recursive approach to the moment-based phase unwrapping method

Recursive approach to the moment-basedphase unwrapping method

Jason A. Langley,1,2 Robert G. Brice,2 and Qun Zhao1,2,*1Bioimaging Research Center, University of Georgia, Athens, Georgia 30602, USA

2Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA

*Corresponding author: [email protected]

Received 12 January 2010; revised 11 April 2010; accepted 12 April 2010;posted 6 May 2010 (Doc. ID 121995); published 28 May 2010

Themoment-basedphase unwrapping algorithmapproximates the phasemapas a product ofGegenbauerpolynomials, but the weight function for the Gegenbauer polynomials generates artificial singularitiesalong the edge of the phase map. A method is presented to remove the singularities inherent to themoment-based phase unwrapping algorithm by approximating the phase map as a product of two one-dimensional Legendre polynomials and applying a recursive property of derivatives of Legendre polyno-mials. The proposed phase unwrapping algorithm is tested on simulated and experimental data sets. Theresults are then compared to those of PRELUDE 2D, a widely used phase unwrapping algorithm, and aChebyshev-polynomial-basedphaseunwrappingalgorithm. Itwas found that theproposedphaseunwrap-ping algorithm provides results that are comparable to those obtained by using PRELUDE 2D and theChebyshev phase unwrapping algorithm. © 2010 Optical Society of AmericaOCIS codes: 100.3020, 100.3190, 100.5070.

1. Introduction

Phase unwrapping is a vital image processing proce-dure in a variety of fields, from medical imaging [1]and optics [2] to geoscience [3]. Phase maps have arange of ½−π; πÞ and the interval ½−π; πÞ is known asthe principal interval. When values in the true phasemap, denoted φðrÞ, fall outside the principal interval,integer multiples of 2π are added to the values in thetrue phasemap.This process is knownas phasewrap-ping, and phasemaps that are wrapped into the prin-cipal interval are known aswrapped phasemaps. Themathematical relation between the true phase map,the wrapped phase map, denoted φwðrÞ, and the inte-ger required to bring the true phase into the principalinterval, denotednðrÞ, isφðrÞ ¼ φwðrÞ þ 2πnðrÞ. Phaseunwrapping is the process of calculating the truephase map from the wrapped phase map.

Manyphaseunwrappingalgorithmshavebeenpro-posed because phase unwrapping is used in many

fields of image processing. Dimensionality is onemethod of classification for phase unwrapping algo-rithms, e.g., recently published two-dimensionalphase unwrapping algorithms [4–7] and three-dimensional phase unwrapping algorithms [8–11].Each phase unwrapping method can be also sortedby its approach to the phase unwrapping problem.The different approaches include Fourier spacemeth-ods [7,12,13],LP-normminimizingphaseunwrappingalgorithms [5,14], region growing algorithms [6,15],Bayesian approaches [4,16], and path followingalgorithms [8,17]. An excellent resource on phase un-wrapping can be found in [18].

The moment-based phase unwrapping algorithms[19,20] fall within the LP-norm minimization cate-gory. The moment-based phase unwrapping algo-rithms model the phase map as products oforthogonal polynomials and utilize the orthogonalityproperty of the polynomials to calculate the coeffi-cients associated with the expansion. The moment-based phase unwrapping algorithms presented in[19,20] were implemented using a type of orthogonalpolynomial known as Gegenbauer polynomials.

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3096 APPLIED OPTICS / Vol. 49, No. 16 / 1 June 2010

Gegenbauer polynomials are a class of polynomialsthat are orthogonal with weight function wðλÞðxÞ onthe domain ½−1; 1�, where λ denotes the type of Gegen-bauerpolynomial considered, and it can takevalues inthe interval ð−1=2;∞Þ. The phase unwrapping algo-rithms based on expansion in terms of Gegenbauerpolynomials donot consider theentirephasemapdur-ing the unwrapping process [19,20]. This is because asingularityexists in theweight functionat theedgesofthe weight functions domain when λ < 1=2. In thiscase, the edges of the domain are neglected duringthe integration step [20]. When λ > 1=2, the weightfunction is zero at the edges of the domain of theweight function and the phase along the edge of thephasemap is not counted during the integration step.

In this work, we extend the moment-based methodto consider the entire phase map by limiting thechoice of basis function in the moment-based phaseunwrapping algorithm to Legendre polynomials.Legendre polynomials are ideal for this algorithm be-cause the weight function for Legendre polynomialstakes a value of 1 over the weight functions in theentire domain, and the derivative of Legendre polyno-mials can be expressed recursively with lower ordersof Legendre polynomials. The modified moment-based phase unwrapping algorithm is tested on simu-lated and experimental data taken from a magneticresonance imaging (MRI) scanner and comparedwiththe Chebyshev phase unwrapping algorithm [19] andPRELUDE 2D [21,22], a widely used phase unwrap-ping algorithm in the neuroimaging community.

2. Proposed Phase Unwrapping Algorithm

In the moment-based phase unwrapping algorithm[19,20], the true phase map is modeled as a two-or three-dimensional product of a one-dimensionalorthogonal polynomial with a remainder term thatincorporates noise and rounding error. Themathema-tical expression for the model with Legendre polyno-mials is φðx; yÞ ¼ QNðx; yÞ þ rðx; yÞ where

QN ¼XNn¼0

Xnm¼0

aðn;mÞPn−mðxÞPmðyÞ; ð1Þ

and rðx; yÞ represents the remainder term. InEq. (1)Ndenotes the expansion order, aðn;mÞ denotes the coef-ficients associated with the expansion, and PiðxÞdenotes the ith order Legendre polynomial. Tounwrap the phase map, the gradient of the phasemap must be considered. Away from wrappingfringes, the gradient of the wrapped phase map, de-noted ∇φw, is identical to the gradient of the truephasemap, denoted∇φ. At wrapping fringes, the gra-dient of the wrapped phase map differs from thegradient of the true phase map by �2π. There existphase gradient computation techniques that removethe discontinuities in the gradient of the wrappedphase map [23–26]. The procedure in [25] was usedto calculate the phase gradient for the proposedalgorithm.

The partial derivative of φðx; yÞ with respect to x is

∂φðx; yÞ∂x

¼XNn¼1

Xn−1m¼0

axðn;mÞP0n−mðxÞPmðyÞ: ð2Þ

The derivative of a Legendre polynomial with an or-der greater than 2 can be defined recursively byP0nþ1ðxÞ − P0

n−1ðxÞ ¼ ð2nþ 1ÞPnðxÞ, where the firsttwo derivatives are P0

0ðxÞ ¼ 0 and P01ðxÞ ¼ P0ðxÞ.

Repeated application of the recursion relation for de-rivatives of Legendre polynomials gives

P0nðxÞ ¼

X⌊n=2⌋k¼0

ð2n − 4k − 1ÞPn−2k−1ðxÞ; ð3Þ

where ⌊n=2⌋ is the floor function and returns the in-teger below n=2. Inserting Eq. (3) into Eq. (2) gives

∂φðx; yÞ∂x

¼XNn¼1

Xn−1m¼0

axðn;mÞPmðyÞ

×X⌊n−m−1

2 ⌋

k¼0

ð2n − 2m − 4k − 3ÞPn−m−2−2kðxÞ: ð4Þ

Expanding the summations in Eq. (4) and regroupingaccording to Legendre polynomials gives

∂φðx; yÞ∂x

¼XNn¼1

Xn−1m¼0

ð2n − 2m − 1ÞPn−m−1ðxÞPmðyÞ

×X⌊N−n

2 ⌋

k¼0

axðnþ 2k;mÞ: ð5Þ

The expansion coefficients inEq. (5) can be found byexploiting the orthogonality of Legendre polynomialson the interval ½−1; 1�. The expansion coefficients cal-culated from ∂φðx; yÞ=∂x when ⌊ðN − nÞ=2⌋ ¼ 0 are

axðn;mÞ¼�2mþ1

4

�Z1

−1

Z1

−1Pn−m−1ðxÞPmðyÞ

∂φðx;yÞ∂x

dydx

ð6Þ

where 1 ≤ n ≤ N and 0 ≤ m ≤ n − 1. The expansioncoefficients calculated from ∂φðx; yÞ=∂x for the case⌊ðN − nÞ=2⌋ ≠ 0 are

axðn;mÞ ¼�2mþ 1

4

�Z1

−1

Z1

−1ðPn−m−1ðxÞ

− Pn−mþ1ðxÞÞPmðyÞ∂φðx; yÞ

∂xdydx; ð7Þ

where 1 ≤ n ≤ N − 1 and 0 ≤ m ≤ n − 1 in both cases.The expansion coefficients calculated from ∂φðx; yÞ=∂y can be calculated using a similar procedure.

In the procedure described above, a significantnumber of expansion coefficients are calculated

1 June 2010 / Vol. 49, No. 16 / APPLIED OPTICS 3097

twice. The redundant expansion coefficients areremoved by averaging the coefficients that were cal-culated twice. The total expansion coefficients are

aðn;mÞ ¼ axðn;mÞ þ ayðn;mÞ2

; ð8Þ

when 1 ≤ n ≤ N and 1 ≤ m ≤ n. When m ¼ 0 and1 ≤ n ≤ N, then the total expansion coefficients areaðn;mÞ ¼ axðn;mÞ. The total expansion coefficientsare aðn;mÞ ¼ ayðn;mÞ when m ¼ n and 1 ≤ n ≤ N.The final steps in the recursive moment-based phaseunwrapping algorithm, calculating fitting surface,QN , and calculating the remainder function rðx; yÞ,follows the Chebyshev phase unwrapping algorithmpresented in [19]. The expansion order, N, is chosenusing the equation

diff ðQNþ1;QNÞ < ε; ð9Þwhere ε is a predetermined convergence parameterset by the user and

diff ðQNþ1;QNÞ ¼ZZ

jQNþ1ðx; yÞ −QNðx; yÞjdxdy:ð10Þ

As with the Chebyshev phase unwrapping algo-rithm, the recursive moment-based phase unwrap-ping method can be extended into three dimensions.The process of extension of the method presented inthis work to three dimensions is similar to that fol-lowed in [20] and will not be repeated in this work.Additionally, as in [19,20], the proposed phaseunwrapping method presented in this paper can beapplied to partitioned phase maps.

3. Results and Discussion

To evaluate the proposed phase unwrapping algo-rithm, the proposed algorithm was tested on experi-mental and simulated data sets. The simulateddata set considered in this work has a matrix sizeof 128 × 128. Three experimental data sets were usedto test the proposed phase unwrapping algorithm.The three experimental data sets consisted of a phan-tom data set, a human brain data set, and a mousetorso data set. Thephantomdata set hadamatrix sizeof 256 × 256, the human brain data set had a matrixsize of 128 × 128, and themouse data set had amatrixsize of 256 × 256. Each experimental data set was ac-quired from aMRI scanner. Each data set consideredin this work, both experimental and simulated, wasunwrapped using the proposed phase unwrapping al-gorithm, theChebyshevphaseunwrapping algorithm[19], and Jenkinson’s PRELUDE 2D phase unwrap-ping algorithm [21,22].

A. Data Processing

The proposed phase unwrapping algorithm describedabove and the Chebyshev phase unwrapping algo-rithm were implemented in MATLAB (The Math-Works, Natick, Massachusetts). The PRELUDE 2D

phase unwrapping algorithm was downloaded as aprecompiled binary file. All algorithmswere executedonaRedHatEnterpriseLinux server equippedwith a2:6 GHz dual-core Intel Xenon CPU.

A mask was created, using a thresholding method,for each data set considered in this work to removeareas of noise surrounding the region of interest. Tounwrap the simulated and experimental phase mapspresented in this work, each data set was partitionedinto 32 × 32 regions, and then each region was un-wrapped using the Chebyshev phase unwrapping al-gorithm and the proposed algorithm. The unwrappedphase maps were finally constructed from the un-wrapped partitions by adjusting the phase offset be-tween the unwrapped partitions [19]. On the otherhand, the PRELUDE 2D phase unwrapping algo-rithm unwrapped the unpartitioned phase maps.

B. Simulation Results

The proposed algorithm was tested using a simu-lated phase map using the equation

φðx; yÞ ¼ 45 expðx2 þ y2Þ þ 2πx; ð11Þ

where x; y ∈ ½−3; 3�. A mask was generated for the si-mulated phase map and the complex valued data setwas constructed from Eq, (9) by

ρðx; yÞ ¼�100 expðiφðx; yÞÞ within the maskexpðiαðx; yÞÞ outside the mask ;

ð12Þwhere αðx; yÞ generates a random value in the inter-val ½−π; πÞ. Gaussian noise was introduced to the realand imaginary parts of the complex valued simulatedphase map in k space. After the addition of Gaussiannoise, the noisy simulated phase map was recon-structed using a fast Fourier transform (FFT). Thesignal-to-noise ratio (SNR) of the magnitude imagewas calculated as S0=ð1:53σNÞ, where S0 is the meanof an area at the center of the simulated phase mapand σN denotes the mean of the standard deviation offour areas of background noise outside the phantom.The standard deviation of the phase noise was foundby using the equation σ ¼ 180°=ðπ × SNRÞ [27]. Thesignal to noise ratio (SNR) of the simulated phasemap was 0.67, which corresponds to a standard de-viation of the phase noise in the simulated phasemap was σ ¼ 86°.

The result of the proposed phase unwrapping algo-rithm on the simulated phase map with Gaussiannoise added is displayed in Fig. 1, where the SNRis 0.67. The phase map unwrapped using PRELUDE2D is displayed in Fig. 1(b). Figure 1(c) shows the re-sult from the Chebyshev phase unwrapping algo-rithm. It is noted that only one voxel in Fig. 1(c)differs from the result obtained using PRELUDE2D by more than 0:1 rad. In Fig. 1(d), only one voxelin the result from the proposed phase unwrappingalgorithm differs from the result obtained usingPRELUDE 2D by more than 0:1 rad.


Figure 2 examines the relationship between Gaus-sian noise and the convergence of the expansion orderN. As with the Chebyshev phase unwrapping algo-rithm [19], theadditionofGaussiannoisedoesnot sig-nificantly affect the expansion order.

C. Experimental Results

Figure 3 displays the application of the proposedphase unwrapping algorithm to themasked phantomdata set. The unwrapped phase map obtained by

using the Chebyshev phase unwrapping algorithmdiffers from the result obtained from PRELUDE 2Dby5pixels, roughly 0.02%of the total number of pixelsin the phantom. The unwrapping procedure with theChebyshev phase unwrapping algorithm took 27:86 sto unwrap the phase map in Fig. 3(a). Figure 3(d) dis-plays the phase map unwrapped using the proposedphase unwrapping algorithm, which required 29:95 sto unwrap the phase map. The result from the pro-posed algorithmdiffers from the result obtainedusingPRELUDE2Dby only 20pixels, representing roughly0.07% of the total number of pixels in the phantom.

Figure 4 shows the application of the proposedphaseunwrappingalgorithmto thehumanbraindataset. The unwrapping procedure took 4:51 s for theChebyshev phase unwrapping algorithm and 4:47 sfor the proposed phase unwrapping algorithm. Theunwrapped phasemap obtained using the Chebyshevphase unwrapping algorithm differs from the resultobtained from PRELUDE 2D by 20 pixels, or roughly0.31% of the total number of pixels in the humanbrain. The result from the proposed algorithm differsfrom the result obtained using PRELUDE 2D by only26 pixels, representing roughly 0.40% of the totalnumber of pixels in the human brain. PRELUDE2D performed an additional masking procedure forthe human brain data set. The discrepancies amongthe unwrapped phase maps in Figs. 4(b)–4(d) aredue to the additional masking procedure.

Figure 5 displays the application of the proposedphase unwrapping algorithm to the mouse torso data

Fig. 1. Evaluation of the performance of the proposed phase un-wrapping algorithm using simulated data set in a noisy environ-ment with an SNR of 0.67: (a) the simulated wrapped phase map,(b) the unwrapped phase map obtained using PRELUDE 2D, (c)the unwrapped phase map obtained using the Chebyshev phaseunwrapping algorithm, and (d) the unwrapped phase map ob-tained using the proposed algorithm. The three algorithm pro-duced identical unwrapped phase maps.

Fig. 2. The difference between consecutive surfaces as defined inEq. (10). The solid curve displays the difference between consecu-tive surfaces in the simulated phase map before Gaussian noisewas added. The dashed curve shows the difference between conse-cutive surfaces in the simulated phase map after Gaussian noisewas added.

Fig. 3. Illustration of the phase unwrapping in a homogeneousenvironment with the phantom data set: (a) the wrapped phasemap, (b) the unwrapped phase map obtained using PRELUDE2D, (c) the unwrapped phase map obtained using the Chebyshevphase unwrapping algorithm, and (d) the unwrapped phase mapobtained using the proposed algorithm.


set. The execution times for the wrapped phase mapin Fig. 5(a) was 28:33 s, while it was 31:21 s for theChebyshev phase unwrapping algorithm and theproposed phase unwrapping algorithm. The un-wrapped phase map obtained using the Chebyshevphase unwrapping algorithm differs from the resultobtained from PRELUDE 2D by 30 pixels, or roughly0.26% of the total number of pixels in the mouse tor-so. The result from the proposed algorithm differsfrom the result obtained using PRELUDE 2D by only38 pixels, representing roughly 0.34% of the totalnumber of pixels in the mouse torso.

Table 1 displays the differences between the phasemaps unwrapped using PRELUDE 2D and the twomoment-based methods, the proposed phase un-wrapping algorithm and the Chebyshev phase un-wrapping algorithm. It should be noted that, foreach phase map considered, less than 1% of the totalpixels in the phase maps unwrapped by the proposedmethod differ from the pixels in the phase maps un-wrapped using PRELUDE 2D by less than 0:1 rad.

The execution times for each of the phase mapsconsidered in this work are summarized in Table 2.In the proposed phase unwrapping algorithm, thecalculation of the phase derivative is the primary fac-tor in determining the execution time. The proceduredescribed in [25] was used to calculate the phase gra-dient for the proposed algorithm and the procedureis based on the FFT. The proposed phase unwrappingalgorithm was implemented using two other phasegradient computation methods: Bakker’s method[24] and the conjugate phase method [25]. UsingBakker’s method and the conjugate phase methodin the proposed phase unwrapping algorithm

Fig. 4. Evaluation of the performance on the human brain dataset (the round object next to the brain was a water bottle used as areference): (a) the wrapped phase map of the human brain, (b) theunwrapped phase map obtained using PRELUDE 2D, (c) the un-wrapped phase map obtained using the Chebyshev phase unwrap-ping algorithm, and (d) the unwrapped phase map obtained usingthe proposed algorithm.

Fig. 5. Illustration of the performance of the proposed unwrap-ping algorithm on an inhomogeneous background: (a) the wrappedphase map of the mouse torso data set, (b) the unwrapped phasemap obtained using PRELUDE 2D, (c) the unwrapped phase mapobtained using the Chebyshev phase unwrapping algorithm, and(d) the unwrapped phase map obtained using the proposedalgorithm.

Table 1. Comparison of the Relative Performance of the TwoPhase Unwrapping Algorithms in Relation to PRELUDEa

Figure Chebyshev Algorithm Proposed Algorithm

Fig. 1 1 1Fig. 3 5 20Fig. 4 20 26Fig. 5 30 38

aThe first column displays the phase map under consideration.The second column shows the total number of pixels of the phasemap unwrapped using the Chebyshev phase unwrapping algo-rithm that differ from the phase map unwrapped using PRELUDEby more than 0:1 rad. The third column displays the total numberof pixels of the phase map unwrapped using the proposed algo-rithm that differ from the phase map unwrapped using PRELUDEby more than 0:1 rad.

Table 2. Comparison of the Relative Performance of the ExecutionTimes (in Seconds) of the Phase Unwrapping Algorithms

Used in This Work

Figure Size PRELUDEChebyshevAlgorithm

ProposedAlgorithm

Fig. 1 128 × 128 0.20 4.32 4.44Fig. 3 256 × 256 0.32 27.86 29.95Fig. 4 128 × 128 0.29 4.51 4.47Fig. 5 256 × 256 0.42 28.33 31.21


significantly reduced the execution time. The execu-tion times for proposed phase unwrapping methodwith the three phase gradient calculation methodsare summarized in Table 3. No significant differencewas found between the phase maps unwrapped usingthe three different phase gradient calculation meth-ods, despite differences in the execution time.

4. Conclusions

The moment-based phase unwrapping algorithmwas modified to allow every pixel in the phasemap to contribute to the calculation of the moments.The performance of the proposed phase unwrappingalgorithm was tested on both experimental data setsand simulated data sets. For each of the data setsconsidered in this work, the proposed phase unwrap-ping algorithm gives results similar to those ac-quired by using PRELUDE 2D in the FunctionalMagnetic Imaging of the Brain Software Library(FSL) and the Chebyshev phase unwrapping algo-rithm. The execution times for the proposed phaseunwrapping algorithm are similar to the Chebyshevphase unwrapping algorithm for all phase maps con-sidered in this work.

Jason Langley would like to thank the John andMary Franklin Foundation for their support.

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Table 3. Comparison of the Relative Performance of the ExecutionTimes (in Seconds) for the Proposed Phase Unwrapping Algorithm with

Three Different Phase Gradient Computation Techniques

Figure Size FFT Bakker’sConjugatePhase

Fig. 1 128 × 128 4.44 1.42 1.50Fig. 3 256 × 256 29.95 5.94 6.85Fig. 4 128 × 128 4.47 1.46 1.62Fig. 5 256 × 256 31.21 6.24 7.67


recursive approach to the moment-based phase unwrapping method

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