To process a fringe-pattern image, one usually has tosolve a sequence of problems that are ill posed in amathematical sense, i.e., problems for which it is notpossible to find a unique solution by use of only theinformation contained in the observed images butthat require the introduction of some form of priorknowledge in the solution algorithms. Thus filter-ing an image can be considered an ill-posed problembecause near its borders the filter output is affectedby unknown information; recovering the phase fromsingle fringe-pattern images is ill posed because ofthe inherent sign ambiguity; phase unwrapping ofnoisy images is ill posed because of noise-generatedinconsistencies, etc.
A very powerful approach for the solution of thiskind of problem can be derived from Bayesian esti-mation theory with prior Markov random-field mod-els.1,2 In this approach, one defines the desiredsolution as the minimizer of a cost ~or energy! func-tion that has two types of terms: data terms ~de-rived from the likelihood function!, which constrain
J. L. Marroquin [email protected]!, M. Rivera, and S.Botello are with the Centro de Investigacion en Matematicas,
pdo. Postal 402, 36000 Guanajuato, Guanajuato, Mexico. R.odriquez-Vera and M. Servin are with the Centro de Investiga-iones en Optica, Apdo. Postal 1-948, 37250 Leon, Guanajuato,exico.Received 11 August 1998; revised manuscript received 9 Novem-
788 APPLIED OPTICS y Vol. 38, No. 5 y 10 February 1999
the solution to be compatible with the available ob-servations, and regularization terms ~derived fromthe prior model!, which constrain the solution to haveertain properties, e.g., to be smooth, to have a pass-and spectrum, to be piecewise constant, etc. Theseethods will be computationally attractive only if
hese prior constraints can be expressed as interac-ions of neighboring pixels ~in formal terms, one re-uires the prior distribution to be Markovian!. Theurpose of this paper is to show that it is possible toonstruct these local-interaction models for almostvery step in the processing of fringe-pattern images.e now present the cost functions associated with
he main fringe-pattern processing operations.
A. Robust Smoothing
Smoothing a fringe pattern is important because thesmoothed image can be considered as an estimate ofthe low-pass illumination component of the pattern,which can then be substracted before further process-ing takes place. The smoothed pattern can be ob-tained by use of a linear, shift-invariant filter ~e.g., aconvolution with a Gaussian kernel!. These types of
lters, however, are limited in that they produce non-eliable results near the borders of the image. As anlternative, one can obtain the smoothed image f ashe minimizer of the cost function2:
U~ f ! 5 (r[L
@ f ~r! 2 g~r!#2 1 l (^r,s&[L
@ f ~r! 2 f ~s!#2. (1)
Here the first sum ~the data term! is taken over allhe sites ~pixels! r 5 ~x, y! of the desired region L of
itff
t
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lt
tc
b
ttwe
@n
the observed image g; the second sum ~the regular-ization term! is taken over all the nearest-neighborpairs of sites ~denoted as ^r, s&!. Note that everypoint r 5 ~x, y! in the interior of L ~i.e., away from itsborders! has four such nearest neighbors, namely, thepoints ~x 2 1, y!, ~x 1 1, y!, ~x, y 2 1!, and ~x, y 1 1!.Boundary points may have only three or two neigh-bors in L. The regularization parameter l controlsthe amount of smoothing to be performed. Notethat, because the sums are taken for only those sitesthat belong to L, one gets a solution that is free ofartifacts near the borders of the image.
B. Robust Passband Quadrature Filters
Passband quadrature filters are useful for the phasedemodulation of carrier fringe patterns when themagnitude of the modulating phase is relativelysmall ~see Ref. 3 for details!. In this case, the outputmage f is complex valued, and the regularizationerm should constrain it so that locally it takes theorm of the analytic signal, f ~r! 5 M exp$i@v0 z r 1~r!#%, where v0 5 ~u0, v0! is the carrier’s spatial
frequency and f is the phase to be recovered. Inother words, one tries to construct a complex image,f ~r! 5 w~r! 1 ic~r!, that satisfies the following con-straints:
~i! The real part w~r! should behave locally like anideal fringe pattern with a dominant frequency v0;that is, if close to a given image point r 5 ~x, y!, onewrites w~r! in the form
w~r! 5 A~r!cos@v0 z r 1 a~r!#.
Then, for a neighboring point s @for example, for s 5~x 2 1, y! or s 5 ~x, y 2 1!#, one should have
w~s! < A~r!cos@v0 z s 1 a~r!#,
where A~r! is a smooth function and ua~r!u is smallwith respect to v0 z r.
~ii! The imaginary part c~r! should be close to thecorresponding quadrature image, that is, it should beapproximately equal to w~r! with a phase shift of 90°:
c~r! < A~r!sin@v0 z r 1 a~r!#.
~iii! The term w~r! should be approximately propor-ional to the observed fringe pattern.
hese constraints may be implemented by a costunction of the form
U~ f ! 5 (r[L
u f ~r! 2 2g~r!u2 1 l (^r,s&[L
Uf ~r!expF i2
v0 z ~s
2 r!G 2 f ~s!expF i2
v0 z ~r 2 s!GU2
, (2)
where g denotes the observed fringe pattern with theow-pass illumination component removed. The fac-or of 2 is introduced into the data term so that, if g~r!
5 cos~v0 z r!, the output has unit magnitude. Theparameter l controls the bandwidth of the filter, withnarrow filters corresponding to large values of l. Af-er the optimal f is computed, the modulating phasean be recovered by use of
f~r! 5 arctanFImf ~r!
Ref ~r!G 2 v0 z r. (3)
C. Adaptive Quadrature Filters and Phase Demodulationfrom Single Images
Phase demodulation from single fringe-pattern im-ages is important, for example, in the study of fasttransient phenomena. In these cases, when themodulating phase of the carrier fringe patterns islarge, and, more importantly, in the case of carrier-free patterns containing closed fringes, it is not pos-sible to obtain good results from single images withordinary quadrature filters because the bandwidthnecessary to accomodate the whole pattern is toowide to be effective for noise suppression. In thesecases, one can still use the cost function equation ~2!ut with a spatially varying tuning frequency v~r!,
which is also constrained to be smooth. The result-ing cost function is
U~ f, v! 5 (r[L
u f ~r! 2 2g~r!u2
1 l (^r,s&[L
Uf ~r!expF i2
v~r! z ~s 2 r!G2 f ~s!expF i
2v~s! z ~r 2 s!GU2
1 m (r,s&
uv~r! 2 v~s!u2, (4)
where l and m are positive parameters. Note thatthe function in Eq. ~4! must be minimized with re-spect to both f and v ~see Ref. 4 for details!. Anotherprocedure that is computationally more efficient andstable consists of first estimating the field v in adecoupled fashion and then minimizing Eq. ~4! withrespect to only f, keeping v fixed ~note that the lastterm is no longer necessary!. To estimate v, onesuccessively estimates the local orientation of thefringes, the frequency sign, and the frequency mag-nitude.5 The complete procedure is as follows:
1. Estimate the local orientation u~r! of the fringes:his can be done by smoothing of the orientation ofhe level curves ~isophotes! of the observed image,hich are orthogonal to the brightness gradient atach point.2. The local orientation u~r!, which is in the interval
0, p!, can correspond to two different directions,amely, u~r! and u~r! 1 p. Thus the second step is to
find a smooth direction field u~r! that is compatiblewith the orientation field found in step 1.
3. Estimate the local frequency magnitude r~r!~proportional to the average number of fringes perunit length! at each location r.
10 February 1999 y Vol. 38, No. 5 y APPLIED OPTICS 789
ˆ
a
nu
sda
Tcq
a
M
7
4. After u~r! and r~r! are known, compute the localfrequency as
v~r! 5 @r~r!cos u~r!, r~r!sin u~r!#.
Each of the above estimation problems can also besolved by use of a regularization approach. To esti-mate the orientation field u, one can use the robustsmoother described by Eq. ~1! to find the sine and thecosine of 2u @denoted as c2~r! and s2~r!, respectively# in
decoupled way; the observations are cos@2ug~r!# andsin@2ug~r!#, respectively, where ug~r! is the direction
ormal to the gradient of the observed image, i.e.,g~r! 5 arctan~2gxygy!, where gx and gy represent the
partial derivatives of g taken with respect to the spa-tial variables. Note that one must double the anglein the estimation process because the orientations uand u 1 p are equivalent. The cost function for thecosine is
U~ f ! 5 (r[L
$c2~r! 2 cos@2ug~r!#%2 1 l (^r,s&[L
@c2~r! 2 c2~s!#2,
with a similar function for the sine.To find the frequency sign, one must first find a
smooth direction field associated with the orientationfield u. This is especially important when thefringes form closed curves because, in this case, asone moves along a closed fringe the orientation anglewill exhibit abrupt jumps ~of size p! near those pointswhere its value is close to either 0 or p. If this angleis used directly to compute the local tuning frequencyof the adaptive quadrature filter, this frequency—hence the recovered phase—will also exhibit spuriousjumps ~sign reversals!. These artifacts are avoidedif the direction field is constrained to be smooth in thesense that two neighboring sites with similar orien-tations should also have similar directions.
If we assume that an orientation field u has alreadybeen computed, then to estimate the direction field u,one needs only to estimate a binary field d, whereeach d~r! can take only the values $1, 0%, and to set u~r!5 u~r! 1 d~r!p. To find the field d, one first definesa probability field p so that pr is the probability thatd~r! 5 1. It is necessary to make an arbitrary choicefor the overall sign of the phase—which for a singlefringe-pattern image is inherently ambiguous—to geta unique solution. This is done by means of fixingthe value of d at an arbitrary site r0 through therequirement that pr0
' 1.One would like u to be smooth; however, the corre-
ponding constraint on d is more complicated: If theifference in orientation at two neighboring sites rnd s is close to 0, the corresponding values of pr and
ps should be similar; if the orientation difference isclose to p, these values should be as different aspossible; and if it is close to py2, they should notinteract. By setting a value of crs 5 cos@u~r! 2 u~s!#,
90 APPLIED OPTICS y Vol. 38, No. 5 y 10 February 1999
we determine that the cost function for this problemcan be written as
U~p! 5 (^r,s&[L
@~1 1 crs!~pr 2 ps!2 1 ~1 2 crs!
3 ~pr 1 ps 2 1!2# 1 m~pr02 1!2, (5)
where m is a large positive parameter ~e.g., m 5 1000!.he minimization of this function with respect to pan be interpreted as the propagation of the fre-uency sign at location r0 ~which is chosen arbitrari-
ly! to the rest of the image in such a way that therere no spurious sign jumps. After the field p that
minimizes Eq. ~5! is found, the direction field is esti-mated with
u~r! 5 u~r!, if pr , 0.5
5 u~r! 1 p, if pr $ 0.5. (6)
After the direction field is obtained, one must esti-mate the frequency magnitude. For this estimate,one discretizes the interval in which r can take values@say, set qk 5 ~k 2 1!py2M, for k 5 1, . . . , M# andfinds M fields $ fk, k 5 1, . . . , M% that minimize Eq. ~4!
times by using each time a fixed frequency field vk,which is computed as
vk~r! 5 @qk cos u~r!, qk sin u~r!#.
Fig. 1. ~a! ESPI image of a metallic plate subjected to thermaldeformation, ~b! demodulated ~wrapped! phase, ~c! unwrappedphase.
tI
wp
Finally, one defines v~r! 5 vm~r!~r!, where m~r! is suchhat u fm~r!~r!u . u fk~r!u for all k Þ m~r!, and finds v~r!
as the minimizer of Eq. ~1! by using v as observations.The whole procedure is described in detail in Ref. 5.It is important to note that with this scheme it ispossible to find the modulating phase correctly, evenfor single interferograms that contain closed fringes,without the spurious sign jumps that are obtainedwith conventional methods.
An illustration of the power of this technique ispresented in Fig. 1. Figure 1~a! shows an electronicspeckle-pattern interference ~ESPI! pattern that cor-responds to a metal plate subjected to thermal defor-mation; Fig. 1~b! shows the demodulated wrappedphase obtained by successive estimation of the orien-tation, direction, frequency magnitude, and complex-filtered fields, as described above. Note the highquality of the reconstruction and the abscense of edgeeffects and spurious sign jumps, in spite of the factthat only a single image is available and that it con-tains closed and high-frequency fringes.
D. Robust Demodulation of Multiple Phase-SteppingImages
The cost function equation ~4! can easily be general-ized to the case of multiple interferograms g1, . . . , gNof the same object obtained with the phase shiftsa1, . . . , aN. It is necessary only to modify the dataterm, which now becomes
(r[L
(k51
N
u f ~r! exp~iak! 2 2gk~r!u2.
Fig. 2. ~a!–~c! Synthetic multiple phase-stepping images generatization method. ~f ! Phase recovered with the standard technique
f precise values of the phase shifts a1, . . . , aN arenot known, it is possible for one to find them bytreating them as unknowns and by minimizing thefunction U, also with respect to these phase shifts.To do this, one should update their values after everyfew iterations of the numerical method that performsthe minimization with respect to the f and the vvariables. To effect this update, one solves for akfrom the equations obtained by setting ]Uy]ak 5 0.The update equations are
ak 5 arctan52(x[L
@Imf ~x!#gk~x!
(x[L
@Ref ~x!#gk~x! 6 .
In this way, when the system converges the truephase shifts are also recovered. The complete pro-cedure is explained in detail in Ref. 6.
To illustrate the power of this method, we presentan experiment, illustrated in Fig. 2, that comparesthe performance of our method with a classical phase-stepping demodulation technique6 for three highlydegraded synthetic images of 128 3 128 pixels gen-erated with the model
gk~x! 5 a~x!cos@f~x! 1 n~x! 1 ak#, k 5 1, 2, 3,
with a1 5 0, a2 5 20.66p, and a3 5 0.66p. Themodulating phase f is obtained by the sum of fourGaussian functions with variances equal to 312, cen-ters at ~25, 25!, ~90, 90!, ~25, 100!, and ~100, 25!, and
eights equal to 20, 20, 220, and 220, respectively,lus uniform phase noise n in the range @25.0, 5.0#.
th ~d! the wrapped phase. ~e! Phase recovered with the regular-ering plus least squares!.
ed wi~filt
10 February 1999 y Vol. 38, No. 5 y APPLIED OPTICS 791
cttpfh
u
w
7
The term a~x! is another Gaussian with a center at~64, 64!, a variance equal to 10000.0, and a weight of1.0.
E. Phase Unwrapping
The phase map obtained by the methods described inSubsection 1.D is, as usual, wrapped into the interval@2p, p#. There are several possible ways of comput-ing the unwrapped phase. In the first method, oneassumes that the observations consist of wrappedfirst-order differences:
gx~x, y! 5 g~x, y! 2 g~x 2 1, y! 1 2pkx,
gy~x, y! 5 g~x, y! 2 g~x, y 2 1! 1 2pky,
where g is the wrapped phase and kx and ky areintegers such that ugx~x, y!u , p and ugy~x, y!u , p,respectively. It is also assumed that the magnitudeof the true unwrapped phase gradient is everywheresmaller than p.
In this method, one computes two fields, fx and fy,that represent the gradient of the unwrapped phase;these fields are required to match the observedwrapped differences, so the data term is
D~ fx, fy! 5 (r[L
$@ fx~r! 2 gx~r!#2 1 @ fy~r! 2 gy~r!#2%. (7)
The regularization term constrains the gradient to beconsistent:
R~ fx, fy! 5 (r[L
@Dx fy~r! 2 Dy fx~r!#2, (8)
where the operators Dx and Dy take the first differ-ences in the x and the y directions, respectively. Thecomplete cost function is, as usual,
U~ fx, fy! 5 D~ fx, fy! 1 lR~ fx, fy!. (9)
Because in this case one wants this consistency con-straint to be strictly enforced, one should set theregularization parameter to a very high value (e.g.,l 5 1000). After the optimal gradient field ~ fx, fy! isomputed, the unwrapped phase can easily be ob-ained by ~path-independent! integration. Note thathis method will not filter out the noise unless itroduces inconsistencies; for this reason it is pre-erred in those cases in which the wrapped phase isighly reliable ~see Ref. 7 for details!. The perfor-
mance of this method on a clean wrapped phase mapis illustrated in Fig. 1~c!.
In some cases, the phase map may have large gra-dients ~e.g., because of noise!, so the basic assumptionnecessary for the above method to work ~i.e., that thegradient magnitude of the unwrapped phase issmaller than p! does not hold. In these cases, onecan still use a regularization formulation that is notbased on this assumption. The idea of this secondmethod is to find a correction field z such that the
nwrapped phase is recovered as f ~r! 5 g~r! 1 2pz~r!.This unwrapped phase is constrained to be smooth,and the field z is required to be close to an integer-
92 APPLIED OPTICS y Vol. 38, No. 5 y 10 February 1999
valued field plus an additive constant; the cost func-tion is thus
U~z! 5 (^r,s&[L
$g~r! 1 2pz~r! 2 @g~s! 1 2pz~s!#%2
1 l$z~r! 2 z~s! 2 Rnd@z~r! 2 z~s!#%2, (10)
here the function Rnd~x! returns the integer that isclosest to x. Note that l controls the amount of noiseabsorbed by the correction field z and thus acts as asmoothing parameter. If l is set to a large value~e.g., l 5 1000!, this phase-unwrapping method willpreserve the integrity of the data ~see Ref. 8 for de-tails!.
The performance of these methods on noisy phasemaps is illustrated in Fig. 3. Figure 3~a! shows asynthetic phase map, given by
f~x, y! 5 a 2 0.0006@~128 2 x!2 1 ~128 2 y!2# 1 n,
Fig. 3. ~a! Unwrapped and ~b! wrapped nosy quadratic phasemaps. ~c! Wrapped phase of ~a! unwrapped with the method dis-cussed in Subsection 1.E that corresponds to the cost functiongiven by Eq. ~9!. ~d! Phase map of ~c! rewrapped to show thereduced dynamic range. ~e! Phase of ~b! unwrapped with themethod discussed in Subsection 1.E that corresponds to the costfunction given by Eq. ~10! and ~f ! rewrapped to show the recon-struction accuracy.
wxa3wsodsnrcp~3s
Euatuptmq
mmd
si
where n is a zero-mean Gaussian random variableith unit variance, a 5 4 for x , 128, and a 5 6 for$ 128. We assume that the available observationsre the corresponding wrapped phase shown in Fig.~b!. Figures 3~c! and 3~d! show the results obtainedith the first method, whereas Figs. 3~e! and 3~f !
how the corresponding results obtained with the sec-nd method. Note how the dynamic range and theiscontinuity at the center of the pattern are pre-erved in this case, in spite of the large amounts ofoise. As one can see, the second method is moreobust to noise than the first; however, because theost function is not quadratic in this case, it is com-utationally more expensive than the first methodfor a 256 3 256 pixel image, the first method takes0 s on a 200-MHz Pentium machine, whereas theecond takes approximately 4 min!.
F. Segmentation ~Mask Finding!
In many cases the fringes of an interferogram aremeaningful in only some portion of a rectangular im-age ~e.g., in a speckle interferogram of a mechanicalpart!. Because the methods we have presented herecan operate on irregularly shaped regions withoutproducing distortions at the edges, it is desirable tofind the masks that define these regions in an auto-matic way. This could be difficult with conventional~e.g., threshold! methods, particularly in the case of
SPI patterns, because the thresholded images aresually too noisy to be useful. This problem can bevoided if the image is filtered and quantized beforehresholding; these operations can be carried out byse of the regularization approach. To apply it, oneroceeds as follows: The dynamic range of the pat-ern ~i.e., the interval between the maximum and theinimum gray levels! is discretized into M values
1, . . . , qM, and the field pr~k!, r [ L, k 5 1, . . . , M,that represents the probability that the desired fieldequals qk at site r, given the observations, is com-puted as the minimizer of
U~p! 5 (r[L
upr 2 pru2 1 l (^r, s&
upr 2 psu2, (11)
where upru2 5 ¥k51
M @pr~k!#2. The likelihood p is com-puted as
pr~k! 51Z
exp$2b@g~r! 2 qk#2%,
where b is a positive parameter. After the likelihoodp is determined, one computes a quantized field f bysetting f ~r! 5 qk*~r!, with k*~r! such that pr@k*~r!# .pr~k! for all k Þ k*~r!. The desired segmentation cannow easily be performed by a threshold operation onthe quantized field f. An illustration of this proce-dure is presented in Fig. 4. Figure 4~a! shows animage of a mechanical part obtained as the superpo-sition of three multiple phase-stepped ESPI images~with a relative phase shift of 120°!. Figure 4~b!shows the result of the segmentation process de-scribed above; this image can be used as a mask forsubsequent filtering and demodulation steps. The
correct values of q for the gray levels were obtainedby the adaptive quantization procedure described inRef. 9.
2. Discussion
We have presented a number of problems related tothe processing of fringe-pattern images that can bevery effectively solved by means of defining the solu-tion as the minimizer of a cost function with data andregularization terms. These cost functions are sum-marized in Table 1.
The application of the regularization techniquespresented here requires the use of efficient algo-rithms for the minimization of the correspondingcost functions. In most cases these functions arequadratic, which means that their minimization isequivalent to the solution of a system of linear equa-tions that are obtained by means of setting thepartial derivatives of the cost function with respectto the field variables equal to zero. If the region Lis a rectangle, it is possible to find direct solutionsby use of the cosine transform10,11; in other cases~e.g., for irregularly shaped regions!, iterative
ethods such as the conjugate-gradient12 methodust be used. If the cost functions are not qua-
ratic @e.g., in the case of demodulation of multiple
Fig. 4. ~a! Image of a mechanical part obtained by the superpo-ition of three ESPI patterns. ~b! Mask obtained by the regular-zed segmentation procedure described in the text.
10 February 1999 y Vol. 38, No. 5 y APPLIED OPTICS 793
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Table 1. Data and Regularization Terms for the Cost Functions Used for Different Operations in Fringe-Pattern Processing
7
phase-stepping images or in the phase-unwrappingcost function given by Eq. ~10!# methods such as the
ewtonian descent13 give good results with reason-able processing times.
The authors were supported in part by grants fromthe Consejo Nacional de Ciencia y Tecnologia, Mexico.
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Data Term Regul
@f~r! 2 g~r!#2 l@uf~r! 2 2g~r!u2 luf~r!exp$iy2@v0 z ~s 2 r!uf~r! 2 2g~r!u2 luf~r!exp$iy2@v~r! z ~s 2
1 muv~r! 2 v~s!u2
(kuf~r!exp~iak!22g~r!u2 luf~r!exp$iy2@v~r! z ~s 21 muv~r! 2 v~s!u2
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tion Term Operation
f~s!#2 Robust smoothingf~s!exp$iy2@v0 z ~r 2 s!#%u2 Robust quadrature filter
f~s!exp$iy2@v~s! z ~r 2 s!#%u2 Adaptive quadrature filter
f~s!exp$iy2@v~s! z ~r 2 s!#%u2 Adaptive quadrature filter forphase stepping
