and thus less critical for saturated colors. As the regions R, W and Y all have an offset induced error, the relative deviations between measurement and simulation, which are required for a uniformity analysis, are even smaller.
In summary, both tested spectral ray files -Red/White and RGBW -are well suited for optical simulations. Apart from the green shadow in the RGBW distribution and slight colorimetric offsets below ∆u v < 0.01, no PMBS induced artifacts were observed. As these spectral reconstructions are more complicated than a single phosphor converted white LED or a RGB LED, it can be concluded that PMBS can be used directly for all these LED systems to simulate spectral distributions in commercial ray tracing software with an appropriate consideration of the spectral near field of these LED systems.

DISCUSSION
In this thesis, a measurement and modeling technique to obtain hyperspectral LED models is proposed, implemented and verified. The first section of this chapter provides a theoretical discussion of the overall approach. The second section focuses on practical aspects. This includes an analysis of the achieved results and the required measurement time -also compared to the state of the art -and concludes with practical applications.

PMBS -APPROACH
The physically motivated basis spectra approach (PMBS) is a method to create hyperspectral LED models based on the assumption that all spectral variations in the near field of a LED source can be described as the weighted sum of their physical basis spectra. The relative spectral distribution of each spectral source is constant. Therefore, PMBS utilizes prior knowledge regarding the device under test.
In the best case, this prior knowledge includes the spectral distributions of the physical sources of the device under test. In the worst case, the basis spectra have to be estimated. Then the only important, but easily obtainable information is the number of different spectral sources in the device. Furthermore, PMBS requires detailed knowledge regarding the spectral sensitivity of the ILMD-based measurement system. Subsequently, all available spectral information is combined to set up a system of linear equations, which is used to reconstruct the individual spectral models from the ILMD-based measurements. The combination of all spectral models is the hyperspectral model.
In a similar way to the general approach, the proposed practical implementations aim to utilize physical boundary conditions. Examples are the assumption that colored LEDs are spatially separated from each other or that negative amplitudes are physically impossible.
The validity of these assumptions is unquestionable. Nevertheless, there are theoretical limitations in the overall approach or specific aspects which have to be discussed in order to assess the developed approach correctly.
The first aspect to be considered is the Monte Carlo simulation used for the filter selection, which aims to minimize the expected reconstruction errors described in section 6.2. By modeling typical uncertainties of the ILMD-based measurements in relation to the optical filters and spectral sensitivity of the ILMD, their influence on the spectral reconstruction is estimated. This information is used to select an optimal filter set. This modeling requires basis spectra, which are at this point not yet optimized. However, as indicated in section 6.4, the filter selection using standard filters and glass edge absorption filters -generally filters with a broad transmission -is stable towards non-optimized basis spectra. The Monte Carlo simulation can therefore be used as it is for choosing an optimal or at least one of the best filter combinations. The determination of the best filter combination cannot be guaranteed, because the number of roughly estimated input parameters influences the result. But performing the filter selection will always minimize the risk of selecting an unsuitable filter combination in the context of PMBS.
Nevertheless, it has to be noted that although the Monte Carlo simulation selects the filters based on estimated reconstruction errors regarding measurement uncertainties, an estimation of the absolute reconstruction uncertainties is far more complex. This would require an analysis according to [93], which requires modeling the input uncertainties and also the device under test in more detail. The third aspect is the basis spectra optimization using spatial separation. The spatial separation utilizes the information that it is very unlikely for the same spatial location to emit, for instance, a red, green and blue LED spectrum at once. It is assumed that colored LED are spatially separated from each other. As outlined in subsection 5.2.3 this technique is only necessary if the basis spectra are not available and are hard to estimate due to a significant spectral overlap. An example is the combination of green LEDs and a yellow phosphor as in subsection 5.2.3 and section 8.2. The basis spectra optimization uses the spectral reconstruction of PMBS to create irradiance images and assesses the results regarding their spatial separation. The optimization changes the basis spectra to achieve a better spatial separation.
However, this will only work up to a certain degree.
The basis spectra optimization only assesses the spatial separation based on a PMBS reconstruction and therefore only validates the reconstruction matrix. The basis spectra are only validated indirectly.
This has three consequences regarding PMBS.
The first is that the relative weightings have to be based on values, which are independent on the basis spectra. An example is the camera luminous flux provided by the RIGO801. The more elegant inverse reconstruction cannot be used because it assumes correct basis spectra.
The second consequence is that deviations from at least one measured spectrum have to be included as an auxiliary condition while effectively changing the reconstruction matrix. This was also considered in all examples which included the basis spectra optimization.
In complex cases adding a low number of additional spectral measurements is recommended as the measurement time is not enlarged significantly while the auxiliary condition becomes more powerful.
However, it must be ensured that all angular positions used in the auxiliary conditions have a suitable signal to noise ratio. This applies for the ILMD-based measurements and for the spectral measurement.
The third consequence affects the filter selection. Filters with a broad transmission are more suited to use the spatial separation as figure of merit for the basis spectra optimization. The worst case for the spatial separation would be a reconstruction matrix which equals the unity matrix because then the spatial separation is already optimal and the non-optimized basis spectra have no effect on the spatial separation.
The fourth aspect is the most general. All modeling concepts require the tacit assumption that the complete system has no additional inaccuracies besides basis spectra deviations or resolution caused artifacts. With this in mind, the last figure of this thesis can be interpreted.  This shows that it would be critical to blame, for instance, only the basis spectra for all artifacts within a reconstruction. A sole "optimization" of the basis spectra would only transfer all ignored aspects directly into the basis spectra. Generally, a sole optimization of the main challenges of PMBS, the basis spectra estimation, the filter selection, and the calculation procedure cannot lead to hyperspectral LED models without any artifacts because they are also caused by factors independent of PMBS. However, hyperspectral LED modeling using PMBS will improve together with near field photometry and may even help to improve this aspect. This will be considered in more detail in the outlook of this thesis.

PMBS -PRACTICAL APPLICATIONS
The PMBS approach has been developed to be a fast modeling technique for spectral near field data of LEDs with a low spectral error.
The measurement and simulation effort has to remain feasible with respect to the current state of the art. Furthermore, the complexity of spectral reconstructions depends on the specific LED system. Another simple case is the RGB LED, which was reconstructed in section 7.5 to verify the calculation procedure using only three ILMDbased measurements with standard filters. The basis spectra modeling without a phosphor is uncritical. Depending on the desired precision of the calculation procedure, the reconstructed spectral models provide details such as the electrode structure of the LEDs.
A more complicated, but practical case is the combination of red LEDs and phosphor converted white LEDs. The polychromatic ray data is the only state-of-the-art method, which covered this case by combining eight ILMD-based measurements with angularly resolved spectral measurements [61]. The PMBS method covered this case in section 8.2 by applying only three ILMD-based measurements and one spectral measurement significantly reducing the measurement time. As in the case of the phosphor converted white and RGB LED, standard filters remain suitable for applying PMBS. The initial modeling provides well suited estimates for the basis spectra. This was also the first case validated by spectral ray tracing. The highest mean chromaticity distance between the simulations and the colorimeter measurements was only ∆u v ≈ 0.013. A large part of the deviations was an offset induced error of ∆u v ≈ 0.009. Therefore, the reconstructed spectral ray files are well suited for spectral ray tracing and uniformity analyses.
The most complex case considered in this thesis is the RGBW example in section 8.2. Although some state-of-the-art methods can be used in theory, a spectral ray file of such a system had not been previously documented. It is the only example in this thesis which requires a basis spectra optimization. In addition, the requirements for the quality of the input data, for example the transmission filter profiles, increase.
Therefore, it is necessary to select an optimal filter combination in order to minimize the influence of uncertainties outside the scope of PMBS. Despite all optimizations, the spectral ray tracing still showed slight PMBS induced artifacts such as a tiny green color shadow within the red part of the distribution. Nevertheless, due to the advanced modeling techniques of PMBS such as the basis spectra optimization, the quantitative results are comparable to those of Red/White and thus well suited for spectral ray tracing and uniformity analyses. Furthermore, the quantitative analysis of the spectral ray tracing considered neither the uncertainties of the array spectroradiometer and the colorimeter, nor the fact that the CIE u v diagram is more reliable for white color stimuli [42], [43]. Physiological studies suggest that the noticeable color differences in the CIE u v diagram increase for saturated colors [107], [108], which would decrease the importance of the offset induced deviations in these regions.

SUMMARY
This chapter summarizes the developed method and achieved results and provides an outlook regarding potential improvements. Furthermore, it suggests applications outside the original scope of the thesis.

SUMMARY
The aim of the thesis was to propose, develop and validate a measurement and modeling technique for obtaining the spectral near field of LEDs in order to improve the optical design of high quality LED based lighting systems. Based on selected concepts of the current state of the art, the physically motivated basis spectra (PMBS) approach, which considers the required measurement effort as well as the usability and generality in practical applications, was proposed and mathematically described.
The main assumption of PMBS is that spectral variations in the near field of an LED system can be described as the weighted sum of all physically occurring spectral sources. This enables the possibility of interpreting a hyperspectral LED model as a set of spectral models, which can each be described individually, for instance as a ray file with one global spectral distribution. The main measurement effort in the form of conventional ILMD-based (imaging luminance measurement device) goniophotometric measurements with different optical filters is determined by the number of spectral sources and is therefore, SUMMARY in contrast to the state of the art, always minimized. The ILMDbased measurements and one global spectral measurement as well as a detailed spectral description of the ILMD-based measurement system are used to reconstruct the near field distribution of the individual spectral models.
The thesis then defined three main challenges concerning the practical implementation of PMBS. These were the potentially mandatory basis spectra estimation, reconstruction errors caused by deviating input data and the application of PMBS on typically available near field data.
The basis spectra estimation uses the global spectral measurement and the number of different spectral sources in the system to obtain phenomenological LED basis spectra. If necessary, the basis spectra can be optimized by quantifying the "spatial separation". This figure of merit assumes that single colored chips are both spatially separated and highly localized in the three dimensional space. In other words, it assumes that a red LED is only red, a green LED is only green, a blue LED is only blue and a white LED is only white or rather blue and yellow. In the case of the spatially non-separated individual phosphor converted white LEDs, a color mixing line, which compares possible color mixtures to actual measurements, can be used. These techniques enable the estimation of the required basis spectra, if they are not available in advance.
Furthermore, it has been shown that an unintelligent filter selection can lead to undesired reconstruction artifacts due to given uncertainties regarding the ILMD-system. To correctly assess the influence of the selected optical filters on PMBS, empirical models for common glass edge absorption and interference filters as provided by the manufacturers were extended to consider physically caused measurement uncertainties. While these models may be evaluated directly in a Monte Carlo simulation, a time saving preselection technique using factorial design was also proposed and validated. During the validation of SUMMARY the filter selection techniques, it was verified that an appropriate filter set is able to minimize reconstruction artifacts. Furthermore, it was shown that the procedure is stable with respect to estimated or rather non-optimized basis spectra, especially for optical filters with a broad transmission. Practical examples are glass edge absorption filters.
To ensure the direct usability of PMBS, the proposed and validated calculation procedure focused on commercially available ray files. The ray files are transformed into histogram based light source models, whose resolution parameters are adapted with respect to a user defined precision parameter. Calculation artifacts, which are caused by the highly discrete nature of ray files, are corrected by applying the nearest neighbor calculation, which utilizes the physical impossibility of physically motivated spectral sources with negative or partly negative amplitudes. This process ensures a correct application of PMBS with high model resolutions. Furthermore, the obtained results can be used directly in state-of-the-art ray tracers.
Using the described implementation, PMBS was first used to reconstruct a conventional phosphor converted white LED. The required measurement effort is the same as in the Blue/Yellow approach, which considers only this type of LED. The hyperspectral LED model, which was obtained with PMBS, showed smaller deviations from validation measurements. The obtained chromaticity distances were even in the region of just noticeable color differences at around ∆u v ≈ 0.002 .
PMBS was further tested by successfully modeling a RGB LED, which covered details such as the electrode structure, as well as spectral ray files of more complex Red/White and Red/Green/Blue/White LED-systems. The spectral ray files were used within a commercial ray tracer to obtain a spectral far field distribution, created by a spectrally sensitive prism in the near field of the measurement object. The spectral simulations were compared to measured colorimetric values of the same setup and showed deviations below ∆u v ≤ 0.016, which SUMMARY led to similar images of these distributions. Therefore, it can be concluded that PMBS is suitable for obtaining hyperspectral LED models, which can be used to improve the optical design of high quality LED based lighting systems.

OUTLOOK
Although PMBS obtains good results for nearly all LED systems, there is still a practical limitation. The application of PMBS on LED systems with different phosphors on different LEDs is only possible, if the basis spectra are known in advance because the initial basis spectra estimation is unsuited for this case. The proposed figure of merit spatial separation remains valid and may still be used in optimizations. However, these optimizations require the development of an appropriate spectral adaptation algorithm, which may for instance try to preserve specific spectral features of the individual phosphor spectra, similar to the spectral adjusting method. Another option to obtain first estimates of the basis spectra is to base the complete initial basis spectra modeling on a single or on a small number of hyperspectral measurements.
The conventional ray files used as input data in this thesis, are already processed data and therefore limit the resolution from the start. Furthermore, the adaptation of the resolution parameters with respect to the input ray files is the bottle neck regarding the computation time.
A higher resolution as well as a faster computation time might be achieved by applying PMBS directly on raw data as for instance compressed luminance images as .ttr-files. While the data compression and adaptive resolution estimation is avoided, the nearest neighbor calculations will still remain the core concept of each calculation procedure.
OUTLOOK However, PMBS is not limited to obtaining spectral near field data for optical simulations. There might be other applications as well.
In particular, the fact that the reconstruction of phosphor converted white LEDs, RGB LEDs and Red/White LEDs requires only standard filters is a promising result. It enables the possibility of estimating a spectral light intensity distribution curve of these LED systems based solely on colorimetric measurements. Furthermore, it is possible to use PMBS to reconstruct a spectral image of LED systems and LED based illuminations with non-goniometric ILMD-based images. If the measurement time is small, it might also be advantageous to consider an over-determined system of linear equations in PMBS.
In this thesis, it was always assumed that all spectral transmissions and sensitivities in the measurement system as well as all measured input data used in the reconstruction were correct. Only the spectral near field of the device under test was unknown. However, this assumption can be inverted by using a spatially, angularly and spectrally well described LED source as developed in the metrology project EMRP ENG62 MESaIL. Then the reconstructions of PMBS, which relate spectral measurements to several ILMD-based near field measurements, might be used to qualify near field measurements and thus near field measurement systems. This may help to advance near field measurements beyond spectral simulations but in the context of photometry in the future. if the arbitrary factor p n is applied (second case). To prove the invariance of the plenoptic function towards an initial basis spectrum weighting, it has to be shown that the term R(X, Y, Z, φ, ϑ, λ) is equal to R (X, Y, Z, φ, ϑ, λ). This is valid if A n is reconstructed with the inverse weighting factor p n such that A n (X, Y, Z, φ, ϑ) = 1 p n A n (X, Y, Z, φ, ϑ) (A. 4) left hand side and the measurement weighting matrix including the relative weightings W R,n is multiplied from the right hand side to the matrix M −1 Sτ . Therefore the correct determination of the measurement weighting matrix is crucial while the basis spectra weighting can be arbitrary as long as it is non-zero.

A.2 DETAILED EXPLANATION OF SPATIAL SEPARATION VALUE P SUM,n
The figure of merit used to quantify the spatial separation P sum,n of the n th individual spectral model according to section 5.3 is explained in detail below. The starting point is a reconstructed radiance image (or irradiance of a projected ray file) A(X, Y), which is normalized to its integral such that ∑ X ∑ Y A(X, Y) = 1. Since a high spatial separation and localization consists of few high values and as many zeros (or dark areas) as possible, the total number of pixels above a threshold Th defined as can be used as an indicator. P tot (Th) is a function of the threshold value and the threshold value lies in the interval Th ∈ (0 1) because all values |A(X, Y)| have to lie between 0 and 1. Since P tot (Th) ≥ 0 ∀ Th, the cumulative sum P cumsum (Th) defined as P cumsum (Th) = Th j=0 P tot (j) (A.15) is a monotone increasing function. Furthermore, it holds that P tot (Th) itself is a monotone decreasing function. Another important attribute of P cumsum (Th) is that its maximum value P cumsum (Th = 1) = const. due to the normalization ∑ X ∑ Y A(X, Y) = 1. The figure of merit for the spatial separation/narrow localization is then defined as P sum,n = Th P cumsum (Th)dTh. "Spatial 1" shows data localized evenly on two pixels (X = 3 and X = 5). "Spatial 2" is localized more strongly at pixel X = 5 and more weakly at pixel X = 3. "Spatial 3" is localized completely at pixel X = 5. The localization decreases from Spatial 1 to Spatial 3. Note that in the case of the PMBS basis spectra validation, each tested distribution would represent a different reconstruction (based on a different set of basis spectra).  show the P tot (Th) for Spatial 1 to Spatial 3. In the case of Spatial 3, P tot (Th) = 1 for each threshold value. In the case of Spatial 2, P tot (Th) = 2 until Th = A(X = 3). Then P tot (Th) = 1 because only pixel X = 5 is above the threshold. After reaching the value Th = A(X = 5), P tot (Th) drops to zero. Spatial 1 is two as pixels X = 3 and X = 5 are above all thresholds until the threshold reaches Th = A(X = 3) = A(X = 5). Subsequently, P tot (Th) = 0. Figure A.2 (d) shows the resulting P cumsum (Th) for all three distributions. It can be seen that all reach the same maximum value. It is 200 since the threshold value was increased evenly 200 times between 0 and 1 in this example. The important point is that it not depends on the normalized A(X). Figure A.2 (e) displays the integrals P sum used as figure of merit for the spatial separation/narrow localization. As expected, Spatial 3, which is localized perfectly has the lowest value, while Spatial 1, which is localized least perfectly has the highest value.
P tot (Th) always decreases with an increasing threshold Th. A low threshold results in a high P tot because many pixels near zero are above the threshold. If Th is increased, it starts to include less pixels and therefore decreases. P cumsum (Th) always increases (or stays constant).
The maximum value depends only on the number of discrete values used for the threshold value Th as described above.
In the case of a low spatial separation the first values of P cumsum (Th) are higher as is the slope of P cumsum (Th). If fewer pixels are above the threshold, the slope gets smaller and may reach zero eventually.
The important fact is that the value P cumsum (Th) of a high spatial separation is larger or equal to P cumsum (Th) of a low spatial separation for each threshold value. This is always true because P cumsum (Th) are monotone increasing functions for all A(X, Y) with the same maximal value and both derivations P tot (Th) are decreasing functions. Therefore, the integral P sum,n = Th P cumsum (Th)dTh can be used as a direct measure of the spatial separation of the n th spectral model.

ALIGNMENT CONSIDERATION
The quantitative evaluation of the spectral ray tracing in section 8.3 did not consider the deviations at the edges of the simulated spectral distribution. Furthermore, it considered only the three major sequences of the ray tracing, which contain 92% of the radiometric power. The remaining sequences were ignored. To justify both steps it has to be shown that neither the stray light nor the deviations are caused by the spectral ray file but occur due to misalignments between the near field measurement and the simulation, misalignments between the far field measurement and the simulation, or deviating material parameters in the simulation. As the problem is identical for the spectral ray tracing of the Red/White example and the RGBW example, it is adequate to consider only one of them below.  To show that these deviations are not caused by the spectral ray files, a simulation with original measured ray files is compared to a simulation with spectral ray files. As no spectrum can be assigned to the original ray files, all wavelengths were set to 555 nm. Figure B.2 shows the normalized intensity obtained with a simulation using the original ILMD-based ray files. The ray files are based on the standard filters x short (λ), x long (λ) and v(λ) because they provide a sufficient signal for all sources. Therefore these deviations are not caused by the spectral ray files.
The same comparison can also be used to verify whether the 8% stray light only occur in the simulation using the spectral ray file set.

Symbol Description
M S,n (X, Y, Z, φ, ϑ) n th sharp M(X, Y, Z, φ, ϑ) M SB (X, Y, Z, φ, ϑ) sharp blue M(X, Y, Z, φ, ϑ) M spatial (φ S , ϑ S ) spatial histogram of measured ray file M spatial,n (φ S , ϑ S ) spatial histogram of n th measured ray file M SY (X, Y, Z, φ, ϑ) sharp yellow M(X, Y, Z, φ, ϑ) M t,n value of M n (X, Y, Z, φ, ϑ) M t,n (X, Y, Z, φ, ϑ) M n (X, Y, Z, φ, ϑ) without any uncertainty n, m index variables, number N angular number of spatial histogram bins used to sample ray files N comb number of possible filter combinations n eff effective index of refraction (used for interference filter) n max number of basis spectra n r refractive index N PS selection number of filter preselection N Ray total numbers of rays in ray file N Ray,n total numbers of rays in the n th ray file N Se selection number of filter selection N spatial number of spatial histogram bins used to sample ray files N Voron number of voronoi regions with constant ADC used to sample ray files NCC normalized cross-correlation NCC n,r NCC of n th model at distance r p initial amplitude of spectral model P cumsum (Th) cumulative sum of P tot (Th) p i (p n ) initial amplitude of i th (n th ) spectral model P(X, Y, Z, φ, ϑ, λ, t) plenoptic function ACKNOWLEDGEMENTS I would like to take this opportunity to thank everyone who has supported me during the time I spend on this thesis. This book presents a fast, accurate and general measurement and modeling technique to obtain spectral near field data of LED systems for optical simulations in order to address the steadily increasing requirements of modern high quality LED systems. The procedure is named physically motivated basis spectra (PMBS) as its main assumption is that each piece of angularly and spatially varying spectral information can be described as the weighted sum of its physical basis spectra. PMBS requires only a minimum of goniophotometric near field measurements and no time-consuming angularly resolved spectral measurements. The complete process is validated and applied to different stateof-the-art LED systems, such as a phosphor converted white LED, a RGB LED, a Red/White LED and a RGBW LED system. The obtained results can be used directly in state-of-the-art ray tracers.