2.1.

Scattering Phase Function

Derivation of optical scattering and absorption parameters in SFDI is usually either performed by look-up-table methods in connection with Monte Carlo simulations^7,9 or by relying on the diffusion approximation to radiative transfer.^2–5 Recently, an analytical solution to the radiative transfer equation for semi-infinite media has been found.^13,14 This solution allows for both fast and accurate modeling and evaluation of SFD reflectance data. Both the use of Monte Carlo simulations and analytical solutions to radiative transfer requires adequate choice of the scattering phase function $p(\theta )$. This function is often unknown or approximated by analytical expressions which are easy to work with, such as the Henyey–Greenstein function.¹⁵ Due to its mathematical simplicity, this is one of the most widely applied scattering phase functions in diffuse optical imaging and beyond.^9,15 Therefore, we start by modeling SFD reflectance using the Henyey–Greenstein function and compare it to that of three different functions with identical asymmetry $\langle \cos (\theta )\rangle$. Afterward, we only focus on the Henyey–Greenstein phase function and investigate the effect of its asymmetry parameter, which can be easily adapted to approximate the light scattering in various media. In this context, the significance of a change in $g$ on the derived optical parameters is presented.

The inset of Fig. 1 depicts four different scattering phase functions on a semi-log-scale with equal asymmetry $g=0.9$. In particular, the Henyey–Greenstein function is compared to Mie scattering and two Reynolds–McCormick (Gegenbauer kernel) scattering functions.¹⁶ Note that the Henyey–Greenstein function equates to a Reynolds–McCormick function with its second input parameter set to $\alpha =0.5$. We therefore choose $\alpha =0.9$ and $\alpha =0.1$ for the two other Reynolds–McCormick functions for modeling of more or less pronounced forward scattering, respectively. The selected Mie phase function corresponds to a sphere of 0.5 μm diameter with a refractive index mismatch of $\mathrm{\Delta }{n}_{\mathrm{r}}=0.1$ between its inside and outside, which is illuminated by a plane wave with a wavelength of $\lambda =650\mathrm{nm}$ in the surrounding medium.

Fig. 1

Modeled spatial frequency domain (SFD) reflectance versus spatial frequency $f$ at ${\mu }_{\mathrm{s}}^{\prime }=1.6{\mathrm{mm}}^{-1}$ and ${\mu }_{\mathrm{a}}=0.03{\mathrm{mm}}^{-1}$ for four different scattering phase functions $p(\theta )$. The corresponding phase functions share the asymmetry $g=0.9$ and are depicted in the inset.

The main graph in Fig. 1 exemplifies SFD reflectance for all four phase functions versus spatial frequency for ${\mu }_{\mathrm{a}}=0.03{\mathrm{mm}}^{-1}$ and ${\mu }_{\mathrm{s}}^{\prime }=1.6{\mathrm{mm}}^{-1}$. These curves are derived from Monte Carlo simulations using point illumination of a semi-infinite media with spatially resolved detection of photons at the medium boundary. A convolution of this spatially resolved reflectance with different spatial frequency patterns (i.e., sine waves) is then used to derive the SFD reflectance signal. Noticeably, the four curves almost coincide for $f=0{\mathrm{mm}}^{-1}$ and their deviation increases with spatial frequency. This deviation is more pronounced in the low scattering regime and almost independent of absorption (assuming $0.003{\mathrm{mm}}^{-1}\le {\mu }_{\mathrm{a}}\le 0.3{\mathrm{mm}}^{-1}$). The deviation in SFD reflectance for differing phase functions at high spatial frequencies can be understood from the corresponding strong sensitivity to short-range photons (i.e., light reflected in proximity to its entry point). Sampling of these photons is enhanced for more intense scattering at larger scattering angles, as observed for the Reynolds–McCormick phase function with $\alpha =0.1$.

In the following, SFD reflectance data are computed for various combinations of ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$, similar to the reflectance curves in Fig. 1. Afterward, data corresponding to Reynolds–McCormick ($\alpha =0.1$ and $\alpha =0.9$) or Mie scattering are evaluated at two spatial frequencies ($f=0{\mathrm{mm}}^{-1}$ and $f=0.24{\mathrm{mm}}^{-1}$) incorrectly assuming Henyey–Greenstein scattering. This and all subsequent fitting procedures are performed using the Gauss–Newton algorithm. Thereby, one arrives at optical parameters which deviate from their original values. This deviation $\mathrm{\Delta }{\mu }_{\mathrm{s}}^{\prime }$ and $\mathrm{\Delta }{\mu }_{\mathrm{a}}$ is strongly dependent on the scattering regime and almost independent of absorption. Figure 2 gives the obtained relative error of ${\mu }_{\mathrm{s}}^{\prime }$ over a wide scattering range for the three different phase functions.

Fig. 2

Relative error of ${\mu }_{\mathrm{s}}^{\prime }$ in percent due to an incorrect scattering phase function. Here, we use the Henyey–Greenstein function to evaluate SFD data which corresponds to Reynolds–McCormick or Mie scattering. The obtained relative error in ${\mu }_{\mathrm{a}}$ (not shown) is almost identical to that of ${\mu }_{\mathrm{s}}^{\prime }$.

The relative error of ${\mu }_{\mathrm{a}}$ is almost identical to that of ${\mu }_{\mathrm{s}}^{\prime }$ and therefore not shown in a separate graph. This similarity in both errors can be qualitatively understood from a scaling property in SFDI. We can observe in Fig. 1 that the variation of the scattering function is very similar to spatial frequency scaling when considering the effect on SFD reflectance. As will be discussed in Sec. 3.4, spatial frequency scaling relates to the scaling of ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ by a common factor and can therefore account for the similarity in both errors.

As can be seen in Fig. 2, large errors arise from the choice of incorrect scattering phase functions in both scattering and absorption. These errors are profound in the low scattering regime, where high spatial frequencies relate to photons, which have undergone only a few scattering events.¹⁷ Therefore, one should carefully consider the phase function of less intensely scattering media. As a consequence, comparative studies across different absorption or scattering regimes (such as dilution series) or spectrally resolved measurements to compute absorption spectra may be falsified by the unawareness of the correct scattering phase function. Note, however, that in order to separate this analysis from our successive study on the scattering asymmetry, we chose all four functions such that $g=0.9$.

In the next step, the influence of the scattering asymmetry $g$ is studied. We do this by calculation of SFD reflectance data for $g=0.9$ using the analytical radiative transfer solution¹³ and the Henyey–Greenstein function. Subsequently, these data are evaluated by means of the same analytical solution assuming incorrect values of $g$. In consequence, incorrect values for ${\mu }_{\mathrm{s}}^{\prime }$ and ${\mu }_{\mathrm{a}}$ are obtained. Figure 3 gives the relative error of ${\mu }_{\mathrm{s}}^{\prime }$ as a function of the assumed asymmetry value $g$ for four different scattering regimes.

Fig. 3

Relative error in ${\mu }_{\mathrm{s}}^{\prime }$ in percent related to an incorrect assumption of the asymmetry value $g$. SFD data were calculated for four reduced scattering values with $g=0.9$ and evaluated assuming different $g$-values. The obtained error scales almost linearly with the incorrectness in $g$. The corresponding relative error in ${\mu }_{\mathrm{a}}$ (not shown) is almost identical to that of ${\mu }_{\mathrm{s}}^{\prime }$.

Variation in scattering asymmetry changes SFD reflectance in a way similar to variations in the scattering phase function (see Fig. 1). Likewise, SFD reflectance is mostly affected at higher spatial frequencies and the relative error in ${\mu }_{\mathrm{a}}$ related to the asymmetry parameter is almost equal to that of ${\mu }_{\mathrm{s}}^{\prime }$ (see Fig. 3) and therefore not shown.

According to Fig. 3, no error in ${\mu }_{\mathrm{s}}^{\prime }$ is obtained for $g=0.9$ as this asymmetry agrees with the previously calculated reflectance data. The more the assumption for $g$ deviates from the actual asymmetry value, the higher the obtained relative error in ${\mu }_{\mathrm{s}}^{\prime }$. The deviation of the four data curves shows larger errors for media with low scattering. This is because a reduction in ${\mu }_{\mathrm{s}}^{\prime }$ (and ${\mu }_{\mathrm{a}}$) corresponds to an upscaling in spatial frequency and thus to a larger deviation in SFD reflectance, as will be further discussed in Sec. 3.4. At the same time, the relative error in ${\mu }_{\mathrm{s}}^{\prime }$ is almost independent of the absorption regime. This is indicated by the error bars of the data points in Fig. 3, which correspond to the standard deviation for analogous computations over a wide absorption regime ($0.003{\mathrm{mm}}^{-1}\le {\mu }_{\mathrm{a}}\le 0.3{\mathrm{mm}}^{-1}$).

2.2.

Diffusion Approximation

Having studied the potential error related to the scattering phase functions, we now concentrate on the evaluation theories for SFDI in general. Particularly, we are interested in the error related to the diffusion approximation to radiative transfer, where the angular scattering characteristics of single scatterers are almost completely neglected.

Data evaluation in SFDI is frequently approximated by diffusion theory due to its simplicity and very high computational speed. In contrast, Monte Carlo simulations can be more accurate, but suffer from relatively long yet decreasing¹⁸ computational times. In order to study the frequently accepted error^2–5 associated with the diffusion approximation, we use our analytical solution to the radiative transfer equation^13,14 for the computation of SFD reflectance data at the frequencies $f=0{\mathrm{mm}}^{-1}$ and $f=0.24{\mathrm{mm}}^{-1}$. Thereby, we assume the Henyey–Greenstein function with $g=0$. This computation is performed systematically for various absorption and scattering combinations and the obtained reflectance data are evaluated using the corresponding diffusion approximation for the same experimental geometry¹² and a nonlinear regression routine. Thereby, one arrives at scattering and absorption values, which deviate from the correct optical parameters. This deviation is shown in two contour plots in Figs. 4 and 5, where different contour levels correspond to different relative errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$, respectively.

Fig. 4

Relative error of the absorption coefficient $\mathrm{\Delta }{\mu }_{\mathrm{a}}/{\mu }_{\mathrm{a}}$ in percent due to the diffusion approximation. The contour lines give the error over a wide scattering and absorption regime.

Fig. 5

Relative error in the reduced scattering coefficient $\mathrm{\Delta }{\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{s}}^{\prime }$ due to the diffusion approximation. The contour lines give the relative error in percent and as a function of the scattering and absorption coefficients.

We observe that the diffusion approximation causes errors which largely depend on the absorption and scattering regime. Expectedly, largest errors are found for media of low scattering and high absorption, where the preconditions for the diffusion approximation are least fulfilled. However, even with highly scattering media, errors in excess of 10% can arise. For high scattering, we learn from Figs. 4 and 5 that diffusion theory underestimates both ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$. This can be understood from the corresponding overestimation of short-range photons in diffusion theory.

Based on this error analysis, the benefit of an analytical radiative transfer model for SFDI can be quantitatively understood. Similarly, also combined Monte Carlo look-up-table approaches have the potential to overcome the presented errors, which may be large for particular combinations of ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ even in the very high scattering and low absorption regime.

2.3.

Refractive Index

The refractive indices of most samples investigated in SFDI are oftentimes only approximately known. Typically, the real part of the refractive index of biological tissue is in the range of $1.35\le n_{\mathrm{r}}\le 1.50$ with some wavelength dependency of about 1% across the optical window ($650\mathrm{nm}\le \lambda \le 1050\mathrm{nm}$).^19–21 Uncertainty in refractive index implies inaccuracy in the derived optical parameters. In the following section, this inaccuracy in determining ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ is quantified as a function of the error in refractive index $\mathrm{\Delta }{n}_{\mathrm{r}}$. It should be noted that we consider the average refractive index of the scattering media and not that of single scattering particles. In consequence, we still make the simplifying assumption of a homogeneously scattering media with scattering and absorption properties, that are independent of the media’s refractive index. Therefore, the assumed refractive index value merely controls the amount of transmitted and reflected light on both sides of the medium surface, as given by the Fresnel equations. In order to study the impact of the refractive index on the derived absorption and scattering coefficients, we calculate SFD reflectance using our analytical radiative transfer solution for semi-infinite media assuming a refractive index of $n_{\mathrm{r}}=1.40$. Subsequently, this data is evaluated by falsely assuming $n_{\mathrm{r},\mathrm{err}}=1.45$ which ultimately produces errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$. Figures 6 and 7 give these relative errors in absorption and scattering, respectively, over a wide scattering and absorption range.

Fig. 6

Relative error of the absorption coefficient $\mathrm{\Delta }{\mu }_{\mathrm{a}}/{\mu }_{\mathrm{a}}$ in percent for incorrect assumption of the refractive index. Here, the refractive index is assumed to be too large by $\mathrm{\Delta }{n}_{\mathrm{r}}=0.05$. SFD reflectance data were calculated for $n_{\mathrm{r}}=1.4$ and evaluated assuming $n_{\mathrm{r},\mathrm{err}}=n_{\mathrm{r}}+\mathrm{\Delta }{n}_{\mathrm{r}}=1.45$.

Fig. 7

Relative error in the reduced scattering $\mathrm{\Delta }{\mu }_{\mathrm{s}}^{\prime }/{\mu }_{\mathrm{s}}^{\prime }$ in percent over a wide absorption and scattering range for an error in refractive index of $\mathrm{\Delta }{n}_{\mathrm{r}}=0.05$.

An increase in the refractive index increases both the amount of specularly reflected light at the medium surface and the number of diffuse photons reflected at the internal medium boundary. In our analysis, we focus only on the latter effect, since specular reflections have a strong dependence on surface properties of the sample and are oftentimes avoided by oblique projection^{2–7,9,12,22} and the use of crossed-linear polarizers.^2–6,9,22 The effect of the internal reflection increases the average lateral propagation distance of diffuse light inside the media. This increases the probability for absorption and reduces SFD reflectance especially for high spatial frequencies. Hence, an overestimation in the refractive index is mostly compensated by an apparent decrease in absorption or an apparent increase in scattering.

Qualitatively, the two contour plots in Figs. 6 and 7 are also correct for different errors $\mathrm{\Delta }{n}_{\mathrm{r}}$. Our computations show that the relative error in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ is almost linearly proportional to $\mathrm{\Delta }{n}_{\mathrm{r}}$ in the range $1.35\le n_{\mathrm{r}}\le 1.50$. It is therefore possible to simply scale the contours in Figs. 6 and 7 by $\mathrm{\Delta }{n}_{\mathrm{r}}/0.05$ to obtain the error for any uncertainty $\mathrm{\Delta }{n}_{\mathrm{r}}$ in the refractive index.

3.1.

Projection Boundary Errors

Experimentally, the projection of spatial frequency patterns is typically restricted to a limited spatial projection area. This gives rise to boundary effects close to the border of the projection field. Underneath projection boundaries, diffusing light scatters away from the projection area without being compensated by diffuse light from outside the projection field. Therefore, reflectance at $f=0{\mathrm{mm}}^{-1}$ [which is typically denoted as direct current (DC) component] decreases and approaches zero at further distance from the projection area.

A similar effect is observed for experiments on phantoms, where the finite phantom size causes a drop in reflectance near the phantom boundaries. This decline is even larger than that of the projection boundaries, since diffusing light which leaves the phantom to the side cannot scatter back into the media.

The SFD reflectance for frequencies above zero is usually termed AC component and is calculated using the following formula:

We are interested in an estimation of the required distance from the projection boundaries for the error in SFD reflectance to be below 1% for both spatial frequencies (DC and AC components). Figure 8 gives this distance in millimeters by means of a contour plot as a function of reduced scattering and absorption. The data are based on Monte Carlo simulations for semi-infinite media with vertical point illumination and spatially resolved detection of photons at the medium boundary. A convolution of this reflectance signal with a finite sinusoidal projection pattern is used to compute the change in SFD reflectance at the projection boundaries. Upon the examination of Fig. 8, it is important to understand that an error of 1% in reflectance may give rise to a relative error in ${\mu }_{\mathrm{a}}$ or ${\mu }_{\mathrm{s}}^{\prime }$ of several percents. Due to strong oscillation of the AC component at the projection boundaries, as calculated by Eq. (1), it is however difficult to directly state the errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$, as they depend on the relative behavior of the DC and the AC components. As a rule of thumb, the relative error in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ for the stated distances from the projection boundaries (see Fig. 8) is mostly within 5%. Our computations show that the absorption parameter is more sensitive to small deviations in reflectance than the reduced scattering parameter ${\mu }_{\mathrm{s}}^{\prime }$. Indeed, the obtained error in ${\mu }_{\mathrm{s}}^{\prime }$ is often below 2% for errors in reflectance below 1%. However, in the very low absorption and high scattering regime, the error in ${\mu }_{\mathrm{a}}$ may rise to values close to 20% for the same accuracy in reflectance. Especially in this regime, the experimentalist is advised not to consider SFD data closer than twice the values stated in Fig. 8, thus achieving an error of $<0.1\%$ in (both DC and AC) reflectance.

Fig. 8

Required distance from the projection boundaries in millimeters in order to achieve a deviation in SFD reflectance of $<1\%$. Doubling of the stated values gives an error in SFD reflectance of $<0.1\%$.

Accuracy in SFD reflectance close to the projection boundaries is to a smaller extent also dependent on the spatial frequency. The presented boundary errors are based on the spatial frequencies $f=0{\mathrm{mm}}^{-1}$ and $f=0.24{\mathrm{mm}}^{-1}$. For higher spatial frequencies, the relative error in SFD reflectance at the projection boundaries increases slightly, in spite of their higher sensitivity to short-range photons.

3.2.

Binning Error

Spatial resolution in diffuse optics is fundamentally limited by the underlying scattering and absorption properties and is clearly outperformed by the resolution of commercial CCD cameras. Therefore, binning of captured SFD images can achieve a reduction in pixel noise at the expense of unsubstantial pixel resolution. At the same time, binning is of additional benefit due to an increase in evaluation speed.

Apart from the reduction in spatial resolution, strong binning of SFD images before the computation of the AC value using Eq. (1) gives rise to errors in SFD reflectance. Similar to the Nyquist sampling criterion for signal transmission, spatial resolution of SFDI data should be large enough to guarantee several sampling points within one spatial oscillation. Otherwise, oscillations in SFD data are averaged out in the binning process. Due to its infinite spatial wavelength, the DC component ($f=0{\mathrm{mm}}^{-1}$) is not affected by this error.

The relative error $\delta R_{\text{Binning}}$ of SFD reflectance based on excessive binning scales inversely with the number of sampling points per spatial oscillation. In order to quantify the binning related error, three phase shifted sine waves were computationally binned with a particular bin size and the AC component was subsequently computed using Eq. (1). Repetition of this calculation for different bin sizes and oscillation frequencies gives the error in the AC component dependent on the oscillation frequency and bin size. Equation (2) is a power law which was fitted to the computational results and which describes the binning related error to a high precision. It gives an estimate of the percentage error in AC reflectance based on the binning x (number of pixels per bin):

As an example, a typical SFD camera image with resolution $N/l=1024/(50\mathrm{mm})$ along the direction of spatial oscillations with $f=0.24{\mathrm{mm}}^{-1}$ allows for binning of $x=8$ pixels in order for the error in SFD reflectance to be below 1.4%. Note that excessive binning always underestimates the real SFD reflectance values. If only one AC component is used, the binning related error corresponds to an error in spatial frequency (see discussion in Sec. 3.4). In this case, the obtained relative errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ are both linearly proportional to the error in AC reflectance.

It is worth pointing out that for particularly high spatial frequencies, diffraction and aberration in the optical imaging system may be additional limiting factors for precise determination of the SFD reflectance signal.

3.3.

Height Adjustment Error

Precise sample and camera positioning are a prerequisite for accurate measurement of SFD reflectance. Although sample movements also affect the projection geometry and may cause spatial frequency errors (see Sec. 3.4), we now focus on the effect of incorrect adjustment of the distance between sample and camera. Thereby, we assume that this error in camera height is static and equally large for the measurement of all spatial frequencies (i.e., $f=0{\mathrm{mm}}^{-1}$ and $f=0.24{\mathrm{mm}}^{-1}$) and phase projections.

Variations in height alter the camera aperture angle and thus the captured light intensity. The additional effect of defocusing of the camera image and also the variation of the detection acceptance angle for points with different lateral separation from the optical axis is neglected in the following analysis. For simplicity, we approximate the angle resolved reflectance by Lambert’s law²³ and thus the intensity of SFD reflectance captured for a particular camera height $h$ and aperture radius $r$ is

Note that Eq. (4) is unit-of-length independent. As an example, for a camera aperture radius of $r=1.5\mathrm{cm}$ and a camera height of $h=20\mathrm{cm}$, a relative error in reflectance of 1% is obtained for height inaccuracy $x=-1\mathrm{mm}$. With respect to this sample calculation, we now present the impact on ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ for a 1% increase in SFD reflectance. In Figs. 9 and 10, the relative errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ are given, respectively.

Fig. 9

Contour plot for the relative error in ${\mu }_{\mathrm{a}}$ in percent based on a sample distance error which causes an increase in reflectance of 1%. According to Eq. (4), this error could be due to an unwanted decrease in sample-camera separation of $x=-1\mathrm{mm}$ (assuming $h=20\mathrm{cm}$ and $r=1.5\mathrm{cm}$).

Fig. 10

Relative error in ${\mu }_{\mathrm{s}}^{\prime }$ in percent based on an incorrect separation of camera and sample. Similar to Fig. 9, the contour lines correspond to an increase in reflectance of 1% due to a reduction in the sample-camera distance.

Generally, an increase in the SFD signal leads to an underestimation of absorption and an apparent increase in scattering. From Fig. 9, we learn that the error in ${\mu }_{\mathrm{a}}$ is heavily dependent on the absorption and scattering regime and much larger than that of ${\mu }_{\mathrm{s}}^{\prime }$ in Fig. 10. Errors in both ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ are almost proportional to the error in SFD reflectance as calculated by Eq. (4). Therefore, Figs. 9 and 10, which correspond to an error in reflectance of 1% can be used to approximate the error in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ based on any height inaccuracy. In this case, the contour values are scaled by the expected error in reflectance according to Eq. (4).

3.4.

Spatial Frequency Error

Correct determination of the dimension of projected spatial frequencies is a prerequisite for accurate evaluation of SFD data. If only one AC frequency component together with the reflectance at $f=0{\mathrm{mm}}^{-1}$ (DC) is used to determine the absorption and scattering, potential errors in spatial frequency are linearly proportional to the obtained relative errors in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$. This is because scaling of ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ by the same factor does not change the probability for photons to exit the media, but only scales the distance of their reflectance. Consequently, the obtained errors are independent of the absorption and scattering regime. This behavior is expressed by the following scaling property for SFD reflectance $R_{\mathrm{SFD}}$

Accordingly, a 5% inaccuracy in the projected spatial frequency pattern gives rise to 5% error in both ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$. This underlines the necessity for accurate calibration of spatial frequencies. However, errors in spatial frequency do not only stem from improper calibration, but also from variations in the projection geometry. Furthermore, especially for oblique projection geometries, careful correction of frequency distortion is necessary.