Phase reconstruction and unwrappingfrom holographic interferogramsof partially absorbent phase objects

Ralf Vandenhouten and Reinhard Grebe

Amethod for automated phase reconstruction from holographic interferograms of nonideal phase objectsbased on a two-dimensional Fourier transform is described. In particular, the problem of phaseunwrapping is solved because earlier techniques are inappropriate for the phase unwrapping frominterferograms of partially absorbent objects. A noise-level-dependent criterion for the binary maskthat defines the unwrapping path for the flood algorithm is derived. The method shows high noiseimmunity, and the result is reliable provided that the true phase is free of discontinuities. The phasedistribution in the outmasked regions is estimated by a linear least-squares fit to the surroundingunwrapped pixels.Key words: Automated interferometry, phase reconstruction, phase unwrapping.

1. Introduction

In recent years automated analysis of interferogramshas become a subject of increasing interest becausefast, cheap computers are available for the largeamount of necessary calculations. For the variousinterferometric techniques a number of methods forunsupervised information retrieval from interfero-grams has been suggested,1 taking into considerationthe inherent problems of, for example, phase wrap-ping and noise. This paper deals with the possibili-ties of restoring the phase distribution from holo-graphic interferograms of transparent phase objects.The object-wave distribution in the recording image

plane can be written in the complex form

u1x, y2 5 A1x, y2exp3if1x, y24, 112

where A1x, y2 is the 1real2 amplitude and f1x, y2 is thephase of the wave according to the optical propertiesof the respective points in the object plane. Theinterference pattern I1x, y2 is the result of the superpo-sition of u and the reference wave h1x, y2 5

The authors are with the Department of Physiology, Laboratoryof Biomedical Systems Analysis, Rheinisch-Westfalische Tech-nische Hochschule Aachen, Pauwelsstrasse 30, Aachen D-52057,Germany.Received 3 March 1994; revised manuscript received 11 October

1994.0003-6935@95@081401-06$06.00@0.

r 1995 Optical Society of America.

R exp3i1kxx 1 ky y24:

I1x, y2 , R2 1 0u1x, y2 02

1 h1x, y2u1x, y2* 1 h1x, y2*u1x, y2. 122

The distance between two 1carrier2 fringes of theinterference pattern is then given by d 5 2p@1kx2 1 ky221@2. We presume that u1x, y2 1or at least therelevant information contained in u2 is band limited,i.e.,

F 3u41kx, ky2 < 0 for 6 1kx, ky26 . k0. 132

Here F denotes the two-dimensional 12-D2 Fouriertransform and 6 6 denotes the 2-D Euclidean norm.We further assume that the Fourier spectrum of thelast term in relation 122 1which we call the primaryinterference term2 is separated in the Fourier domainfrom the spectrum of all other terms in this equation.The latter condition can easily be achieved, e.g., bychoice of a fringe distance d , 2p@3k0 during therecording.Relation 122 describes idealized interference with-

out noise. Under realistic conditions, it has to besupplemented by an additive noise term n1x, y2, whichusually 1as high-frequency speckle or white noise2infects all 1separated2 parts of the Fourier spectrum.Our aim is a method for automated reconstruction ofthe phase distribution f1x, y2 from an interferogramrecorded under realistic conditions.

10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1401

2. Reconstruction of the Object Wave

The Fourier-transform analysis of the fringe patternwe use for the extraction of the complex object waveu1x, y2 is similar to the technique proposed by Takedaet al.2 and its 2-D extension presented by Bone et al.3The main difference in our approach is that we do notuse a simple cutoff filter for the separation of theprimary spectrum; wemultiply the spectrumwith theGaussian function exp12k2@2k022 instead 3Fig. 11a24.On the one hand, the Gaussian can serve as a cuttingfunction because it is approximately one inside the k0circle and fades rapidly to zero outside. On the otherhand, a sharp cutoff of signal components lyingbeyond the band limit k0, even if they are only minorand less than the noise level, leads to undesired echopeaks in the reconstructed signal, which is not thecase with the Gaussian filter because the multiplica-tion with a Gaussian in the Fourier domain is equiva-lent to a convolution with the conjugated Gaussian inthe original space domain. Thus although in idealcases in which the object signal has a sharp band limitthe sharp cutoff filter mayminimize the rms error, theGaussian filter yields better results in the generalcase with respect to the conservation of the objectform. This is of particular interest in applications inwhich structural properties are to be analyzed 1e.g., inbiomedical microscopy2. The reconstructed objectwave can be written as

u1x, y2 5 Re1x, y2 1 i Im1x, y2, 142

with Re1x, y2 and Im1x, y2 containing the real and theimaginary parts of the image, respectively. Theerror 0u1x, y2 2 utrue1x, y2 0 due to noise is usually 1e.g.,in the case of white noise2 independent from the

Fig. 1. 1a2 Processing steps for the reconstruction of the objectwave from the interferogram, 1b2 the subsequent calculation andunwrapping of the phase distribution.

1402 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995

location 1x, y2 in the image. Therefore we define thenoise level Du as the expectation value of the absolutedifference between the computed and the true objectwave:

Du [ E3 0u1x, y2 2 utrue1x, y2 0 4. 152

3. Phase Unwrapping

Because of the periodic behavior of the exponentialfunction along the imaginary axis, the true phasedistribution f1x, y2 cannot be determined directly fromthe complex object wave in Eq. 142. However, itsprincipal value fw1x, y2 can be calculated, e.g., as

fw1x, y2

5 5arcsin3Im!1Re2 1 Im221@24 for Re . 0

p 2 arcsin3Im!1Re2 1 Im221@24 otherwise. 162

fw is the true phase wrapped into the range from 2pto p; i.e., fw ; f mod 2p. We use Eq. 162 instead ofthe usual formula f 5 arctan1Im@Re2 in order to avoidthe denominator becoming 1close to2 zero and becausethe expression 1Re2 1 Im221@2 must be calculated any-way for later purposes 1see Subsection 3.B2. If thephase image is free of noise and the true phase doesnot contain steps $p between neighboring samplepoints, the true phase can trivially be restored by thedefinition of an arbitrary path through the image 1e.g.,row by row, in turn42 and the addition of a multiple of2p to the wrapped phase fw1x, y2 of each pixel suchthat the difference to the preceding pixel becomessmaller than p. However, even if undersampling ofthe data can be excluded, the result of the 2-Dunwrapping depends on the chosen path if phaseerrors due to noise occur; thus a more sophisticatedstrategy has to be applied.

3. A. Existing Algorithms

In recent years a number of methods have beensuggested to solve the problem of 2-D phase unwrap-ping.5–10 The cellular-automata algorithm of Ghigliaet al.5 and the regional-analysis technique of Gierloff 6are computationally expensive even for completelyconsistent phase fields, and both fail to localize theinconsistencies correctly in the general case.9 NeitherGhiglia et al.5 nor Gierloff 6 make use of explicitmodels for the nature and the localization of inconsis-tencies and can be classified as heuristic approaches.Goldstein et al.7 and Huntley8 showed that the path

dependence of the unwrapping result can be attrib-uted to local inconsistencies, called residues. For anarbitrary closed path 51xk, yk2, 1 # k # N6 consisting ofN points 1xk, yk2 in the phase field 1each point is eitherthe horizontal or the vertical neighbor of its predeces-sor2 the unwrapping error G can be defined as

G 5 ok51

N

53fw1xk, yk2 2 fw1xk21, yk2124@2p6, 172

where the braces mean that the argument is roundedto the nearest integer 1with the convention that 0.5

and 20.5 are both rounded to zero2. The smallestpossible closed path connects four corners of a squareof neighboring points. For such a path 1with thepoints enumerated in the clockwise sense2 the unwrap-ping error G is either21, 0, or 1.9 If it is not zero, it iscalled a residue. The unwrapping error for an arbi-trary path is the sum of the residues enclosed by thepath.7,8 Goldstein et al.7 and Huntley8 constructedbranch cuts 1i.e., connected groups of points that theunwrapping path is not permitted to cross and inwhich the sum of all residues is zero2 by connectingpairs of nearest-neighbor residueswith different signs,which leads to a consistent phase unwrapping in thesense that the resulting phase is free of ambiguities orpath dependencies. However, Bone9 recognized thata consistent phase unwrapping need not necessarilybe a correct unwrapping with respect to the truephase distribution of the recorded sample. For ex-ample, it is likely that a branch cut connecting tworesidues due to random noise separates two parts ofthe same physical object, which results in a partiallyincorrect phase unwrapping for this object althoughthe object itself is free of residues. Another shortcom-ing in this approach is that discontinuous jumps inthe true phase cannot be distinguished from inconsis-tencies. Bone9 focused on the latter problem andtried to solve it by introducing second differences ofthe locally unwrapped phase as a masking criterionfor the unwrapping path. None of the techniques ofGoldstein et al.,7 Huntley,8 or Bone,9 however, is ableto avoid incorrect reconstruction due to random noise.Their techniques operate only on the phase valuesfw1x, y2 and do not use a criterion for the reliability ofthese values, which is particularly important forinterferograms with a partially high degree of absorp-tion 1i.e., a low signal-to-noise ratio2. Vrooman andMaas10 used a binary mask as branch cuts based on amodulation intensity threshold that can in fact beconsidered a reliability criterion but is directly appli-cable only to phase-shifted speckle interferograms.Judge et al.11 suggested an algorithm for the analysisof double-exposure holographic interferograms usinga one-dimensional Fourier transform and a tiling ap-proach12 for the unwrapping step. A modulationcriterion similar to that of Vrooman and Maas10 wasemployed detecting the absence of the carrier fringeson the tile level 1not on the pixel level2.

3. B. Masking Criterion

In most holographic applications the phase distribu-tion can be regarded as continuous, so for the follow-ing considerations we assume that no true phasejumps occur in the object. This is not a real con-straint because the Bone9 algorithm should be able todetect true discontinuities in a separate processingstep.All phase inconsistencies can then be attributed to

random noise. Figure 2 illustrates the relation be-tween the object wave u1x, y2 and the noise level Dufrom relation 152 in the complex domain. If the

computed value for u has the distance r from the truevalue, then a certain error d of the phase is obtained3Eq. 1624. Obviously the greater the Du 1which is theexpectation value of r2 and the smaller the amplitudeA1x, y2 of the complex signal are, the greater becomethe phase error d and the absolute expectation valueDsind of sin d. More precisely, Dsind can be estimatedby

Dsind1x, y2 [ E1 0sin d 0 2 52Du

pA1x, y2. 182

Aderivation of this equation is given inAppendixA.To avoid inconsistencies in the phase unwrapping,

we must make Dsind significantly smaller than one.If we postulate Dsind , 1@4 1which corresponds to aphase error range 2d of roughly p@62, we obtain acondition for the reliability of a pixel 1x, y2; i.e., thephase fw1x, y2 is regarded as valid if

A1x, y2 .8Du

p< 3Du, 192

where

A1x, y2 5 3Re1x, y22 1 Im1x, y2241@2 1102

is the amplitude calculated from the restored objectwave. Inequality 192 is a thresholding condition thatcan be used to generate a binary mask of invalidpoints. In this mask each pixel that does not matchrelation 192 is set to one. All the other pixels are set tozero and should belong to reliable and consistentphase values. All residues will be located inside theregions of invalid points if the threshold has been setcorrectly. Thus if phase unwrapping takes placeexclusively on pixels not excluded by the binarymask,the result will be correct and free of ambiguities andinconsistencies.Alternatively, consistent unwrapping can be

achieved by construction of branch cuts between pairsof residues with different signs inside the sameconnected region of the binary mask, preferably withbranch-cut paths lying in a valley of the amplitude

Fig. 2. Phase error d can be estimated from the complex error Du

and the amplitude A1x, y2 1seeAppendixA2.

10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1403

distribution. This has the advantage that the num-ber of pixels for which the phase can be calculated isincreased, and the result will be the same for allnonmasked pixels, but large errors may occur forpixels lying in an outmasked region.

3. C. Unwrapping Algorithm

The unwrapping algorithm must be able to traceconnected regions of arbitrary shape, so a simplerow-by-row approach would be inappropriate. Weused an algorithm based on a standard nonrecursiveflood-fill algorithm, which is often used for coloringconnected pixels of a bit map uniformly. A similaralgorithm has already been used by Goldstein et al.,7which has also been adopted by Huntley8 and Bone.9Instead of recursivity, the algorithm uses a first-in,first-out queue for keeping track of all connectedpixels in a valid region. Validity is determined fromthe binary mask described above. A second binarymask is employed for ticking off the processed pixels.If the noise level is not too high, all valid pixels areusually contained in one region; however, they may bedivided by the invalid pixels into a few separatepieces. In this case, for each piece a new run of theflood fill must be performed consecutively.After the unwrapping has finished, all pixels lying

in outmasked regions are left unprocessed. It doesnot seem adequate to use the wrapped phase values ofthese pixels for restoring the correct phase becausetheir phase uncertainty is considerably high owing torelation 182. We use a 2-D linear least-squares 1LLS2fit to the surrounding valid pixels instead. For eachoutmasked pixel 1x, y2 a rectangular neighborhoodW11x, y2 is considered 1usually the size of W is from5 3 5 to 9 3 92. W1x, y2 contains a subset W*1x, y2 ofall valid points in the neighborhood. For the LLS thereal parameters a, b, and c are chosen to minimize theexpression

G 5 o1k,l2[W*

3a1k 2 x2 1 b1l 2 y2 1 c 2 f1k, l242, 1112

where f1k, l2 is the unwrapped phase of the respectivevalid pixel. The value of c is then the estimate of thecorrect phase at the location 1x, y2. A scheme of thewhole unwrapping process is shown in Fig. 11b2.

4. Discussion

Provided the respective threshold for the binary maskof invalid pixels is chosen correctly, the presentedmethod yields consistent and correct phase restora-tion for a wide range of noise. This holds true even ifthe object is partially highly absorbent. Figure 3shows the interferogram of a microvessel and thedifferent steps in the phase-restoration process. Theamplitude image shows regions of a stronglyreduced transparency, in particular, near the bound-aries of the vessel. These regions are marked in thebinary mask for defining a consistent unwrappingpath. Figure 4 shows the final result of the phasereconstruction.

1404 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995

The main problem with this technique is to find theadequate threshold that defines the binary mask.If the noise level of the interferogram is known, it canbe well estimated by relation 192, but usually the noiselevel is unknown and cannot be derived directly fromthe image. The user then needs to know the ad-equate threshold from experience or has to do a roughestimation. If the threshold is too low, some inconsis-tencies may not be covered by the mask, and theunwrapping will be defective. If it is too high, theunwrapping remains consistent, but some of thepixels are masked out although they have reliablephase values. Thus the LLS approximation is usedto restore the phase of these pixels, although aconsistent and correct unwrapping would have beenpossible.A reasonable method for finding the correct thresh-

old is the following: Take the smallest thresholdvalue for which all residues are contained in the maskof invalid pixels and for which the sum of residues in

Fig. 3. 1a2 Holographical interferogram of a microvessel 1venule2,1b2 the wrapped phase, 1c2 the amplitude distribution of the objectwave, and 1d2 the binary mask obtained by thresholding of theamplitude image.

Fig. 4. Reconstructed phase distribution of the microvessel.

each connected region of the mask is zero. Thiscriterion yields consistent unwrapping and can beeasily implemented for automatic detection of thethreshold.Our current technique does not work properly if

discontinuities occur in the true phase distributionand if the transparency of the object in the vicinity ofthe jump is good. For a large number of applicationsthis is not important; when necessary, the problemcan be overcome by supplementing the binary maskwith all pixels meeting the second differences crite-rion of Bone.9It should be mentioned that the mask of invalid

pixels is useful not only for the unwrapping processitself but also exposes the amount and the location ofuncertainty in the image, thus giving the user impor-tant information for assessing the interferogram andthe quality of the analysis. Yet the computationalexpense for the presented method is not higher thanit is for any of the known algorithms. For example,it is roughly twice as fast as the Judge et al.12approach, which was one of the fastest reliable algo-rithms.

5. Conclusions

A method for automated phase restoration fromholographic interferograms using 2-D Fourier trans-forms has been presented. For the phase-unwrap-ping problem a reliability condition has been derivedthat depends on the noise level and the complexamplitude of the reconstructed object wave. On thebasis of this condition a binary mask can be definedthat is used to exclude uncertain pixels from theunwrapping path. Provided that no discontinuitiesin the true phase distribution occur, the unwrappingis consistent and approximately correct. Invalid pix-els can be restored by a LLS fit to the unwrappedneighborhood.Reliable results can be obtained even for high noise

levels and interferograms with partially high absorp-tion. The algorithm is fast and may also be suitablefor time-critical and on-line applications.

Appendix A

Let f 1r, a2 be the probability that the measured com-plex object wave u1x, y2 has the distance r under theangle a from the true value 1see Fig. 22. Because f is aprobability distribution, we have

e0

`

dr 1r2 e0

2p

da f 1r, a2 5 1. 1A12

In case of random high-frequency noise, f should besymmetric with respect to a, i.e., f 1r, a2 5 f 1r2. Thenoise level as the expectation value of r 3see relation

1524 is then given by

Du 5 E1r2 5 e0

`

dr r2 e0

2p

da f 1r2

5 2p e0

`

dr r2 f 1r2. 1A22

For given values of r, a, and amplitude A, the phaseerror d can be calculated with simple trigonometricconsiderations:

sin d 5r

1r2 1 A2 2 2Ar cos a21@2sin a. 1A32

The absolute expectation value Dsind of sin d is thengiven by

Dsind 5 E1 0sin d 0 2

5 e0

`

dr rf 1r2 2 e0

p

dar sin a

1r2 1 A2 2 2Ar cos a21@2

5 2 e0

`

dr rf 1r2 e21

1

dx1

11 1 1A2@r22 2 21A@r2x21/2

52

A e0

`

dr r2 f 1r2 e112A@r22

111A@r22

dy1

2Œy

52

A e0

`

dr r2 f 1r211 1A

r2 01 2

A

r 02

54

A e0

A

dr r2 f 1r2 1 4 eA

`

dr rf 1r2, 1A42

where the substitutions x 5 cos a and y 5 1 1 A2@r2 2

2Ax@r are used. The second term in Eq. 1A42 is thetail of the probability integral and can therefore beneglected, whereas the upper integration limit in thefirst term can be replaced approximately with `.So we finally obtain

Dsind1x, y2 <4

A1x, y2 e0`

dr r2 f 1r2 52Du

pA1x, y2. 1A52

We thank Dorothea Seebode for supplying theinterferogram and Louis Philippe for the integrationof the algorithm into the image-processing environ-ment Khoros.

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