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A comparison of phase shifting and fourier methods in the analysis of discontinuous fringe patterns
Optics and Lasers in Engineering 19 (1993) 269-284
A Comparison of Phase Shifting and Fourier Methods in the Analysis of Discontinuous Fringe Patterns
K. E. Perry,
Jr & J. McKelvie
Department of Mechanical Engineering, Division of Mechanics of Materials, University of Strathclyde, 75 Montrose St, Glasgow, UK, Gl 1XJ
ABSTRACT Phase Shifting Interferometry is a highly accurate data acquisition technique that eficiently utilizes several frames of information for each measurement. In this work, the advantages of phase shifting have been applied to a conventional moire interferometer, yielding a system capable of recording phase shifted fringe patterns for both in-plane displacement components. Using this method, the phase of a wavefront of interest can be determined at each detector location, so that the resolution of the phase measurements is limited primarily by the detector discrimination and geometry. Unlike traditional Fourier fringe analysis, the noise rejection of phase shift processing algorithms does not degrade image fidelity in the presence of edges and discontinuities. A general discussion of both the phase shifting technique and the Fourier fringe analysis method is included to provide insight into the problems of processing discontinuous fringe patterns.
1 INTRODUCTION Phase shifting interferometry (PSI) is becoming an increasingly popular technique for measuring optical path differences in a variety of applications. The technique has been successfully used with holographic, multiple-wavelength and speckle interferometry to produce surface contours and deformation measurements,’ and Han* has utilized a phase shifting technique in conjunction with a microscopic moire interferometer. The method of PSI dates back to at least the work of 269 Optics and Lasers in Engineering 0143-8166/93/$06.00 Ltd, England. Printed in Northern Ireland
0 1993 Elsevier
Science Publishers
270
K. E. Perry
Jr, J. McKelvie
Car&” and has received much attention more recently, due to the increased availability of inexpensive piezo-ceramics and fast video frame buffer technology. An extensive review of phase shifting can be found in Ref. 4. In this work, the advantages of phase shifting have been successfully applied to a conventional, three-mirror moire interferometer, permitting the acquisition of phase shifted fringe patterns that can be processed by either Fourier domain filtering or conventional PSI algorithms. The technique of moire interferometry involves interfering two diffracted beams from a deformed diffraction grating, where the resulting fringe pattern is a direct encoding of the in-plane strains of the grating.” Of particular interest is the application of moire interferometry to problems arising in the field of experimental stress analysis. With detailed knowledge of interferometrically determined displacement fields, parameters that characterize a material or structure can be deduced. Major difficulties arise with the analysis of these types of fringe patterns when they contain discontinuities and/or internal boundaries. Therefore, a method of acquiring and processing fringe patterns that performs well under these circumstances is needed, and provides the motivation for this work. Traditional analysis of interferograms, such as the identification and ordering of fringe centres, results in a precision well below that obtainable from most interferometric data. Even with the most advanced digital image processing techniques, precise determination of fringe centres is difficult and subject to many sources of error. Furthermore, these methods generally require the manual ordering of fringes, since the convexity or concavity of the wavefront can not be automatically deduced from a single interferogram. Thus, the more powerful methods of Fourier fringe analysis and PSI are considered. The Fourier method has the advantage of requiring only one interferogram for the analysis. This makes it particularly attractive for dynamic applications, or instances where vibrations make interferometric measurements difficult. It is also possible to recover both in-plane displacements simultaneously by superimposing the two interference patterns with different carrier frequencies, as was done by Huntley & Field.‘j However, difficulties with this method are inevitable near edges and discontinuities. An important aspect of the PSI analysis is that the phase of the wavefront under study is determined independently at each pointwithout the need to consider data at other points in the image. This means that arbitrary discontinuities in the fringe pattern will be
Comparison
271
of phase shifting and Fourier methods
correctly interpreted and the ‘sense’ of the wavefront will be unambiguously resolved--automatically. Both methods are based on virtually an identical analysis-the main difference is that the PSI algorithms perform computations in the spatial domain, while the computations for the fourier method are performed in the spatial frequency domain.4 Following a general discussion of each technique, two examples taken from current research on displacement fields in composite materials are presented to illustrate the relative advantages and disadvantages of each method.
2 SOME EXPERIENCE
WITH FOURIER
FRINGE
ANALYSIS
The basic Fourier pattern analysis algorithm proposed by Takeda et al.,’ was based on a one-dimensional Fourier transform. The method was refined by employing a two-dimensional Fourier transform by Bone et al.,” greatly improving the ability to isolate the desired signal in the frequency domain. With the Fourier method, a spatial carrier is first established by tilting one of the interfering beams. The resulting intensity distribution, f(~, y), of a two-dimensional fringe pattern can be represented by the following equation
f(%
Y>
= 44
Y>
+%
Y> cos [274f,x + f, Y >+
44%Y )I
(1)
where a(x, y) is the background intensity, b(x, y) is the modulation amplitude of the fringes fx and f, are the linear components of the imposed tilt, and 4(x, y) is the phase information of interest. By performing a two-dimensional Discrete Fourier Transform (DFT), usually in the form of an FFT, the spatial frequency domain representation of the pattern becomes F(& ‘I) =A(=& rl) + C(& rl) + C*(f; V)
(2)
where A(<, 77) is the transform of a(x, y), and C(<, 77) and C*(<, 7) are the positive and negative frequency spectra of the modulated carrier fringes. It is essential for the three terms in the above equation to be completely isolated from one another in the spatial frequency domain. This is satisfied provided the signal of interest is sufficiently band limited, and enough carrier is introduced to separate the positive and negative spectra. Once the function F(& n) is available, a power spectrum, IF(& ~)1’, can be formed by simply taking complex magnitudes of F(& 7). Using the power spectrum as an indicator of where the energy of the signal
272
K. E. Perry Jr, J. McKelvie
lies, a suitable filter can be constructed that selectively attenuates the undesired portion of the spectrum. A simple rectangular bandpass filter, with a value of unity in the region containing the signal, and zero elsewhere, will usually yield satisfactory results. A more precise method of designing the filter is presented below. The strength of the Fourier method lies in the ability, afforded by the DFT, to identify the signal and separate it from the noise. However, difficulties are encountered due to certain implicit assumptions involved with frequency domain analysis using the DFI. The discrete nature of the DFT assumes that the function f(x, y) can be identically decomposed into a sum of complex exponentials having frequencies that are an integer multiple of the fundamental frequency. The fundamental frequency implicitly determined by the sampling of the input function f(x, y). In the case of video data acquisition, the individual pixels perform the sampling, so the fundamental frequency can be determined from the number of pixels per scan line. By the Sampling Theorem,” if the detector array is M X N pixels, then the transform will provide a decomposition of the input function into frequencies ranging from 0 to M/2 and 0 to N/2 fringes per screen. Frequencies higher than these values will be aliased into this range, and frequencies that are not exact integers of the fundamental will be rounded to the nearest integer frequency. A more serious problem with using the DFT results from Discontinuities in f(x, y). The DFT assumes that the original function is cyclicly defined-that is, it should have the same value, and continuous derivatives across the left and right, and top and bottom edges. Thus, the edges of a fringe pattern are an inherent source of discontinuities. If left unchecked, the process of transforming the function will spuriously distribute large amounts of energy over a wide range of frequencies. This not only obscures the signal power, but also makes isolation of pure signal impossible. The common solution to this problem is to first remove the arbitrary dc component of the fringe pattern, and then taper the data at the edges using a numerical window, such as the Hamming or Blackman Window. A further improvement can be made by first artificially creating continuity in the function and the first few of its derivatives by replacing the data around the edges with values obtained from cubic spline interpolates. Still, errors of the order of 10% can be expected near the edges. Additional problems arise because the input function is reduced to a very low intensity by the numerical window. Thus, data near the edges become acutely sensitive to noise, and more importantly, to leakage from nearby regions of the image.
Comparison of phase shifring and Fourier methods
273
In addition to the inherent edges of a fringe pattern, discontinuities can also occur in many practical applications of interferometry, such as with interferograms of shock waves in supersonic flow,” or the analysis of crack-tip displacement fields. Even if the discontinuities do not involve large regions of ‘dead’ zones, where there are no fringes, the analysis is made difficult by the typically large variation in fringe density between different parts of the observation field. There is little that one can do to prevent leakage from occurring in such cases. Limited success has been achieved in such cases by forcing continuity across ‘dead’ zones by replacing the data using interpolating splines, similar to what can be done around the edges. Another method as effective, and easier to implement is to simply replace the ‘dead’ zone with appropriately scaled random data. When transformed, the random data are spread uniformly over the frequency domain, so that corruption of the data near the ‘dead’ zone will be minimized. Obviously, the best way to handle such problems is simply to process the data on a region-by-region basis, in such a way as to completely avoid the discontinuities. This is difficult to automate however, and it will still not yield reliable results near the discontinuities. Another difficulty with the Fourier analysis method occurs when the signal of interest is no longer confined to a narrow band of frequencies. In such cases, it may not be possible to provide sufficient carrier to entirely separate the positive and negative spectra. Even if the signal can be properly isolated, a more carefully designed filter may still be required to sufficiently suppress the noise. A simple approach to filtering in frequency space that can be used for large bandwidth signals is to construct an optimal Weiner filter.” Since the noise encountered in typical fringe patterns is generally random in nature (white noise), the power spectrum will clearly indicate which portions of the frequency domain the signal occupies, so that a simple thresholding will suffice to isolate the signal. Assuming that the fringe pattern noise is additive, the function f(x, y) can be described as
f(x7 Y> = 4x9Y) + n(x, Y) where s(x, y) is the pure signal to be measured, and n(x, y) the noise. The power spectrum of f(x, y), can then be expressed as (3) The terms involving cross products can be neglected because it is assumed that the signal and the noise are uncorrelated. The ideal filter I({, q), is one which, when applied to F(x, y), would
K. E. Perry Jr, J. McKelvie
274
yield the spectrum of the signal of interest S(r; 59 = WJ rl)F(L n) Simple algebraic manipulation optimal filter as
of the previous two equations
yields the
The optimal filter is easily constructed by first smoothing then thresholding a normalized grey-scale version of the (long magnitude) power so that the peaks representing the signal IS(& n)l* spectrum IF(L r1)12, become isolated. Once IS({, n)l’ has been determined, IN(& n)l’ can be deduced from eqn (3). The optimal filter can then be constructed and extended to negative frequencies as necessary. Figc;re 1 is an example of a fringe pattern obtained by taking the difference of two acq.uired images, with a phase shift of 180” between them. A carrier pattern of approximately 17 fringes/screen was introduced, to facilitate isolation of the signal in frequency space. The signal modulating the carrier pattern represents horizontal displacements of the surface of a deformed composite laminate specimen,
Fig. 1.
Differenced
fringe pattern
of a composite load.
laminate
under a three-point
bending
Comparison of phase shifting and Fourier methods
Fig. 2.
27.5
Wrapped Fourier filtered fringe pattern.
under a bending load. The field of view is a two millimetre square, and the fringe contour interval is O-25 pm. The specimen is composed of layers of different materials, and the relative displacements between each layer is of particular interest. After removing the arbitrary DC component and applying a Hamming window, a two-dimensional FFT was performed, resulting in a frequency space representation of the fringe pattern. An optimal filter was constructed from the power spectrum using the method discussed above, and a wrapped filtered fringe pattern was obtained, and is shown in Fig. 2. The wrapped and filtered pattern was unwrapped using the phase unwrapping method presented below to yield a continuous phase map (displacement field). A topological plot of this result is presented in Fig. 3, where each contour represents a change of 0.15 pm. This simple example illustrates how the Fourier method can be used to drastically improve the signal-to-noise ratio of an image, and produce a continuous phase map. However, the improved fringe contrast comes at the expense of errors introduced particularly at the corners and edges of the image. This can be seen in Fig. 2. An analysis of this same fringe pattern using the four-step PSI algorithm is presented below for comparison.
K. E. Perry Jr, J. McKelvie
276
Fig. 3.
Unwrapped topological plot of the phase extracted from wrapped fringe pattern (contour interval, 0.15 pm).
3 PHASE
SHIFTING
MOIRG
Fourier
filtered
INTERFEROMETRY
The filtering associated with the Fourier method is applied on a global basis-no information about the spatial description of the signal is retained in frequency space. In contrast, PSI provides a point-by-point measurement of wavefront phase, and the problems of discontinuities and boundaries do not exist. The PSI technique is based upon recording several intensity distributions arising from the interference of two beams while a known, uniform phase shift is introduced across one of them. The resulting intensity distribution can then be processed in a relatively simple manner to recover the original phase relationship between the two interfering beams. There are numerous ways to induce the necessary phase shifts and record the resulting intensity distributions. Discrete or continuous phase modulation can be achieved by moving a mirror or a grating, tilting a glass plate, rotating a half-wave plate or analyser, heating or deforming an optical fibre, or using an acousto-optic or electro-optic modulator. Images can be recorded at video frame rates using standard frame grabber hardware installed in a personal computer, which can also be used to drive the phase shifting device, and process the data. One can either ramp the phase continuously, or provide discrete phase steps. The method adopted for this work uses discrete phase stepping, where four frames are sequentially acquired with a phase
277
Comparison of phase shifting and Fourier methods
shift of n/2 between each one:
w, Y> =
4% Y> + a
Y>
cosMb YN
51
I’(.%y> = a(4
Y>
+ w, Y) cos 4(x, Y> +
Lk
Y>
+ WGY> cosC4h Y) + 4
Y)
= 4%
[
[
13(x,y> = 4-T y> + b(x, Y> cos 44%Y I+
$
1
The images are captured and stored at a rate of 25 frames/second using a standard frame grabber. A piezo-electric ceramic pusher used to produce the phase shifts is controlled by a standard personal computer D/A card. This allows for integrated control and careful calibration of the entire data acquisition procedure. One difficulty with using piezo-ceramic devices is that they are inherently nonlinear. All PSI algorithms require a consistent phase shift between frames, although the exact phase step need not be known.” There are several methods that have been proposed for verifying the phase shift procedure,4 most of which assume that a separate control unit is used to control the piezo-electric device. The average frame-toframe phase shifts is obtained from a series of acquired images, and appropriate corrections are applied to the gain control of the piezo voltage control device until the desired shift is achieved. The method used for this work takes a different approach, and provides for the verification of the precise shift for each individual frame of intensity values. An undeformed diffraction grating is placed in the interferometer, and a reference tilt is introduced to yield a uniform carrier pattern. An array of digital values is constructed by interpolating a cubic spline fit to a set of control points (usually lo), which are originally chosen to produce a linear voltage ramp. These values are then fed to the D/A converter that drives the piezo-electric pusher, and four frames of intensity values are sequentially acquired. A representative horizontal strip of the observation field is selected by the user, and a one-dimensional Fourier transform is performed on the data in this region for each of the acquired images. In frequency space, the exact phase at the carrier frequency is determined for each frame, and reported to the user. The control points are then modified by the user, a new cubic spline interpolation performed, and another set of images is acquired. This process is continued until the precise phase step is achieved for each frame. Once the appropriate control points have been determined, they are stored and used to initialize a voltage
278
K. E. Perry Jr, J. McKelvie
ramp to drive the piezo-electric pusher during subsequent data acquisition procedures. Even with a carefully calibrated voltage ramp, it is necessary to have some immediate measure of the quality of the phase shifts during the data acquisition process, since vibrations can severely upset the process. For this reason, an array processor installed in the personal computer is used to quickly calculate the wrapped phase values and a modulation mask, using the equations
[,,x, 4(x, Y> -
+tx7 y,=arctan y)
w, Y>
Z,(x, y) 1
The modulation mask provides an estimate of the relative signal-tonoise ratio for each pixel location in the image and is indispensable in assessing the quality of a wrapped fringe pattern. Other algorithms are available for calculating the phase from shifted images, particularly the five-step self-calibrating method.12 However, it has been found that better results are obtained by limiting the time of data acquisition by using the four-step algorithm. The wrapped fringe pattern will consist of values between +X and -r, so they are scaled between 0 and 255 and transferred back to the frame grabber for viewing and possible hard disk storage. The modulation mask must also be scaled into the range O-255, where low values indicate regions where relatively little modulation was detected, and high values indicate where the fringe pattern was strongly modulated. Both the modulation mask and the wrapped fringe pattern are processed and available for inspection in approximately five seconds. The largest source of error is due to high frequency vibrations during the data acquisition process. These vibrations result in deviations from the nominal phase shift, and are thus manifest as phase shift errors. It is not easy to tell from a wrapped fringe pattern alone if the acquisition process was affected by extraneous vibrations. However, it has been shown (for example in Ref. 4) that the error term resulting from phase shift error will be modulated by the function cos [2+(x, y)]. This modulated error term can be easily detected in the modulation mask by the user, providing an immediate assessment of the acquisition process. Once the user is satisfied that a particular wrapped fringe pattern is vibration free, both the modulation mask and the wrapped fringe pattern can be stored on the hard disk of the personal computer. The modulation mask is not necessarily needed for the remainder of the processing, but it can be helpful in certain instances, such as when the
Comparison of phase shifting and Fourier methods
279
image contains internal boundaries or other regions where the modulation strength is low or non-existent. By simply thresholding the modulation mask, boundaries become automatically defined, and lowmodulation regions can be tagged for additional pre-processing. The mask can be used to assist in the unwrapping process in very much the same way as Bone” used the locally unwrapped second differences of the phase. Fringe patterns produced by both Fourier fringe analysis and the conventional phase shifting method require unwrapping, due to the nature of the arctan calculation. One major difference between the Fourier filtered wrapped fringe pattern, and the wrapped fringe pattern produced by eqn (4), is that a large degree of smoothing is implicit in the former. Thus, unwrapping Fourier filtered wrapped fringes is generally a trivial procedure, while unwrapping images determined using eqn (4) can become very difficult when the signal-to-noise ratio becomes low, such as when the fringe gradient becomes large. In these cases, it is necessary to provide a pre-processing filtering routine to reduce the noise level in regions where the fringe gradient is high. A special method has been developed for this task, which is based on replacing the data on a point-by-point basis with smoothed values obtained from locally unwrapped least squares cubic surfaces. The method is similar to simple median filtering, except a more reliable replacement criteria can be established, since more effort is used to characterize the region surrounding each point. The basic idea behind phase unwrapping, is that a multiple of 2n must be added or subtracted to each wrapped phase value. This is needed to restore continuity to the phase values that are produced by the arctan function in eqn (4). In order to produce a globally correct unwrapped phase map, it is necessary to first determine residues, and form appropriate branch-cuts, as discussed by Huntley.‘” Residues indicate where locally unwrapped phase values exhibit an anomalous phase jump, so the branch-cuts are used to mask regions of the fringe pattern that would otherwise introduce global errors into the resultant unwrapped phase map. Once the residues have been determined, and appropriate branchcuts made, the unwrapping procedure is straightforward. The unwrapping procedure developed for this work is based on a recursive flood-fill algorithm, and proceeds as follows. A starting point is selected, and its phase is arbitrarily set to some value. Each neighbouring pixel is then checked in turn to see if it needs unwrapping. When a neighbouring pixel needs unwrapping, it is unwrapped relative to the first pixel, and the process of checking the neighbourng pixels will begin again, this
K. E. Perry Jr, J. McKeluie
280
time centred on the recently unwrapped pixel. In this manner, all pixels connected to the starting pixel will be unwrapped to produce a continuous phase distribution. The algorithm proceeds very quickly and is robust. Occasionally, regions may become isolated by the branch cuts. These regions can still be unwrapped, but an arbitrary value must be used to begin the flood-fill algorithm. Thus, these small regions still need to be ‘connected’ to the larger main region. This is achieved using a simulated annealing algorithm to determine the arbitrary offset between the two regions, and then adding or subtracting this value to each pixel within the smaller region. After the complete image has been unwrapped, it is necessary to ‘repair’ the phase map by replacing the data points that were identified as being residue or branch-cut locations. This is accomplished using neighbouring unwrapped phase values to construct and interpolate cubic splines to replace the data as needed. An example of a wrapped fringe pattern is presented in Fig. 4. This fringe pattern was obtained from the same deformed grating as the previous example, except that the carrier fringes were removed by tilting one of the mirrors in the interferometer. A topological plot of
Fig. 4.
Wrapped
fringe pattern
generated
using the four-step
phase
shifting
equation.
281
Comparison of phase shifting and Fourier methods
Fig. 5.
Unwrapped
topological
phase
plot of wrapped O-15 pm).
fringe pattern
(contour
interval,
the unwrapped phase map is shown in Fig. 5, and can be compared directly to Fig. 3. Both topological plots provide similar results, except that the phase shifting example is not degraded near the edges and in the corners. Figure 6 is an example of another wrapped fringe pattern taken from a similar specimen, this time however, the fringe pattern contained a discontinuity due to a crack. This discontinuity made an analysis using the Fourier filtering method impractical, but there was no difficulty with using the phase shift method. As can be seen in the topological plot of the continuous phase, Fig. 7, the discontinuity was correctly identified. The wrapped fringe patterns in Figs 4 and 6 were relatively simple to process. Neither required additional pre-processing, and only one had isolated regions that were automatically corrected using the simulated annealing algorithm.
4 SUMMARY Despite the power of the Fourier method, there still remain grave difficulties associated with edge effects and spectral leakage. Problems arise when the frequency space representation of the fringe pattern is modified, and inversely transformed. Regions that were necessarily reduced in intensity to enforce cyclic continuity become increasingly sensitive to noise and leakage from other parts of the fringe pattern.
K. E. Perry
282
Fig. 6.
Fig. 7.
Wrapped
Unwrapped
fringe
pattern
topological
Jr, J. McKelvie
of a double
phase
cantilever,
plot of wrapped 0.15 pm).
cracked
fringe pattern
composite
(contour
beam.
interval,
Comparison of phase shifting and Fourier methods
283
Discontinuous fringe patterns arise in many practical circumstances, and their analysis is complicated by a number of factors. Obvious problems arise in coping with ‘dead-zones’ or regions that contain no data. In addition, strong transitions in the relative density and orientation of fringes surrounding discontinuities pose further difficulties for the Fourier filtering approach, by increasing the bandwidth of the signal of interest. It is our experience that even with the most judicious implementation, the Fourier method is not capable of providing satisfactory measurements near boundaries and discontinuities. In contrast, the phase shifting technique has no problem with internal boundaries or discontinuities, nor with large variations in fringe density and orientation. Another advantage of the PSI algorithm is that it requires virtually no assistance from a user, whereas the Fourier filtering method requires at least a user-supervised filter selection process. The PSI method also has the advantage that each fringe pattern can represent absolute phase measurements, while the Fourier method will require an additional step to remove the carrier fringes that were added to separate the spectra. Of course, limitations of the phase shifting method do exist. The primary difficulty is associated with coping with large fringe gradients, since the unwrapping becomes increasingly more difficult. The residue calculation and branch-cut procedure guarantees a consistent unwrapping, but when many branch-cuts are present, many regions will be isolated, and an attempt to connect these regions becomes intractable. Work continues to develop more robust pre-processing routines to be used when the fringe gradient becomes large, thus increasing the usefulness of this method for problems arising in the field of experimental stress analysis.
REFERENCES 1.
interferometry techniques. Progress in (1986) 349-93. sensitivity moire interferometry for micromechanics
Creath, K., Phase-measurement Optics XXVI,
l(1) Higher
2. Han, B., studies. Optical Engineering, 31(7) (1992) 1517-25. 3. P. CarrC, Installation et utilisation due compateur photoelectrique et interferential du bureau international des poids et mesures. Metrologia, 2(13) (1966) 12-23. 4. Greivenkamp, J. E. & Bruning, J. H., Phase Optical Shop Testing (ed. D. Malacara). Wiley
Optics,
John
Wiley,
1986 Ch. 14, pp. 501-98.
shifting interferometry. In Series in Pure and Applied
K. E. Perry Jr, J. McKelvie
284
5. Post, D., Moire interferometry.
In, Handbook on Experimental Mechanics (ed. A. S. Kobayashi). Prentice-Hall Inc., New Jersey, 1987, 314-83. 6. Huntley J. M. 62 Field, J. E., High resolution moire photography: application to dynamic stress analysis. Optical Engineering, 28(8) (1989) 926-33. 7. Takeda,
pattern
M., Ina, H. & Kobayashi, S., Fourier transform method of fringe analysis for computer-based topography and interferometry.
Journal of the Optical Society of America, 72(l) (1982) 156-60. 8. Bone, D. J., Bachor, H.-A. & J. Sandeman. Fringe pattern analysis using a 2-d Fourier transform. Applied Optics, 25(10) (1986) 1653-60. 9. Gaskill, J. D., Linear Systems, Fourier Transforms, and Optics. John Wiley
& Sons, New York, 1978. 10. Bone, D. J., Fourier fringe analysis: the two-dimensional phase unwrapping problem. Applied Optics, 25(30) (1991) 3627-32. 11. Press, W. H., Flannery, B. P., Teukosky, S. A. & Vetterling, W. T., Numerical Recipes in C. Cambridge University Press, UK, 1988. 12. Hariharan, P., Oreb, B. F. & Eiju, T., Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm. Applied Optics, 26(13) (1987) 2504-5. 13. Huntley, J. M., Noise-immune phase Optics, 28(15) (1989) 3268-70.
unwrapping
algorithm.
Applied
A Comparison of Phase Shifting and Fourier Methods in the Analysis of Discontinuous Fringe Patterns
K. E. Perry,
Jr & J. McKelvie
Department of Mechanical Engineering, Division of Mechanics of Materials, University of Strathclyde, 75 Montrose St, Glasgow, UK, Gl 1XJ
ABSTRACT Phase Shifting Interferometry is a highly accurate data acquisition technique that eficiently utilizes several frames of information for each measurement. In this work, the advantages of phase shifting have been applied to a conventional moire interferometer, yielding a system capable of recording phase shifted fringe patterns for both in-plane displacement components. Using this method, the phase of a wavefront of interest can be determined at each detector location, so that the resolution of the phase measurements is limited primarily by the detector discrimination and geometry. Unlike traditional Fourier fringe analysis, the noise rejection of phase shift processing algorithms does not degrade image fidelity in the presence of edges and discontinuities. A general discussion of both the phase shifting technique and the Fourier fringe analysis method is included to provide insight into the problems of processing discontinuous fringe patterns.
1 INTRODUCTION Phase shifting interferometry (PSI) is becoming an increasingly popular technique for measuring optical path differences in a variety of applications. The technique has been successfully used with holographic, multiple-wavelength and speckle interferometry to produce surface contours and deformation measurements,’ and Han* has utilized a phase shifting technique in conjunction with a microscopic moire interferometer. The method of PSI dates back to at least the work of 269 Optics and Lasers in Engineering 0143-8166/93/$06.00 Ltd, England. Printed in Northern Ireland
0 1993 Elsevier
Science Publishers
270
K. E. Perry
Jr, J. McKelvie
Car&” and has received much attention more recently, due to the increased availability of inexpensive piezo-ceramics and fast video frame buffer technology. An extensive review of phase shifting can be found in Ref. 4. In this work, the advantages of phase shifting have been successfully applied to a conventional, three-mirror moire interferometer, permitting the acquisition of phase shifted fringe patterns that can be processed by either Fourier domain filtering or conventional PSI algorithms. The technique of moire interferometry involves interfering two diffracted beams from a deformed diffraction grating, where the resulting fringe pattern is a direct encoding of the in-plane strains of the grating.” Of particular interest is the application of moire interferometry to problems arising in the field of experimental stress analysis. With detailed knowledge of interferometrically determined displacement fields, parameters that characterize a material or structure can be deduced. Major difficulties arise with the analysis of these types of fringe patterns when they contain discontinuities and/or internal boundaries. Therefore, a method of acquiring and processing fringe patterns that performs well under these circumstances is needed, and provides the motivation for this work. Traditional analysis of interferograms, such as the identification and ordering of fringe centres, results in a precision well below that obtainable from most interferometric data. Even with the most advanced digital image processing techniques, precise determination of fringe centres is difficult and subject to many sources of error. Furthermore, these methods generally require the manual ordering of fringes, since the convexity or concavity of the wavefront can not be automatically deduced from a single interferogram. Thus, the more powerful methods of Fourier fringe analysis and PSI are considered. The Fourier method has the advantage of requiring only one interferogram for the analysis. This makes it particularly attractive for dynamic applications, or instances where vibrations make interferometric measurements difficult. It is also possible to recover both in-plane displacements simultaneously by superimposing the two interference patterns with different carrier frequencies, as was done by Huntley & Field.‘j However, difficulties with this method are inevitable near edges and discontinuities. An important aspect of the PSI analysis is that the phase of the wavefront under study is determined independently at each pointwithout the need to consider data at other points in the image. This means that arbitrary discontinuities in the fringe pattern will be
Comparison
271
of phase shifting and Fourier methods
correctly interpreted and the ‘sense’ of the wavefront will be unambiguously resolved--automatically. Both methods are based on virtually an identical analysis-the main difference is that the PSI algorithms perform computations in the spatial domain, while the computations for the fourier method are performed in the spatial frequency domain.4 Following a general discussion of each technique, two examples taken from current research on displacement fields in composite materials are presented to illustrate the relative advantages and disadvantages of each method.
2 SOME EXPERIENCE
WITH FOURIER
FRINGE
ANALYSIS
The basic Fourier pattern analysis algorithm proposed by Takeda et al.,’ was based on a one-dimensional Fourier transform. The method was refined by employing a two-dimensional Fourier transform by Bone et al.,” greatly improving the ability to isolate the desired signal in the frequency domain. With the Fourier method, a spatial carrier is first established by tilting one of the interfering beams. The resulting intensity distribution, f(~, y), of a two-dimensional fringe pattern can be represented by the following equation
f(%
Y>
= 44
Y>
+%
Y> cos [274f,x + f, Y >+
44%Y )I
(1)
where a(x, y) is the background intensity, b(x, y) is the modulation amplitude of the fringes fx and f, are the linear components of the imposed tilt, and 4(x, y) is the phase information of interest. By performing a two-dimensional Discrete Fourier Transform (DFT), usually in the form of an FFT, the spatial frequency domain representation of the pattern becomes F(& ‘I) =A(=& rl) + C(& rl) + C*(f; V)
(2)
where A(<, 77) is the transform of a(x, y), and C(<, 77) and C*(<, 7) are the positive and negative frequency spectra of the modulated carrier fringes. It is essential for the three terms in the above equation to be completely isolated from one another in the spatial frequency domain. This is satisfied provided the signal of interest is sufficiently band limited, and enough carrier is introduced to separate the positive and negative spectra. Once the function F(& n) is available, a power spectrum, IF(& ~)1’, can be formed by simply taking complex magnitudes of F(& 7). Using the power spectrum as an indicator of where the energy of the signal
272
K. E. Perry Jr, J. McKelvie
lies, a suitable filter can be constructed that selectively attenuates the undesired portion of the spectrum. A simple rectangular bandpass filter, with a value of unity in the region containing the signal, and zero elsewhere, will usually yield satisfactory results. A more precise method of designing the filter is presented below. The strength of the Fourier method lies in the ability, afforded by the DFT, to identify the signal and separate it from the noise. However, difficulties are encountered due to certain implicit assumptions involved with frequency domain analysis using the DFI. The discrete nature of the DFT assumes that the function f(x, y) can be identically decomposed into a sum of complex exponentials having frequencies that are an integer multiple of the fundamental frequency. The fundamental frequency implicitly determined by the sampling of the input function f(x, y). In the case of video data acquisition, the individual pixels perform the sampling, so the fundamental frequency can be determined from the number of pixels per scan line. By the Sampling Theorem,” if the detector array is M X N pixels, then the transform will provide a decomposition of the input function into frequencies ranging from 0 to M/2 and 0 to N/2 fringes per screen. Frequencies higher than these values will be aliased into this range, and frequencies that are not exact integers of the fundamental will be rounded to the nearest integer frequency. A more serious problem with using the DFT results from Discontinuities in f(x, y). The DFT assumes that the original function is cyclicly defined-that is, it should have the same value, and continuous derivatives across the left and right, and top and bottom edges. Thus, the edges of a fringe pattern are an inherent source of discontinuities. If left unchecked, the process of transforming the function will spuriously distribute large amounts of energy over a wide range of frequencies. This not only obscures the signal power, but also makes isolation of pure signal impossible. The common solution to this problem is to first remove the arbitrary dc component of the fringe pattern, and then taper the data at the edges using a numerical window, such as the Hamming or Blackman Window. A further improvement can be made by first artificially creating continuity in the function and the first few of its derivatives by replacing the data around the edges with values obtained from cubic spline interpolates. Still, errors of the order of 10% can be expected near the edges. Additional problems arise because the input function is reduced to a very low intensity by the numerical window. Thus, data near the edges become acutely sensitive to noise, and more importantly, to leakage from nearby regions of the image.
Comparison of phase shifring and Fourier methods
273
In addition to the inherent edges of a fringe pattern, discontinuities can also occur in many practical applications of interferometry, such as with interferograms of shock waves in supersonic flow,” or the analysis of crack-tip displacement fields. Even if the discontinuities do not involve large regions of ‘dead’ zones, where there are no fringes, the analysis is made difficult by the typically large variation in fringe density between different parts of the observation field. There is little that one can do to prevent leakage from occurring in such cases. Limited success has been achieved in such cases by forcing continuity across ‘dead’ zones by replacing the data using interpolating splines, similar to what can be done around the edges. Another method as effective, and easier to implement is to simply replace the ‘dead’ zone with appropriately scaled random data. When transformed, the random data are spread uniformly over the frequency domain, so that corruption of the data near the ‘dead’ zone will be minimized. Obviously, the best way to handle such problems is simply to process the data on a region-by-region basis, in such a way as to completely avoid the discontinuities. This is difficult to automate however, and it will still not yield reliable results near the discontinuities. Another difficulty with the Fourier analysis method occurs when the signal of interest is no longer confined to a narrow band of frequencies. In such cases, it may not be possible to provide sufficient carrier to entirely separate the positive and negative spectra. Even if the signal can be properly isolated, a more carefully designed filter may still be required to sufficiently suppress the noise. A simple approach to filtering in frequency space that can be used for large bandwidth signals is to construct an optimal Weiner filter.” Since the noise encountered in typical fringe patterns is generally random in nature (white noise), the power spectrum will clearly indicate which portions of the frequency domain the signal occupies, so that a simple thresholding will suffice to isolate the signal. Assuming that the fringe pattern noise is additive, the function f(x, y) can be described as
f(x7 Y> = 4x9Y) + n(x, Y) where s(x, y) is the pure signal to be measured, and n(x, y) the noise. The power spectrum of f(x, y), can then be expressed as (3) The terms involving cross products can be neglected because it is assumed that the signal and the noise are uncorrelated. The ideal filter I({, q), is one which, when applied to F(x, y), would
K. E. Perry Jr, J. McKelvie
274
yield the spectrum of the signal of interest S(r; 59 = WJ rl)F(L n) Simple algebraic manipulation optimal filter as
of the previous two equations
yields the
The optimal filter is easily constructed by first smoothing then thresholding a normalized grey-scale version of the (long magnitude) power so that the peaks representing the signal IS(& n)l* spectrum IF(L r1)12, become isolated. Once IS({, n)l’ has been determined, IN(& n)l’ can be deduced from eqn (3). The optimal filter can then be constructed and extended to negative frequencies as necessary. Figc;re 1 is an example of a fringe pattern obtained by taking the difference of two acq.uired images, with a phase shift of 180” between them. A carrier pattern of approximately 17 fringes/screen was introduced, to facilitate isolation of the signal in frequency space. The signal modulating the carrier pattern represents horizontal displacements of the surface of a deformed composite laminate specimen,
Fig. 1.
Differenced
fringe pattern
of a composite load.
laminate
under a three-point
bending
Comparison of phase shifting and Fourier methods
Fig. 2.
27.5
Wrapped Fourier filtered fringe pattern.
under a bending load. The field of view is a two millimetre square, and the fringe contour interval is O-25 pm. The specimen is composed of layers of different materials, and the relative displacements between each layer is of particular interest. After removing the arbitrary DC component and applying a Hamming window, a two-dimensional FFT was performed, resulting in a frequency space representation of the fringe pattern. An optimal filter was constructed from the power spectrum using the method discussed above, and a wrapped filtered fringe pattern was obtained, and is shown in Fig. 2. The wrapped and filtered pattern was unwrapped using the phase unwrapping method presented below to yield a continuous phase map (displacement field). A topological plot of this result is presented in Fig. 3, where each contour represents a change of 0.15 pm. This simple example illustrates how the Fourier method can be used to drastically improve the signal-to-noise ratio of an image, and produce a continuous phase map. However, the improved fringe contrast comes at the expense of errors introduced particularly at the corners and edges of the image. This can be seen in Fig. 2. An analysis of this same fringe pattern using the four-step PSI algorithm is presented below for comparison.
K. E. Perry Jr, J. McKelvie
276
Fig. 3.
Unwrapped topological plot of the phase extracted from wrapped fringe pattern (contour interval, 0.15 pm).
3 PHASE
SHIFTING
MOIRG
Fourier
filtered
INTERFEROMETRY
The filtering associated with the Fourier method is applied on a global basis-no information about the spatial description of the signal is retained in frequency space. In contrast, PSI provides a point-by-point measurement of wavefront phase, and the problems of discontinuities and boundaries do not exist. The PSI technique is based upon recording several intensity distributions arising from the interference of two beams while a known, uniform phase shift is introduced across one of them. The resulting intensity distribution can then be processed in a relatively simple manner to recover the original phase relationship between the two interfering beams. There are numerous ways to induce the necessary phase shifts and record the resulting intensity distributions. Discrete or continuous phase modulation can be achieved by moving a mirror or a grating, tilting a glass plate, rotating a half-wave plate or analyser, heating or deforming an optical fibre, or using an acousto-optic or electro-optic modulator. Images can be recorded at video frame rates using standard frame grabber hardware installed in a personal computer, which can also be used to drive the phase shifting device, and process the data. One can either ramp the phase continuously, or provide discrete phase steps. The method adopted for this work uses discrete phase stepping, where four frames are sequentially acquired with a phase
277
Comparison of phase shifting and Fourier methods
shift of n/2 between each one:
w, Y> =
4% Y> + a
Y>
cosMb YN
51
I’(.%y> = a(4
Y>
+ w, Y) cos 4(x, Y> +
Lk
Y>
+ WGY> cosC4h Y) + 4
Y)
= 4%
[
[
13(x,y> = 4-T y> + b(x, Y> cos 44%Y I+
$
1
The images are captured and stored at a rate of 25 frames/second using a standard frame grabber. A piezo-electric ceramic pusher used to produce the phase shifts is controlled by a standard personal computer D/A card. This allows for integrated control and careful calibration of the entire data acquisition procedure. One difficulty with using piezo-ceramic devices is that they are inherently nonlinear. All PSI algorithms require a consistent phase shift between frames, although the exact phase step need not be known.” There are several methods that have been proposed for verifying the phase shift procedure,4 most of which assume that a separate control unit is used to control the piezo-electric device. The average frame-toframe phase shifts is obtained from a series of acquired images, and appropriate corrections are applied to the gain control of the piezo voltage control device until the desired shift is achieved. The method used for this work takes a different approach, and provides for the verification of the precise shift for each individual frame of intensity values. An undeformed diffraction grating is placed in the interferometer, and a reference tilt is introduced to yield a uniform carrier pattern. An array of digital values is constructed by interpolating a cubic spline fit to a set of control points (usually lo), which are originally chosen to produce a linear voltage ramp. These values are then fed to the D/A converter that drives the piezo-electric pusher, and four frames of intensity values are sequentially acquired. A representative horizontal strip of the observation field is selected by the user, and a one-dimensional Fourier transform is performed on the data in this region for each of the acquired images. In frequency space, the exact phase at the carrier frequency is determined for each frame, and reported to the user. The control points are then modified by the user, a new cubic spline interpolation performed, and another set of images is acquired. This process is continued until the precise phase step is achieved for each frame. Once the appropriate control points have been determined, they are stored and used to initialize a voltage
278
K. E. Perry Jr, J. McKelvie
ramp to drive the piezo-electric pusher during subsequent data acquisition procedures. Even with a carefully calibrated voltage ramp, it is necessary to have some immediate measure of the quality of the phase shifts during the data acquisition process, since vibrations can severely upset the process. For this reason, an array processor installed in the personal computer is used to quickly calculate the wrapped phase values and a modulation mask, using the equations
[,,x, 4(x, Y> -
+tx7 y,=arctan y)
w, Y>
Z,(x, y) 1
The modulation mask provides an estimate of the relative signal-tonoise ratio for each pixel location in the image and is indispensable in assessing the quality of a wrapped fringe pattern. Other algorithms are available for calculating the phase from shifted images, particularly the five-step self-calibrating method.12 However, it has been found that better results are obtained by limiting the time of data acquisition by using the four-step algorithm. The wrapped fringe pattern will consist of values between +X and -r, so they are scaled between 0 and 255 and transferred back to the frame grabber for viewing and possible hard disk storage. The modulation mask must also be scaled into the range O-255, where low values indicate regions where relatively little modulation was detected, and high values indicate where the fringe pattern was strongly modulated. Both the modulation mask and the wrapped fringe pattern are processed and available for inspection in approximately five seconds. The largest source of error is due to high frequency vibrations during the data acquisition process. These vibrations result in deviations from the nominal phase shift, and are thus manifest as phase shift errors. It is not easy to tell from a wrapped fringe pattern alone if the acquisition process was affected by extraneous vibrations. However, it has been shown (for example in Ref. 4) that the error term resulting from phase shift error will be modulated by the function cos [2+(x, y)]. This modulated error term can be easily detected in the modulation mask by the user, providing an immediate assessment of the acquisition process. Once the user is satisfied that a particular wrapped fringe pattern is vibration free, both the modulation mask and the wrapped fringe pattern can be stored on the hard disk of the personal computer. The modulation mask is not necessarily needed for the remainder of the processing, but it can be helpful in certain instances, such as when the
Comparison of phase shifting and Fourier methods
279
image contains internal boundaries or other regions where the modulation strength is low or non-existent. By simply thresholding the modulation mask, boundaries become automatically defined, and lowmodulation regions can be tagged for additional pre-processing. The mask can be used to assist in the unwrapping process in very much the same way as Bone” used the locally unwrapped second differences of the phase. Fringe patterns produced by both Fourier fringe analysis and the conventional phase shifting method require unwrapping, due to the nature of the arctan calculation. One major difference between the Fourier filtered wrapped fringe pattern, and the wrapped fringe pattern produced by eqn (4), is that a large degree of smoothing is implicit in the former. Thus, unwrapping Fourier filtered wrapped fringes is generally a trivial procedure, while unwrapping images determined using eqn (4) can become very difficult when the signal-to-noise ratio becomes low, such as when the fringe gradient becomes large. In these cases, it is necessary to provide a pre-processing filtering routine to reduce the noise level in regions where the fringe gradient is high. A special method has been developed for this task, which is based on replacing the data on a point-by-point basis with smoothed values obtained from locally unwrapped least squares cubic surfaces. The method is similar to simple median filtering, except a more reliable replacement criteria can be established, since more effort is used to characterize the region surrounding each point. The basic idea behind phase unwrapping, is that a multiple of 2n must be added or subtracted to each wrapped phase value. This is needed to restore continuity to the phase values that are produced by the arctan function in eqn (4). In order to produce a globally correct unwrapped phase map, it is necessary to first determine residues, and form appropriate branch-cuts, as discussed by Huntley.‘” Residues indicate where locally unwrapped phase values exhibit an anomalous phase jump, so the branch-cuts are used to mask regions of the fringe pattern that would otherwise introduce global errors into the resultant unwrapped phase map. Once the residues have been determined, and appropriate branchcuts made, the unwrapping procedure is straightforward. The unwrapping procedure developed for this work is based on a recursive flood-fill algorithm, and proceeds as follows. A starting point is selected, and its phase is arbitrarily set to some value. Each neighbouring pixel is then checked in turn to see if it needs unwrapping. When a neighbouring pixel needs unwrapping, it is unwrapped relative to the first pixel, and the process of checking the neighbourng pixels will begin again, this
K. E. Perry Jr, J. McKeluie
280
time centred on the recently unwrapped pixel. In this manner, all pixels connected to the starting pixel will be unwrapped to produce a continuous phase distribution. The algorithm proceeds very quickly and is robust. Occasionally, regions may become isolated by the branch cuts. These regions can still be unwrapped, but an arbitrary value must be used to begin the flood-fill algorithm. Thus, these small regions still need to be ‘connected’ to the larger main region. This is achieved using a simulated annealing algorithm to determine the arbitrary offset between the two regions, and then adding or subtracting this value to each pixel within the smaller region. After the complete image has been unwrapped, it is necessary to ‘repair’ the phase map by replacing the data points that were identified as being residue or branch-cut locations. This is accomplished using neighbouring unwrapped phase values to construct and interpolate cubic splines to replace the data as needed. An example of a wrapped fringe pattern is presented in Fig. 4. This fringe pattern was obtained from the same deformed grating as the previous example, except that the carrier fringes were removed by tilting one of the mirrors in the interferometer. A topological plot of
Fig. 4.
Wrapped
fringe pattern
generated
using the four-step
phase
shifting
equation.
281
Comparison of phase shifting and Fourier methods
Fig. 5.
Unwrapped
topological
phase
plot of wrapped O-15 pm).
fringe pattern
(contour
interval,
the unwrapped phase map is shown in Fig. 5, and can be compared directly to Fig. 3. Both topological plots provide similar results, except that the phase shifting example is not degraded near the edges and in the corners. Figure 6 is an example of another wrapped fringe pattern taken from a similar specimen, this time however, the fringe pattern contained a discontinuity due to a crack. This discontinuity made an analysis using the Fourier filtering method impractical, but there was no difficulty with using the phase shift method. As can be seen in the topological plot of the continuous phase, Fig. 7, the discontinuity was correctly identified. The wrapped fringe patterns in Figs 4 and 6 were relatively simple to process. Neither required additional pre-processing, and only one had isolated regions that were automatically corrected using the simulated annealing algorithm.
4 SUMMARY Despite the power of the Fourier method, there still remain grave difficulties associated with edge effects and spectral leakage. Problems arise when the frequency space representation of the fringe pattern is modified, and inversely transformed. Regions that were necessarily reduced in intensity to enforce cyclic continuity become increasingly sensitive to noise and leakage from other parts of the fringe pattern.
K. E. Perry
282
Fig. 6.
Fig. 7.
Wrapped
Unwrapped
fringe
pattern
topological
Jr, J. McKelvie
of a double
phase
cantilever,
plot of wrapped 0.15 pm).
cracked
fringe pattern
composite
(contour
beam.
interval,
Comparison of phase shifting and Fourier methods
283
Discontinuous fringe patterns arise in many practical circumstances, and their analysis is complicated by a number of factors. Obvious problems arise in coping with ‘dead-zones’ or regions that contain no data. In addition, strong transitions in the relative density and orientation of fringes surrounding discontinuities pose further difficulties for the Fourier filtering approach, by increasing the bandwidth of the signal of interest. It is our experience that even with the most judicious implementation, the Fourier method is not capable of providing satisfactory measurements near boundaries and discontinuities. In contrast, the phase shifting technique has no problem with internal boundaries or discontinuities, nor with large variations in fringe density and orientation. Another advantage of the PSI algorithm is that it requires virtually no assistance from a user, whereas the Fourier filtering method requires at least a user-supervised filter selection process. The PSI method also has the advantage that each fringe pattern can represent absolute phase measurements, while the Fourier method will require an additional step to remove the carrier fringes that were added to separate the spectra. Of course, limitations of the phase shifting method do exist. The primary difficulty is associated with coping with large fringe gradients, since the unwrapping becomes increasingly more difficult. The residue calculation and branch-cut procedure guarantees a consistent unwrapping, but when many branch-cuts are present, many regions will be isolated, and an attempt to connect these regions becomes intractable. Work continues to develop more robust pre-processing routines to be used when the fringe gradient becomes large, thus increasing the usefulness of this method for problems arising in the field of experimental stress analysis.
REFERENCES 1.
interferometry techniques. Progress in (1986) 349-93. sensitivity moire interferometry for micromechanics
Creath, K., Phase-measurement Optics XXVI,
l(1) Higher
2. Han, B., studies. Optical Engineering, 31(7) (1992) 1517-25. 3. P. CarrC, Installation et utilisation due compateur photoelectrique et interferential du bureau international des poids et mesures. Metrologia, 2(13) (1966) 12-23. 4. Greivenkamp, J. E. & Bruning, J. H., Phase Optical Shop Testing (ed. D. Malacara). Wiley
Optics,
John
Wiley,
1986 Ch. 14, pp. 501-98.
shifting interferometry. In Series in Pure and Applied
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5. Post, D., Moire interferometry.
In, Handbook on Experimental Mechanics (ed. A. S. Kobayashi). Prentice-Hall Inc., New Jersey, 1987, 314-83. 6. Huntley J. M. 62 Field, J. E., High resolution moire photography: application to dynamic stress analysis. Optical Engineering, 28(8) (1989) 926-33. 7. Takeda,
pattern
M., Ina, H. & Kobayashi, S., Fourier transform method of fringe analysis for computer-based topography and interferometry.
Journal of the Optical Society of America, 72(l) (1982) 156-60. 8. Bone, D. J., Bachor, H.-A. & J. Sandeman. Fringe pattern analysis using a 2-d Fourier transform. Applied Optics, 25(10) (1986) 1653-60. 9. Gaskill, J. D., Linear Systems, Fourier Transforms, and Optics. John Wiley
& Sons, New York, 1978. 10. Bone, D. J., Fourier fringe analysis: the two-dimensional phase unwrapping problem. Applied Optics, 25(30) (1991) 3627-32. 11. Press, W. H., Flannery, B. P., Teukosky, S. A. & Vetterling, W. T., Numerical Recipes in C. Cambridge University Press, UK, 1988. 12. Hariharan, P., Oreb, B. F. & Eiju, T., Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm. Applied Optics, 26(13) (1987) 2504-5. 13. Huntley, J. M., Noise-immune phase Optics, 28(15) (1989) 3268-70.
unwrapping
algorithm.
Applied