In fringe projection technique, system calibration is a tedious task to establish the mapping relationship between the object depths and the fringe phases. Especially, it is not easy to accurately determine the parameters of the projector in this system, which may induce errors in the measurement results. To solve this problem, this paper proposes a new calibration by using the cross-ratio invariance in the system geometry for determining the phase-to-depth relations. In it, we analyze the epipolar eometry of the fringe projection system. On each epipolar plane, the depth variation along an incident ray induces the pixel movement along the epipolar line on the image plane of the camera. These depth variations and pixel movements can be connected by use of the projective transformations, under which condition the cross-ratio for each of them keeps invariant. Based on this fact, we suggest measuring the depth map by use of this cross-ratio invariance. Firstly, we shift the reference board in its perpendicular direction to three positions with known depths, and measure their phase maps as the reference phase maps; and secondly, when measuring an object, we calculate the object depth at each pixel by equating the cross-ratio of the depths to that of the corresponding pixels having the same phase on the image plane of the camera. This method is immune to the errors sourced from the projector, including the distortions both in the geometric shapes and in the intensity profiles of the projected fringe patterns.The experimental results demonstrate the proposed method to be feasible and valid.
In fringe projection profilometry, the phase sensitivity of a fringe pattern to depth variation of the measured surface is vital to measurement accuracy and resolution. This paper represents the implementation of the optimal fringe pattern with the best phase sensitivities over the whole fringe pattern, and deduces an efficient calibration method to determine the relationship between the phase-difference distribution and the depth variation. In it, first we find the epipole location by projecting sets of horizontal and vertical fringe patterns on several depth-known reference planes, and meanwhile determine the parameters of the measurement system calibration by analyzing the geometry of measurement system. And then project the optimal fringe pattern onto the object to measure. Experimental results demonstrate that this method is very efficient and easy to implement.
Spatial carrier fringe pattern analysis is an effective tool in optical measurement, e.g. in interferometry and fringe projection technique. With it, the very large phase deformations in a spatial carrier fringe pattern may increases the bandwidth of fringe component thus leading to difficulties in retrieving its phase map. In order to overcome this problem, many local-adaptive methods have been developed for processing the spatial carrier fringe pattern with large phase variations, and in fact, the local spatial frequency estimation is central to these methods. This paper introduces a simple algorithm for estimating the local frequencies of a fringe pattern with spatial carrier. First, the intensity gradients of the fringe pattern are calculated, and then the standard deviations (SDs) of the intensity gradients at each pixel are estimated from its neighborhood. Finally the local frequencies are estimated from the SDs just calculated simply using an arccosine function. This algorithm is potential in developing effective techniques for retrieving phases from a spatial carrier fringe pattern with large phase variations. For example, we can recover the phase map by directly integrating the local frequencies or by use of an adaptive spatial carrier phase shifting algorithm (SCPS) with the local frequencies being the local phase shifts. It can also be used in Fourier transform method for exactly determining the carrier frequencies, or for extrapolating aperture in order to reduce the boundary effect. Combined with time-frequency techniques such as windowed Fourier transform and wavelet transform methods, it is helpful for alleviating the computational burdens.
In fringe projection profilometry, the measurement system generally consists of a projector for casting fringes onto a measured object and a monochrome camera for capturing the deformed fringe patterns. In addition to these components, we can add a color camera for capturing the texture of the object simultaneously. For implementing texture mapping on the reconstructed three-dimensional (3-D) surface, the parallax between the views of the texture camera and the measuring camera has to be corrected. For this purpose, we analyze the geometry of the fringe projection system with a color texture camera and further suggest a system calibration method. Using this method, the corresponding relationship between the texture and the 3-D data and the mapping relationship between the depths and the fringe phases are determined simultaneously, so that the duration time for the system calibration implementation is saved. The data processing with this method is of a low computational complexity because it involves only linear minimizations. Using the calibration results, we can transform the texture image from the view of the color camera to that of the measuring camera and precisely map it on the reconstructed object surface. Experimental results demonstrate that this method is effective in correcting the parallax of the texture image.
A novel multi-view connection method is proposed for whole field three-dimensional (3D) profile measurement. Firstly
3D profiles of tested object from different views can be measured by using the digital fringe projection profilometry, and
then these profiles are transformed into the common coordinate by applying transform matrix. With the help of a new
determination method for the revolution axis of rotary stage, the direction vector and one point of the axis are achieved.
To improve the computability and feasibility of coordinate transformation, the quaternion method is also used to get the
transform matrix. Considering the error movements between the views, an effective method based on multi-aperture
overlap scanning technique (MAOST) is presented, which can determine the relationship between two adjacent views
from their overlapping areas. The connection of adjacent views is performed accurately in 3D space. Both computer
simulation and experimental results are presented to verify the feasibility of proposed method.
This paper presents a novel technique for measuring the three-dimensional shapes of specular surfaces. Differing from
the conventional techniques, the diffusive light source in our technique can be moved vertically to two different
positions, and at each position the phase distribution of the deformed fringe pattern is measured, so that the orientation of
incident light for each pixel is tracked through the corresponding phases. The 3-D coordinates of points on the specular
surface are further determined. In so doing, the restrictions and limitations of the existing techniques in computational
complexities, phase ambiguities and error accumulations are eliminated. The validity of this technique is demonstrated
by experimental results.
Digital fringe projection profilometry employs a digital video projector as a structured light source and thus gains great
flexibility. However, the luminance nonlinearity of the video projector may decrease measurement accuracy and
resolution. To overcome this problem, we propose a nonlinearity correction technique for digital fringe projection
profilometry. This technique allows determining the response curve of a digital video projector by matching the
histogram of the fringe images with that of a standard sinusoid signal. By iterating the two steps, histogram matching
and phase evaluation, the phase distribution of the fringe pattern is finally solved with higher accuracy. In so doing,
neither photometric calibration nor knowledge about the device is required. Both computer simulation and experiment
are carried out to demonstrate the validity of this technique.
A novel aperture connection method for measurement of surface with rotation axis is presented. This work is an extension of “Multi-aperture overlap-scanning technique (MAOST)” in a cylindrical coordinate system, by which a surface with rotation axis such as 360-deg shape can be accurately measured. The principle of MAOST is to make the adjacent sub-apertures partially overlapped, and then the relationship between each couple of adjacent sub-apertures can be obtained through the overlapped area to restrain the error propagation in patching these sub-apertures into an overall measurement result. So the high-precision mechanism and time-consuming system alignment is not required. Based on the inherent mapping relationship between the coordinate transformation operator and error vector, an iterative algorithm is presented, at the same time by which we can use a set of linear equations instead of non-linear equations to calculate the actual error vector. The computer simulation test shows the excellent results. The result of experimental test of a semi-cylinder surface is also presented.
A new concept of self-consistency is proposed in this paper and then a new phase-stepping algorithm based on it is presented. If the phase steps between the grabbed data frames are known, the phase distributions of every data frame can be calculated by the modified conventional phase-stepping algorithm. On the other hand, the phase steps between data frames can be calculated through the calculated phase distributions of every data frame above. The known phase steps and the calculated phase steps should be consistent. This is so called self-consistent concept .By means of iterative method, we can easily gain the accurate phase steps while calculating the measured phase distribution. Numerical simulations and experimental results verify the insensitivity of new algorithm to the phasestepping error and automatic calibration of phase shifter.
In phase-shifting interferometry, a phase shifter usually has tilt shift error along with translational shift error during shifting. The pixels on the same interferogram can not be shifted by an equal amount. Thus the phase measurement errors can not be avoided, even when the translational shift error has been corrected effectively. However, based on the fact that phase shifts of all the pixels on the same interferogram are still kept on the phase shift plane. So by solving this plane the phase errors can be eliminated significantly In this paper, a new algorithm insensitive to both the translational and tilt shift errors of a phase shifter for phase-stepping interferometers is presented. The first order Taylor series expansion replaces the nonlinear equations in solving the phase shift plane, and by iterative, the accuracy can be guarantied. The simulative and experimental results show that phase measurement errors caused by both translation and tilt shift-error can be compensated significantly.
In phase-shifting interferometer, software processing method is a very important means to achieve high accuracy. By analyzing and simulating, it is indicated that only removing the effect of the error which has a spatial frequency of twice the fringe frequency in the result is not enough, especially in high accuracy measuring. The paper presents an averaging method to reduce the effect of errors. Testing results testify this averaging method is effective and robust.
In optical 3-D measurements, two steps are generally required to obtain the whole-body 3-D shapes of objects: measuring the 3-D shape from different views, and afterwards connecting them together. The multiview overlapping scanning connection technique in a cylindrical coordinate system is an effective method for measuring a surface with a rotation axis, e.g., a 360-deg shape. However, there are great difficulties in measuring a more complex surface, such as those with concavities or composed of several discontinuous patches, because a complex surface generally cannot be explicitly represented in cylindrical coordinates. To solve these problems, a novel multiview connection method based on virtual cylinders for measurement of 3-D surfaces is proposed. In a Cartesian coordinate system, the virtual cylinders are determined by least-squares fitting to the local overlapping surface patches. The error movements are obtained from a linear equation system based on the virtual cylinders. The connection of adjacent views is then performed by coordinate transformation in 3-D space. Both computer simulation and experimental results are presented to verify the effectiveness of the suggested method.
A frequency encoding approach for fringe projection profilometry is proposed. First, a series of fringe patterns are generated in computer where the pixels are encoded with the temporal frequencies. Second, the patterns are cast on the object surface by an LCD projector and then the distorted patterns are captured by a CCD camera. Third, a least squares algorithm based on a linear prediction is deduced to estimate the frequencies, and then the depth map is further reconstructed by using the mapping relationship between the temporal frequency and the depth of the object surface. Experimental results are presented to demonstrate the validity of this technique.
A novel calibration method for fringe projection 3-D measurement system is presented. To get the data serving the calibration, a calibration gauge with white-black checker pattern is transferred to different positions with known depths. At each position, the phase values in the black squares are regarded as invalid data for their lower modulation, and the phase distribution of the whole calibration gauge is obtained by the least-square fitting to the phases in the white squares according to the theoretical distribution function. The phase-to-depth and pixel to lateral coordinate mapping relationship are simultaneously calibrated. The validityof the proposed method is demonstrated by experimental results.
This paper presents a novel calibration approach for determining the mapping relationship between the depth map and the phase difference in fringe projection profilometry. This approach is based on a simple nonlinear function, which is deduced by analyzing the geometry of measurement system and hence perfectly describes the mapping between the depth map and the phase-difference distribution. The calibration is implemented by translating a target plane to a sequence of given positions with known depths, and measuring its phase distributions. A least-squares estimation algorithm with linear computation is deduced to retrieve the related parameters and to reconstruct the mapping function. Both computer simulation and experiment are carried out to demonstrate the validity of this technique.
This paper presents a novel least squares calibration approach for fringe projection profilometry. This approach is based on a simple nonlinear function, which is deduced by analyzing the geometry of measurement system and perfectly describes the mapping relationship between the depth map and phase distribution. The calibration is implemented by translating a target plane to a sequence of given positions with known depths, and measuring its phase distributions. Based on least squares estimation, an algorithm with linear computation is deduced to retrieve the related parameters, by which the burden of computational complexity is effectively alleviated. In experiment, a plaster statue is measured to demonstrate the validity of the principle.
In this paper, we analyze the phase-shifting algorithm by utilizing Fourier transformation theory. In fact, the phase-shifting algorithm is corresponding to a discrete Fourier transformation (DFT), the image capturing operation is a temporal sampling procedure, and the purpose of phase-shifting technique is to retrieve the phase of one frequency component. According to the sampling theory, if the number of phase steps is too less that means a too low sampling frequency is adopted, the frequency of interest will be mixed with high order spectra. By mathematical analysis, the relationship between the number of phase steps and the effect of harmonics is deduced, and the criterion of selecting phase step number is discussed. The applications of the Fourier analysis in digital fringe projection profilometry are described.
In optical three-dimensional measurement, fringe projection technique has found more and more applications. However, in 3D measurement of complex objects, due to the inherent limitation of triangulation, occultation is unavoidable. Meanwhile, the surface of the measured object usually contains discontinuities or the object profile is often sharply steep. Another problem is that for the measurement of a surface with complex reflectivity or quasi-specular, the resulting phase values are unreliable. Therefore, there must be some invalid areas caused by shadow, phase errors or discontinuities, etc. in just a single-view. For compensating the lacked 3-D coordinates of the points in the areas, the multi-frequency fringe projection method is used and temporal phase unwrapping is applied. Invalid areas are marked and further cancelled according to the modulation and phase fitting reliability. To obtain the whole 3D world coordinates of the measured object, a novel connection method based on the principle of the virtual cylinder is presented to accurately integrate the 3D coordinates of every single-view into a global coordinate system. The connection method derives from a technique named Multi-view (aperture) Overlap-scanning Technique (MAOST) based on overlapping areas. The measurement results of an automobile headlamp reflector are experimentally presented to demonstrate the validity.
In this paper, the multi-aperture overlap-scanning technique (MAOST) and its recent developments are presented. In the first instance, MAOST is used in interferometry, and the principle of MAOST for interferometry is that, a tested large-scale plane is covered by an array of interferometric subapertures, and the relationship between each couple of adjacent subapertures is determined from their overlapping areas by a least squares method, and then the profile of tested plane is obtained by connecting all the subapertures together. Recently, to meet the requirements of advanced manufacturing, the idea of MAOST has been extended to precision three-dimensional (3-D) measurement. In practice, a whole-body 3-D shape is acquired by two steps: first measuring the 3-D shape from different views and afterwards connecting all the views together. In order to accurately determine the position and orientation of every single-view in a common coordinate system, an iterative algorithm based on MAOST concept is utilized.
Phase-stepping interferometry has been extensively applied in optical metrology. But the phases calculated by the phase-stepping algorithm are wrapped into the range [-it, it] because of the arctangent function. The procedure of recovering the real phase from thd range [-it, it] is known as phase-unwrapping. The phase unwrapping may be a trivial problem with the cases of noisy, low modulation, corrupted regions etc. in the interferograms. The conventional algorithm always failed in those cases. Many algorithms were developed to solve the problem. A new phase unwrapping algorithm is proposed by our group. It is very suitable for the flat measurement. The new algorithm first fits the measured flat data to an ideal flat by a group of new formulas derived by ourselves. Then do operation of plus or minus 2it for the wrapped phases calculated by phase-stepping algorithm according to the ideal flat. Since the measured flat data do not affect each other in the phase unwrapping procedure, the conventional phase unwrapping problem is avoided. A flat metal plane is measured with many corrupted regions in a Linnik interference microscope. The experimental results indicate that our new algorithm is robust and fast.
Many phase-stepping algorithms have been developed for last decade. Some of them may be more attractive to practical measurement because of the insensitivity to phase-stepping errors. But it should be noticed that all existing phase- stepping algorithm need to know or calculate the phase steps. Why cannot we extract the measured phase without knowing or calculating phase steps. A new phase-stepping algorithm is proposed by our group. The algorithm can be implemented without knowing or calculating the phase steps. Actually, the new algorithm gives important improvements for the algorithms developed by C.T. Farrell and M.A. Player. Based on using Lissajous figures and ellipse fitting, the new algorithm introduces a simple transform for the intensity of the interferograms. New phase extraction expressions are derived. The simulating results indicate that the new algorithm is insensitive to phase-stepping errors and the accuracy of the new algorithm is only limited by the computation truncation errors.