Evaluation of a Preconditioned Conjugate-Gradient Algorithm for Weighted Least-Squares Unwrapping of...

Evaluation of a preconditionedconjugate-gradient algorithm forweighted least-squares unwrapping ofdigital speckle-pattern interferometry phase maps

Guillermo H. Kaufmann, Gustavo E. Galizzi, and Pablo D. Ruiz

Inasmuch as current fringe analysis techniques used in digital speckle-pattern interferometry ~DSPI!yield a phase map modulo 2p, phase unwrapping is the final step of any data evaluation process. Theperformance of a recently published algorithm used to unwrap DSPI phase maps is investigated. Thealgorithm is based on a least-squares minimization technique that is solvable by the discrete cosinetransform. When phase inconsistencies are present, they are handled by exclusion of invalid pixels fromthe unwrapping process through the assignment of zero-valued weights. Then the weighted unwrap-ping problem is solved in an iterative manner by a preconditioned conjugate-gradient method. Theevaluation is carried out with computer-simulated DSPI phase maps, an approach that permits thegeneration of phase fields without inconsistencies, which are then used to calculate phase deviations asa function of the iteration number. Real data are also used to illustrate the performance of the algo-rithm. © 1998 Optical Society of America

OCIS codes: 100.5070, 120.2650, 120.5050, 120.6160.

1. Introduction

Digital speckle-pattern interferometry ~DSPI! is awell-known laser-based technique for the measure-ment of mechanical displacements and strains overthe surfaces of objects.1–3 As with other optical tech-niques, it has the advantage of being noncontact, pro-vides whole-field information with submicrometersensitivity, and does not require any preparation ofthe specimen surface. Its main strength, however,resides in its ability to display fringe patterns in realtime by use of video detection and digital processing.Because of these characteristics, DSPI is becoming afavorite candidate for use for nondestructive inspec-tion in industrial environments.

Because DSPI fringe patterns are contaminatedwith high levels of speckle noise, displacement fields

are usually determined through the evaluation of theoptical phase distribution. However, current meth-ods of phase evaluation such as phase stepping4 andfrequency-domain processing5 produce a distributionmodulo 2p. For this reason phase unwrapping isthe final step of any DSPI fringe analysis process.

In the literature many methods appear that claim tobe capable of automated unwrapping of DSPI phasemaps. For a review of the existing methods until1993, see Refs. 6 and 7. The simplest methods ofphase unwrapping are the path-dependent ones thatare based on the evaluation of the phase differencebetween adjacent pixels along columns and rows oralong more complex paths.8 The process is carriedout by addition or subtraction of 2p each time thephase map presents a 2p discontinuity. Even thoughthese methods are well suited for unwrapping noise-free data, the process becomes more difficult when theabsolute phase difference between adjacent pixels atpoints with no discontinuities is greater than p.These spurious jumps can be caused by noise, phasediscontinuities, phase-deficient filtering, or local un-dersampling in the fringe pattern. If any such localphase inconsistency is present, an error appears that ispropagated along the unwrapping path. For avoidingthis problem more powerful methods based on the ideaof ordering the pixels were recently proposed, so the

The authors are with the Instituto de Fısica Rosario, ConsejoNacional de Investigaciones Cientificas y Tecnicas, UniversidadNacional de Rosario, 2000 Rosario, Argentina. G. H. Kaufmannis also with the Departamento de Fısica, Facultad de CienciasExactas, Ingenierıa y Agrimensura, Universidad Nacional deRosario.

Received 1 October 1997; revised manuscript received 7 January1998.

0003-6935y98y143076-09$15.00y0© 1998 Optical Society of America

3076 APPLIED OPTICS y Vol. 37, No. 14 y 10 May 1998

noisiest areas of the phase image are unwrapped last.9However, these methods do not ensure the detection ofall errors, so the undetected errors can still propagatealong the unwrapping path.

For overcoming these difficulties, robust path-independent phase-unwrapping methods were devel-oped in recent years. Different approaches includethose based on the detection of phase discontinuitiesby integration of the phase along a closed path of fourneighboring pixels arranged in a 2 3 2 cluster.10 Asthe integration along any closed path in the imagemust yield a result of zero, nonzero positive and neg-ative residues can easily be found. Then branch cutsthat link positive and negative residues are defined toavoid unwrapping paths that cross them. Althoughsome methods exist to find the best branch-cut net-work when there are many residues in the image,11,12

these methods are computationally intensive, andthere are no criteria for identifying the true solution.

Another algorithm based on temporal rather thanspatial unwrapping was recently proposed.13 Record-ing several time-lapsed fringe patterns allows unwrap-ping to proceed along the time axis by use of theinformation contained in all the patterns. However,this method is not practical for large displacements, asit would require recording many images; neither can itbe used to analyze fast transient events.

Tile processing is a robust method that combinesunwrapping by regions and pixel queuing.14 In thismethod the data array is divided into square tilesthat contain no phase ambiguities, and each tile isunwrapped independently. Then the tiles are ar-ranged in order according to the confidence attachedto the consistency of the phase data at the edges ofthe tiles. It can be shown that the performance ofthis method depends on the size of the tiles withrespect to the average fringe spacing. Ideally, theprocess of tile-size determination would be done au-tomatically, but this is not possible in all cases.Also, fringe discontinuities, which can occur acrosstwo tiles, can be missed, and phase errors will prop-agate through the array of tiles.

Another quite effective approach is that based on acellular automata, although this method requireslong computation times.15 For noisy wrapped im-ages invalid pixels in nonzero residues are automat-ically masked out before each global iteration, givingas a result isolated inconsistent regions. Unfortu-nately, these masked areas extend progressively withthe iterations, so the method fails when the numberof invalid pixels is moderately high.

Ghiglia and Romero16 have proposed a new ap-proach to phase unwrapping that uses a least-squaresminimization technique solvable by the discrete cosinetransform. Phase inconsistencies are handled by ex-clusion of invalid pixels from the unwrapping processthrough the assignment of zero-valued weights. Inthis way phase unwrapping is performed without theneed for interjection of branch cuts to resolve pathintegral discrepancies. For solving the weightedproblem, two schemes were proposed: one based on aPicard iteration and a second based on a precondi-

tioned conjugate-gradient ~PCG! method. The latterapproach has the advantage of giving a faster andguaranteed convergence but requires much morememory storage than does the Picard scheme.

More recently, Marroquin and Rivera17 proposed theinclusion of a regularization term in the least-squaressolution, which has the effect of reducing noise. How-ever, as the noise is smoothed out, the slopes of theoriginal phase are also reduced, thus producing a flat-tening effect in the solution. Because of this draw-back Rivera et al.18 proposed a new quadratic functionthat does not introduce a distorting effect into the so-lution. Thus there is no need to introduce smooth-ness assumption for any unwrapped-phase region withvalid information.

Kerr et al.19 developed their own implementation ofthe Picard iterative method proposed in Ref. 16 to testits performance in real DSPI fringe patterns. Eventhough their results proved that the algorithm is ro-bust and suitable for unwrapping DSPI phase maps,they concluded that an excessive number of iterationsmay be required for certain phase maps, such asthose that contain phase shears.

In this paper we investigate the performance of thephase-unwrapping iterative algorithm based on thePCG method, using computer-simulated DSPI phasemaps. This approach permits the generation of phasefields without inconsistencies, which are then used toevaluate the algorithm’s reliability and its conver-gence. Because the PCG algorithm smoothly interpo-lates the phase over inconsistencies and regions withmissing data, the magnitude of phase errors and itsvariation with the number of iterations are determinedfor different cases. Other issues investigated includethe behavior of the algorithm when it is applied tounwrap phase maps containing multiple isolated arbi-trary regions and also real DSPI phase data.

2. Preconditioned Conjugate Gradient for WeightedPhase Unwrapping

In this section we briefly revisit the PCG algorithmused for phase unwrapping.16 If cm,n is the originalwrapped-phase data for pixel ~m, n! of an N 3 Nimage, it can be shown that the unwrapped phasefm,n in the least-squares error sense can be obtainedas the solution of the following discrete Poisson equa-tion with Neumann boundary conditions:

~fm11,n 2 2fm,n 1 fm21,n! 1 ~fm,n11 2 2fm,n

1 fm,n21! 5 rm,n, (1)

where rm,n is the Laplacian phase vector:

rm,n 5 ~Dm,nx 2 Dm21,n

x ! 1 ~Dm,ny 2 Dm,n21

y !, (2)

and the horizontal and vertical phase-differenceterms are defined respectively as

Dm,nx 5 W~cm11,n 2 cm,n!, m 5 0, . . . , N 2 2,

n 5 0, . . . , N 2 1, (3)

Dm,ny 5 W~cm,n11 2 cm,n!, m 5 0, . . . , N 2 1,

n 5 0, . . . , N 2 2, (4)

10 May 1998 y Vol. 37, No. 14 y APPLIED OPTICS 3077

and 0 otherwise, where W is the wrapping operatorthat wraps all phase differences into the range ~2p,p!.

Equation ~1! can be rewritten as

Pf 5 r, (5)

where P is a matrix that performs the discrete Lapla-cian operation on the vector f and r is a vector con-taining the discrete Laplacian operation on thewrapped-phase differences. Equation ~5! is solvedby use of the two-dimensional discrete cosine trans-form, as detailed in Ref. 20. This process is referredto in Ref. 16 as algorithm 1.

As was already mentioned, the wrapped-phase dis-tribution usually contains areas of phase inconsisten-cies and bad data, which generate localized errorsaround them. Moreover, from the physics of theproblem one can know that certain regions should beunwrapped independently from others. Ghiglia andRomero16 suggested an approach to deal with suchsituations by weighting the original wrapped-phasedata cm,n with a matrix W with elements wm,n 5 0, 1.A value of 0 in the weighting matrix removes thecorresponding wrapped-phase difference value fromthe differential equation.

The introduction of the weighting matrix modifiesEq. ~5!:

Qf 5 c, (6)

where Q is a matrix that performs the modified dis-crete Laplacian operation on the weighted phase dif-ferences and c is a vector formed from the modifieddiscrete Laplacian of the weighted wrapped phasedifferences.

The weighted least-squares problem described byEq. ~6! is solved by application of a fast convergingalgorithm based on the method of preconditioned con-jugate gradients ~algorithm 3 in Ref. 16!. This algo-rithm is summarized as follows:

1. For the iteration counter k 5 0, values of f0 5 0and r0 5 c are chosen.

2. For rk Þ 0, Pzk 5 rk is solved with the two-dimensional discrete cosine transform, as detailed inRef. 20.

3. The iteration counter k 5 k 1 1 is updated.4. If k 5 1, p1 5 z0 is computed.5. If k . 1, then

bk 5 rk21T zk21yrk22

T zk22,

pk 5 zk21 1 bkpk21.

6. One scalar and two vector updates are per-formed:

ak 5 rk21T zk21ypk

TQpk,

fk 5 fk21 1 akpk,

rk 5 rk21 1 akQpk.

7. If k $ kmax the calculation is stopped. Other-wise, it goes to step 2.

It is important to note that the two-dimensionalarrays to be computed, zk and rk, must be interpretedas one-dimensional vectors for calculation of the sca-lars bk and ak and the vector pk. The one-dimensional vectors are easily obtained byconcatenation of columns of the corresponding two-dimensional array.

The PCG algorithm was written in C and run on aPentium Pro computer at 200 MHz with the public-domain 32-bit GNU compiler. This C compiler al-lows access to the full extent of RAM of the computer.It must be noted that a slight disadvantage of theunwrapping algorithm lies in the fact that it requiresexcessive storage. For a 512 3 512 8-bit image thealgorithm requires approximately 12 Mb of RAM tostore a one-byte, two-integer array and nine floating-point two-dimensional arrays to avoid swapping.

3. Numerical Experiments

As we mentioned above, the accuracy of the PCGalgorithm was evaluated with synthetic wrapped-phase maps. To obtain these distributions we gen-erated computer-simulated DSPI fringe patternsusing a method reported in Ref. 21. This methodmodels the process of generating DSPI fringes for anout-of-plane interferometer by the correlation of twospeckle interferograms. The intensity Im,n of each ofthe speckle interferograms to be correlated was cal-culated by means of

Im,n 5 uAeiu 1 Am,nu2

5 uAeiu 1 FT21$Hu,vFT@exp~iam,n!Um,n#%u2, (7)

where A and u are, respectively, the amplitude andthe phase of the reference beam, Am,n is the ampli-tude of a single-exposure speckle pattern, Um,n is anN 3 N complex image with its phase uniformly dis-tributed in the interval ~2p, p!, i is the square root of21, am,n is the phase introduced by the deformation,FT denotes the Fourier transform evaluated with atwo-dimensional fast Fourier transform algorithm,and Hu,v is a circular low-pass filter in the Fourierplane with radius r that is used to define the averagespeckle size. To obtain good contrast fringes weadded the reference beam with twice ~A 5 2! themean amplitude of the object beam.

The average speckle size a is defined by

a 5 Ny2r. (8)

We obtained each DSPI fringe pattern by subtractinga reference speckle interferogram Im,n

1 generatedwith am,n 5 0 from another one, Im,n

2 , calculated witha known phase difference. Then fringe patternswere processed with a low-pass Fourier filter to re-duce speckle noise, and finally the wrapped-phasedistribution was obtained by means of a Fouriertransform method that uses bandpass filtering in the


1u and 1v half-planes of the spatial-frequency do-main.22

To introduce phase inconsistencies in the wrapped-phase map we introduced a small amount of speckledecorrelation by adding an independent speckle pat-tern Am,n

i to the reference speckle interferogram inEq. ~7!. Therefore Im,n

1 was calculated as

Im,n1 5 uAeiu 1 ~1 2 dm,n!Am,n 1 dm,n Am,n

i u2, (9)

where dm,n is a decorrelation matrix with its elementsranging from 0 to 1, representing cases of completecorrelation and decorrelation, respectively.

To assess the performance of the PCG algorithmwe compared the unwrapped-phase map fm,n withthe continuous phase distribution f9m,n determinedwhen no inconsistencies were present. For this com-parison two parameters were used. One parameteris the rms deviation s, given by

s 5 F 1M 2 1 (~fm,n 2 f9m,n!

2G1y2

, (10)

where M is the number of pixels included in the sum.It should be noted that, as different cases were ana-lyzed, M was not always equal to the number of pixelsof the whole image. The other parameter used forthe evaluation is the mean relative error e, given by

e 51M (

ufm,n 2 f9m,nuf9m,n

100%, (11)

where u. . .u means the absolute value.

4. Results and Discussion

The PCG algorithm was tested first on a wrapped-phase map obtained by generation of a DSPI fringepattern with a linear phase along the horizontal di-rection that is modulated by a linear phase along theradial direction. An average speckle size of a 5 2pixels was used. The first phase term simulates 15vertical carrier fringes generated by a tilt introducedbetween the reference and the object beams of theinterferometer. The second phase term, describedby the linear phase along the radial direction, simu-lates six circular fringes that are due to the objectdeformation. This fringe pattern is a typical exam-ple of patterns whose phases are evaluated by meansof Fourier transform methods. As the carrierfringes are vertical, the wrapped-phase map was de-termined by means of bandpass filtering in the fre-quency domain along the 1u half-plane. A lowercutoff frequency of 2 pixels and an upper one of 25pixels were chosen to filter out most of the specklenoise. To simulate an experimental case we chose adecorrelation matrix dm,n 5 0.35 over a centeredsquare of 128 3 128 pixels. Figure 1 shows thewrapped-phase map generated by this procedure.

The weighting array used to remove the effect ofthese phase inconsistencies is shown in Fig. 2 andwas determined by integration of the phase along theclosed paths of four neighboring pixels arranged in2 3 2 clusters. In Fig. 2 black pixels correspond to

zero-valued weights and the remainder of the imageis set to unity weight and is shown as white. Theblack border merely shows the array’s boundaries.

It should be noted that, as inconsistent clustersappear in pairs, the phase values between these res-idues are locally perturbed. For this reason smallregions including adjacent inconsistencies should bemasked out. When this process is carried out, theunwrapped phase is interpolated from the bound-aries of the masked regions in a least-squares senseto exclude inconsistent clusters and the local phaseperturbations that appear between them. For com-parison, two unwrapped-phase distributions were de-termined: ~i! when only the inconsistent clusterswere masked out and ~ii! when the regions betweenthese residues were also discarded. For the secondcase rectangular masks were generated by operatorinspection with the computer mouse to cover adjacentpairs of inconsistent clusters. We selected adjacentclusters by pairing each cluster to the nearest un-

Fig. 1. Wrapped-phase map determined by the Fourier-transform method for modulated DSPI carrier fringes.

Fig. 2. Weighting array used to unwrap the phase map shown inFig. 1.


paired one that had the opposite phase-differencesign. The rms deviation s and the relative meanerror e of the unwrapped phase inside the rectangularmasked regions were s 5 0.75 and e 5 22.08% for thefirst case and s 5 0.39 and e 5 15.95% for the secondone. It should be emphasized that, besides quanti-fying the phase differences that are due to the pres-ence of inconsistencies, both parameters include theerrors that are due to the speckle-decorrelation termintroduced in Eq. ~9!. These results clearly showthat the procedure of masking the regions betweencluster pairs is more accurate. However, when ahigh density of corrupt data is present in the phasemap this procedure becomes impractical.

To compare the unwrapped-phase maps obtainedfor different numbers of iterations with the continu-ous phase without inconsistencies f9m,n, we also eval-uated the rms deviation s and the relative mean errore for all consistent pixels of the whole image. Figure3 is a plot of the rms deviation versus the iterationnumber when the weighting array shown in Fig. 2was used. It is seen that the unwrapped phaseacross consistent pixels ~continuous curve! requiresonly three iterations to converge. For this iterationnumber the rms deviation s is equal to 1.69 3 1021

and the relative mean error e is 1.77%. For compar-ison, the curve obtained by the Picard iterativemethod described in Ref. 19 is also included in thesame plot. From Fig. 3 it is clearly seen that thePCG method has a much faster convergence thandoes the Picard approach.

Figure 4 shows the quality of the unwrapped-phasedistribution obtained after three iterations. It is im-portant to note that the PCG algorithm took approx-imately 8 s to evaluate this image.

To simulate a case of missing data we introduced avertical rectangular region of 100 3 10 pixels of un-correlated speckle noise into the reference wrapped-phase map. This region was located to produce themost unfavorable case, which occurred when the de-formation phase had a maximum and the phase var-ied nonmonotonically along the vertical direction.The weighting array used in this case is displayed in

Fig. 5. Figure 6 shows a plot of the rms deviationversus the iteration number, and one can see that theunwrapped phase outside the noise region ~continu-ous curve! still requires a few iterations to converge.For five iterations the rms deviation s is equal to1.9 3 1023 and the relative mean error e is 2.8 3 1022

%. In this case the value of M used in calculatingthe rms deviation was taken as the number of pixelsoutside the noisy region. It is important to note thatthese values are much lower than those previouslyobtained for Fig. 1. This is so because in this casethe whole inconsistent region, not just the individualclusters as in Fig. 1, is masked out. Then the locallyperturbed phase values between the inconsistentclusters is replaced by a continuous phase distribu-tion interpolated in the least-squares sense. For theinterpolated phase determined across the noisy re-gion @dashed curve ~c! in Fig. 6#, higher deviationsvalues were obtained ~s 5 7.8 3 1022 and e 5 1.6%also for five iterations!. In this case the value of M

Fig. 3. Plot of the rms phase deviation versus the iteration num-ber with the weighting array shown in Fig. 2: solid curve, PCGalgorithm; dashed curve, Picard iterative method. Fig. 4. Unwrapped-phase distribution of Fig. 1 after three itera-

tions with the PCG algorithm.

Fig. 5. Weighting array for a vertical rectangular region of miss-ing data.


used to calculate the rms deviation was taken as thenumber of pixels inside the noisy region.

For comparison, the curves obtained by the Picarditerative method presented in Ref. 19 are also in-cluded in the same figure. From the figure one cansee that the Picard approach needs at least 8 timesmore iterations than for the PCG method to reachsimilar values of the rms deviation and mean relativeerror.

The phase of a fringe pattern can be evaluated bymeans of the Fourier transform method developed byKreis,22 which does not need the introduction of carrierfringes. This phase-recovery method is useful whenpulsed DSPI is used to study fast transient phenom-ena, so the application of multiphase-stepping tech-niques is precluded. If such a wrapped-phase map isdetermined by means of bandpass filtering along the1u half-plane in the frequency domain, this filtermasks out negative spatial-frequency components inthe u direction while letting the negative and positivespatial-frequency components pass through in the or-thogonal direction. As a result, a sign change in thewrapped-phase map appears when the pattern is

formed by closed fringes. The case of a wrapped-phase map with a phase-reversed boundary is simu-lated by the image shown in Fig. 7. Note that asimilar phase shear can be generated when the objectto be tested contains a crack.

The weighting array used to unwrap both halves ofthe wrapped-phase map independently, shown inFig. 7, is composed of a zero-valued vertical line lo-cated along the phase shear. From the solid curve ofFig. 8 it is seen that the convergence of the PCGalgorithm is still quite fast. For 25 iterations therms deviation s is 1.2 3 1023 and the relative meanerror e is 8.4 3 1022%. For comparison, the dashedcurve shows that the Picard iterative method needsmore than 450 iterations to reach convergence. Thislast result clearly shows the great advantage of usingthe PCG algorithm. Figure 9 shows the quality ofthe unwrapped-phase distribution obtained after 25iterations.

A more complex case is given by the wrapped-phasemap shown in Fig. 10~a!. Superimposed phase mapssuch as this one can be obtained when one is studyingthe deformation of a bolt head and an underlyingmetal plate and the nut is tightened. This phase

Fig. 6. Phase rms deviation versus the iteration number with theweighting array shown in Fig. 5 obtained across ~c! consistentpixels and ~i! a region of missing data: solid curves, PCG algo-rithm; dashed curves, Picard iterative method.

Fig. 7. Wrapped-phase map determined by the Fourier transformmethod for a circular DSPI fringe pattern.

Fig. 8. rms phase deviation versus the iteration number for thewrapped-phase map shown in Fig. 7: solid curves, PCG algo-rithm; dashed curve, Picard iterative method.

Fig. 9. Unwrapped-phase distribution of Fig. 7 with the PCGalgorithm.


map presents simultaneously two isolated fringe pat-terns, regions with interrupted fringes, and objectedges. A 1-pixel-wide circumference with zero valuewas used to unwrap independently the two isolatedregions, as shown in Fig. 10~b!. The black region atthe right was used to weight the corresponding areain the wrapped-phase map. Note also that only theinconsistent clusters, which were automatically de-termined by the procedure described above, weremasked out. The unwrapped-phase map obtainedwith the PCG algorithm after 25 iterations is shownin Fig. 11~c!. It can be seen that both fringe regionsand all physical discontinuities are preserved, eventhough this success is due to the correct definition ofthe weighting array shown in Fig. 10~b!.

Finally, a last example is drawn from a real appli-cation. Figure 11~a! shows a typical DSPI fringepattern generated by the residual deformation in-duced in a steel plate when the plate is subjected topulsed ion implantation.23 The wrapped-phase mapobtained by means of a three-step phase-shiftingtechnique after speckle noise reduction is shown inFig. 11~b!. From this image it can be seen that thereare several regions where the phase is broken up and

noisy. Figure 11~c! shows the weighting array usedto remove the effect of the phase inconsistencies andthe background, where again the perturbed regionsbetween adjacent residues were not masked out.The phase map displayed in Fig. 11~d! shows that theweighting array permits a correct unwrapping resultto be produced over the entire steel plate after 25iterations.

5. Conclusions

The performance of a previously published phase-unwrapping algorithm that uses a least-squares min-imization technique solvable in an iterative mannerby a PCG method has been investigated by use ofcomputer-simulated DSPI phase maps. This ap-proach permits the generation of inconsistency-freephase fields that are used to calculate phase devia-tions as a function of the number of iterations. In-asmuch as phase inconsistencies are handled byexclusion of corrupted pixels from the unwrappingprocess through the assignment of zero-valuedweights, results demonstrate that the PCG algorithmis robust and suitable for unwrapping DSPI phasemaps. To assess the performance of the PCG algo-

Fig. 10. Unwrapping of a phase map presenting multiple isolated regions:~a! original phase map; ~b! weighting array; ~c! unwrapped-phase map.


rithm we have compared unwrapped-phase mapswith the phase of the DSPI fringes by evaluating twoparameters: the rms deviation s and the mean rel-ative error e.

First, the case of a wrapped-phase map generatedby modulated carrier DSPI fringes, which constitutesa typical example of those patterns whose phases areevaluated by Fourier transform methods, has beenanalyzed. Phase inconsistencies generated byspeckle decorrelation were introduced by the additionof an independent speckle pattern to one of the inter-ferograms used to generate the synthetic fringes.Results show that it is more accurate to mask outinconsistent cluster pairs and their neighborhoods.It has also been shown that the PCG algorithm is

faster than another unwrapping algorithm based ona Picard iterative approach.

The case of missing data has been simulated byintroduction of uncorrelated noise across a rectangu-lar region. It has been demonstrated that the num-ber of iterations needed to reach convergence is alsoquite small.

The main advantage of the PCG algorithm occurswhen wrapped-phase maps containing a signchange or a phase shear are unwrapped. It hasbeen shown that the PCG algorithm can correctlyunwrap phase data containing multiple isolated re-gions and that at least 18 times fewer iterations areneeded to reach convergence than with the Picarditerative method.

Fig. 11. Unwrapping of a real phase map: ~a! experimental DSPI fringes generated upon a steel plate subjected to pulsed ionimplantation; ~b! original phase map; ~c! weighting array; ~d! unwrapped-phase map.


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