convex optimization-based windowed fourier filtering with

Convex optimization-based windowed Fourierfiltering with multiple windows for wrapped-phasedenoisingKOHEI YATABE* AND YASUHIRO OIKAWA

Department of Intermedia Art and Science, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan*Corresponding author: [email protected]

Received 1 March 2016; revised 12 May 2016; accepted 15 May 2016; posted 16 May 2016 (Doc. ID 260304); published 7 June 2016

The windowed Fourier filtering (WFF), defined as a thresholding operation in the windowed Fourier transform(WFT) domain, is a successful method for denoising a phase map and analyzing a fringe pattern. However, it hassome shortcomings, such as extremely high redundancy, which results in high computational cost, and difficultyin selecting an appropriate window size. In this paper, an extension of WFF for denoising a wrapped-phase map isproposed. It is formulated as a convex optimization problem using Gabor frames instead of WFT. TwoGabor frames with differently sized windows are used simultaneously so that the above-mentioned issues areresolved. In addition, a differential operator is combined with a Gabor frame in order to preserve discontinuityof the underlying phase map better. Some numerical experiments demonstrate that the proposed method is ableto reconstruct a wrapped-phase map, even for a severely contaminated situation. © 2016 Optical Society of America

OCIS codes: (100.5070) Phase retrieval; (100.3175) Interferometric imaging; (100.3190) Inverse problems.

http://dx.doi.org/10.1364/AO.55.004632

1. INTRODUCTION

Interferometry-based measurement techniques generallyobserve information of interest as a wrapped-phase map. Theso-called wrapped phase is a restriction of the original phase tothe interval �−π; π� by adding or subtracting an integer which isa multiple of 2π. The process of recovering the original phasefrom its wrapped version by removing the modulus 2π ambi-guities, called phase unwrapping, is necessary to acquire theinformation. If the magnitude of phase difference betweenadjacent pixels is less than π (i.e., if the Itoh condition [1] issatisfied) the unwrapped phase can be obtained exactly by apath-following algorithm. However, the presence of measure-ment noise often violates the condition, and phase unwrappingbecomes an ill-posed problem. Thus, many unwrapping tech-niques have been proposed during the last two decades to over-come such difficulties [2–8].

An effective strategy for eliminating the issue is to integrate adenoising filter before or within the unwrapping process[9–18]. Such a denoising process is usually desirable becausethe unwrapping problem becomes considerably easier as thenoise level becomes lower. One successful approach is the win-dowed Fourier transform (WFT)-based filter, which is oftencalled windowed Fourier filtering (WFF) [19–24]. The pro-cedure of WFF is as follows. First, a wrapped-phase map φis converted into the exponential phase field (EPF): eiφ, wherei � ffiffiffiffiffi

−1p

. Second, EPF is represented by a linear combination

of windowed sinusoidal functions via WFT. Then, a threshold-ing operator is employed to enforce sparsity on the noisy EPF inthe WFT domain. Finally, the inverse WFT provides denoisedEPF, and the corresponding wrapped phase is obtained bycalculating the principal value of its complex argument.Conversion of wrapped phase to EPF is necessary to removethe difficulty caused by the 2π discontinuity of the wrappingeffect. Since EPF is a sinusoidal function, a Fourier-type trans-formation, including WFT, allows sparse representation ofnoiseless EPF. At the same time, noisy EPF is generally notsparse in the WFT domain. This sparsity-based characteriza-tion admits WFF to be a reasonable and effective approachfor denoising the wrapped phase.

Although WFF has achieved great success, there are someshortcomings. The first point is high computational complex-ity. WFF is usually based on the full WFT, which results inextremely redundant representation. In general, redundancyin the transformed domain should be controlled properly inorder to avoid unnecessary computation. Another point, whichis closely related to the first one, is that WFF has less freedomon a choice of a window function. There are infinitely manypairs of a window function and its dual counterpart, whichshould be chosen depending on application. Nevertheless, theordinary WFF seems to adopt only the Gaussian window,whose support is not compact. The last but most importantpoint is that there is a trade-off between the effectiveness of

4632 Vol. 55, No. 17 / June 10 2016 / Applied Optics Research Article

1559-128X/16/174632-10 Journal © 2016 Optical Society of America

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http://dx.doi.org/10.1364/AO.55.004632

noise removal and preservation of details [25,26]. When thewindow size is large, random noise can be effectively removed.However, essential details of a phase map, such as rapid changeor discontinuity, may be damaged easily due to oversmoothing.On the other hand, a small window, which can preserve suchdetails, is usually less effective for noise reduction. Althoughsome methods using adaptive windows have been proposedto resolve this trade-off [13,14], adaptation of window size maysuffer from instability because adaptation itself is anotherdifficult task in the presence of noise.

In this paper, a denoising method for a wrapped-phase mapbased on sparse convex optimization techniques is proposed. Itcan be viewed as an improved version of WFF, since it reliesupon the same sparsity principle, while the proposed methodallows simultaneous use of multiple windows with controllabil-ity of computational cost. Also, it allows one to ignore low-frequency components that often degrade performance ofWFF. The main contributions of this paper can be summarizedas follows:

• Use of the Gabor frame instead of WFT. The two-dimen-sional Gabor frame representation, which is a mathematicalframework for representing a two-dimensional function bycombination of the windowed Fourier functions, allows usto control redundancy and to choose any window functionwhose dual window is reasonably available. Therefore, our for-mulation can easily balance computational cost and effective-ness of noise reduction by varying the redundancy.

• Formulation of the wrapped phase denoising problem as aconvex optimization problem involving an infimal convolutiontype regularization term. The regularizer is designed to enforcesparsity on EPF in the Gabor frame domains computed withmultiple windows. Thus, the above-mentioned trade-off onperformance due to the window size can be solved by usingwindows with different sizes simultaneously. It also allows usto compose some linear transformations, a difference operatorin the case of this paper, which can enhance the performance ofnoise reduction.

• Providing an efficient iterative algorithm for solving theoptimization problem. The proposed algorithm is based on arecently developed accelerated linearized preconditioned alter-nating direction method of multipliers (ALP-ADMM) so thatno matrix inversion is involved in each step. Hence, the pro-posed method can be implemented as a matrix-free scheme,which greatly saves computational resources and time com-pared to a full matrix implementation. If we ignore the com-putation of the Gabor frame, computational complexity of theproposed algorithm is O�N � where N is the number of totalpixels. Therefore, the Gabor frame representation, which maybe computed withO�K log M � by using the fast Fourier trans-form, is the main computational cost, and that can be reducedby decreasing its redundancy, where K and M depend on thenumber of frequency channels and shifting width.

This paper is organized in the following way. First, aninterpretation of WFF in terms of sparse optimization is intro-duced in Section 2. Then, the proposed method is described asan extension of WFF in Section 3, with a brief explanation ofthe Gabor frame and convex optimization. Its experimental re-sults are presented in Section 4, and finally, the paper concludeswith Section 5.

2. INTERPRETATION OF WFF AS OPTIMIZATION

In this section, the standard WFF is explained as a proceduresolving a sparse optimization problem. This optimization pointof view will clarify the relationship between WFF and theproposed method in the subsequent section.

A. Two-Dimensional WFTThe two-dimensional WFT is a Fourier-based integral trans-form often defined as

�Wg f ��y; k� �ZR2

f �x�g�x − y�e−2πihk;xidx; (1)

where x; y ∈ R2 are position vectors, g is a window functionwhose L2-norm is normalized to one, z is the complex conju-gate of z, h· ; ·i, is the standard inner product, and k ∈ R2 is aspatial frequency vector (for simplicity, we will omit somemathematical details that are not important for describingthe proposed method such as f ; g ∈ L2�R2�). Its inversetransform,

f �x� �ZR2

ZR2

�Wg f ��y; k�g�x − y�e2πihk;xidydk; (2)

allows reconstruction of f from its representation Wg f . Fornotational convenience, Eq. (2) will be denoted as

f � WginvW

g f : (3)

Awindow function g , usually the Gaussian function in the con-text of WFF, provides spatially local information of f .Compared to the global nature of the Fourier transform, thislocalization property is preferable for most applications becausefrequency components of practical data usually vary in position.

B. WFFWFF is a denoising method for a fringe pattern and wrappedphase. It is defined as a thresholding operation in the WFTdomain:

f̃ � WginvT

λhard�Wg f �; (4)

where for a threshold value λ > 0,

T λhard�z�n �

�zn �jznj ≥ λ�0 �jznj < λ� (5)

is the element-wise hard-thresholding operator. In the casewhen WFF is applied to a wrapped-phase map φ, conversionbetween the wrapped phase and EPF is required before andafter the filtering, i.e.,

φ̃ � Arg�WginvT

λhard�Wg eiφ��; (6)

where the symbol Arg�z� denotes the principal value of thecomplex argument of z. The thresholding operator convertsa vector into a sparser one: a vector consists of more zeros.Therefore, thresholding has an ability to recover clean EPF,which is sparsely represented in the WFT domain becauseof its sinusoidal nature, from noisy EPF, whose WFT domainrepresentation is usually not sparse.

C. Optimization View of WFFThe hard-thresholding operator provides a solution to thefollowing minimization problem [27]:

Research Article Vol. 55, No. 17 / June 10 2016 / Applied Optics 4633

T λhard�z� ∈ arg min

x�‖x − z‖22 � λ2‖x‖0�; (7)

where ‖z‖0 is number of nonzero elements of a vector z.Therefore, with suitable discretization of WFT, WFF can beviewed as an optimization procedure solving Eq. (7) in the dis-crete WFT domain.

This optimization problem naturally arises in sparse model-ing, since ‖ · ‖0 is the most direct realization of a sparsity-enforcing penalty function. More generally, replacing ‖ · ‖0in Eq. (7) with any function promoting sparsity can recovera sparse signal to some extent. Among such penalty functions,probably the most famous and popular one is l1-norm∥z∥1 � Σnjznj, which is the convex relaxation of ‖ · ‖0, result-ing in the following optimization problem: finding

x⋆ ∈ arg minx

�1

2‖x − z‖22 � λ‖x‖1

�; (8)

whose solution is unique and obtained by the element-wisesoft-thresholding operation,

T λsoft�z�n �

� �1 − λ∕jznj�zn �jznj ≥ λ�0 �jznj < λ� : (9)

3. CONVEX OPTIMIZATION-BASED WFF

In this section, the proposed method is described as an exten-sion of WFF. Its building blocks, Gabor frame, and convexoptimization techniques are also introduced here briefly.

A. Gabor Frame RepresentationCalculation of WFT is time-consuming because it is anextremely redundant representation. Instead, a Gabor frameis adopted for the proposed method so that unnecessary com-putation can be eliminated by controlling redundancy.

A two-dimensional Gabor system for fixed step size a; b > 0is defined as

fg�x − an�e2πibhm;xign;m∈Z2 : (10)

It is a Gabor frame if there exists a dual window gd and thefollowing reconstruction formula holds [28]:

f �x� �X

n;m∈Z2

�F g f ��n; m�gd�x − an�e2πibhm;xi; (11)

where

�F g f ��n; m� �ZR2

f �x�g�x − an�e−2πibhm;xidx: (12)

Similar to Eq. (3), Eq. (11) will be denoted as

f � F gdinvF

g f : (13)

By comparing Eqs. (12) and (11) with Eqs. (1) and (2), it isclear that the Gabor frame can be interpreted as a discretelysampled version of WFT. Therefore, redundancy of Gaborframe representation can be decreased by changing the sam-pling intervals as long as a reasonable dual window exists. Acomputationally efficient version of WFF can be written as

f̃ � F gdinvT

λhard�F g f �; (14)

which utilizes a Gabor frame instead of WFT.

For a window function g , there are infinitely many choicesfor its dual gd satisfying Eq. (11). A tight window is a specialclass of window function whose dual window is itself. In thisstudy, the canonical tight window g t, whose frame bound isnormalized to one, is adopted [28].

B. Convex OptimizationConvex optimization is a fundamental tool for solving manyengineering problems in signal and image processing. It is oftenformulated as the following minimization problem: finding

x⋆ ∈ arg minx�Fd �x� � λG�Lx��; (15)

where Fd and G are convex functions, respectively called datafidelity term and regularization term; λ > 0 is a tuning param-eter balancing the importance of them, and L is a linear oper-ator. The regularization term is used to implement some priorknowledge, such as sparsity in the transformed domain, whilethe data fidelity term tries to keep the solution not too differentfrom the observed data d . Recent advances in the field allow Gand/or Fd to be nondifferentiable. Therefore, the above formu-lation can include hard constraints to convex sets as well asstructured priors necessary for modern signal processingmethods [29].

Among several advantages of convex optimization-based for-mulation, probably the most important one is its ease in achiev-ing a global solution. When objective functions and constraintsare all convex, a local minimum of a minimization problembecomes a global minimum. Therefore, convex formulationis less sensitive to an initial value and noise compared to a non-convex one. Convex relaxation is a well-accepted strategy toreceive benefit from this advantage. Since globally solving anonconvex optimization problem is practically impossible, ex-cept in some very special cases, the convexly approximatedproblem of the original problem is solved instead. An exampleof such relaxation is l1-norm, which is a convex functionapproximating ∥·∥0, as introduced in Section 2.C.

C. Proposed MethodTo extend WFF for overcoming some difficulties mentioned inSection 1, we first start with a convexly relaxed version of WFF:finding

x⋆ ∈ arg minx

�1

2‖x − d‖22 � λ‖F g tx‖1

�; (16)

which is a solution x⋆ that has sparse Gabor frame representa-tion F g tx⋆ and is close to the original observed data d . Thedegree of sparsity is controlled by the regularization parameterλ > 0; larger λ provides a sparser solution. (Strictly speaking,Eq. (16) and the soft-thresholding version of WFF are similarbut not equivalent, since data fidelity is measured in a differentdomain; see [30]).

In order to incorporate multiple windows, the sparse regu-larization term is divided into two terms:

minx;y

�1

2‖x − d‖22 � λfα‖F g t1�x − y�‖1 � �1 − α�‖F g t2y‖1g

�;

(17)

where y is an auxiliary variable, and α ∈ �0; 1� is a balancingparameter typically chosen around 0.5. This splitting using the


auxiliary variable is in the form of infimal convolution [31];thus, we shall call it “infimal convolutional WFF” with twowindows. Such decomposition results in a solution which ispenalized by either of the regularization terms whose valueis smaller. The penalty is designed so that the smooth and de-tailed parts are handled separately in each term. For example,when g t1 is a large window and g t2 is a smaller one, the smoothpart of a phase map will be penalized with ∥F g t1 ·∥1, and thedetailed part will be penalized with ∥F g t2 ·∥1. Therefore, thewindow size is automatically and continuously balanced be-tween the two, depending on the position, in the process ofoptimization. Obviously, it can be easily extended to more thanthree windows in a similar manner.

Although the above formulation seemingly works, we willmodify the second part of the regularizer slightly in order toachieve better results. A known issue of WFF is that its perfor-mance will be degraded if low frequency components are con-tained in each local area because their side lobe violates thesparsity assumption of a clean EPF in the windowed frequencydomain. However, such low frequency components will arisequite frequently due to windowing. Therefore, a high-pass fil-ter, or difference operator D, is combined with the penaltyterm; our proposed method is to find

x⋆ ∈ arg minx;y

�1

2‖x − d‖22

� λfα‖F g t1�x − y�‖1 � �1 − α�‖F g t2Dy‖1g�: (18)

Each term of the regularizer now ignores low-frequency com-ponents, as they will be canceled by y, whose low-frequencycomponents are not penalized by any term. Thus, the sparsityassumption should be fulfilled in every local area, and theeffectiveness of noise reduction is ensured. In addition, thecombination of the high-pass filter can treat discontinuityand detail of a clean phase map better because it enhances suchnonsmooth patterns, which are dominated by high-frequencycomponents.

D. ALP-ADMMAmong many convex optimization algorithms [32,33], anALP-ADMM [34], which is a recently proposed acceleratedvariant of the well-known ADMM [35], is chosen for solvingEq. (18) in this paper. Here, as usual in convex analysis [31],Γ0�CN � denotes the set of all proper lower-semicontinuousconvex functions over the Euclidean space CN .

Let us restate the convex optimization problem in Eq. (15):finding

x⋆ ∈ arg minx∈B⊂CN

�F�x� � G�Lx��; (19)

where F ∈ Γ0�CN � is differentiable with the β-Lipschitziangradient (there exist a positive constant β ∈ �0;∞� such that‖∇F �x�−∇F �y�‖≤β‖x −y‖ for all x; y ∈ CN ), G ∈ Γ0�CM �,L ∈ CM×N , and B is a set of box constraints bounding eachelement of x: jxnj ≤ bn �bn > 0�. Then, its solution can beobtained approximately by iterating

666666666664

x̃ ←�1 − 2

k�1

�x̂ � 2

k�1 x

x ← PB

hx − k

τ

nL��ρKk �Lx − z� � r

�� ∇F �x̃�

oiz ← prox k

ρKG

hLx � k

ρK ri

r ← r � ρkK �Lx − z�

x̂ ←�1 − 2

k�1

�x̂ � 2

k�1 x

(20)

from initial values x, z �� Lx�, x̂ �� x�, and r � 0, where k �1;…; K is the iteration index, ρ is a parameter typically chosenaround 1, τ � 2β� ρK ∥L∥2

S, ‖L‖S is the spectral norm of L,

L� denotes the adjoint of L, proxG �·� denotes the proximityoperator [33],

proxλG �x� � arg miny∈CN

�G�y� � 1

2λ‖x − y‖22

�; (21)

and PB is the projection operator to the convex set B,

PB �x�n ��bnxn∕jxnj �jxnj ≥ bn�

xn �jxnj < bn� : (22)

We should note that Eq. (20) is a simplified version modified sothat it is easier to apply to the proposed method. ALP-ADMMcan solve more general problems than Eq. (19) with more free-dom on choice of the step-size parameters [34].

Algorithm 1: Proposed algorithm (ALP-ADMM)

1. Procedure INFCONVWFF d ; x; λ; α; ρ; K2. x̂ ← x3. z ← F �1�x4. τ ← 9ρK � 25. y ← 0, ζ ← 0, r ← 0, γ ← 06. for k � 1;…; K do7. t ← x − y − F �1�

inv�z − kρK r�

8. x ← P1�x − ρK

τ t − kτ ��1 − 2

k�1�x̂ � 2k�1 x − d�

9. y ← P1

�y − ρK

τ �−t � D�fDy − F �2�inv�ζ − k

ρK γ�g�10. p ← F �1��x − y�11. q ← F �2�Dy12. z ← T λαk∕ρK

soft �p� kρK r�

13. ζ ← T λ�1−α�k∕ρKsoft �q � k

ρK γ�14. r ← r � ρk

K �p − z�15. γ ← γ � ρk

K �q − ζ�16. x̂ ← �1 − 2

k�1�x̂ � 2k�1 x

17. return x̂

E. Proposed AlgorithmIn order to apply ALP-ADMM to the proposed method ofEq. (18), we will restate the problem more precisely.

The observation d ∈ CN is EPF eiφ converted from a noisywrapped-phase map φ that consists of N � N v × N h pixels,where N v and N h are, respectively, the numbers of verticaland horizontal pixels. Let the two Gabor frame transformationsF g t1∶CN → CR1N and F g t2∶CN → CR2N be simply written asF �1� andF �2�, where R1 and R2 are the redundancy of the trans-formations. The first order difference operator with theNeumann boundary condition D∶CN∋x↦�xv; xh� ∈ C2N

and its adjoint D� are defined as usual [36], where xv and xhare the vertical and horizontal difference of x. By defining a


linear operator L∶C2N∋�x; y�↦�F �1��x − y�;F �2�yv;F�2�yh� ∈

CR1N�2R2N and two convex functions F∶C2N∋�x; y�↦12∥x − d∥22 ∈ R and G∶CR1N�2R2N∋�x; y�↦λfα∥x∥1��1 − α�∥y∥1g ∈ R, the proposed formulation of Eq. (18) canbe viewed as the convex optimization problem, Eq. (19).Thus, Algorithm 1 can be derived by directly applying ALP-ADMM in Eq. (20), where P1 is in Eq. (22) with bn � 1 forall n. After iterating the algorithm for some K , the denoisedwrapped-phase map can be obtained as φ̃ � Arg�x̂�.F. Some RemarksSome notes on the proposed method are presented in thissubsection.

First, it is well-known that a sparse solution obtained froman l1-norm regularized problem contains the bias error: eachnonzero element has a smaller modulus compared to the origi-nal observation. This phenomenon triggered a great deal ofresearch on reducing this error [27,37]. The proposed methodalso contains such bias, and thus a solution x̂ has a smaller mag-nitude than one obtained by a hard-thresholding type method.However, this bias will not be a matter for the wrapped-phasedenoising problem, because its objective is to recover the com-plex argument, which does not depend on the magnitude (seeexperiments in Section 4). Therefore, the proposed method canreceive the benefit of l1-norm, for example, robustness toinitial values and noise thanks to its convexity, without muchconcern about its disadvantage in usual applications.

Second, the proposed method treats the gradient terms yvand yh separately, which results in anisotropic thresholding,which is often out of favor in many applications. An isotropicvariant of the proposed method can be obtained by simplyreplacing the soft-thresholder T soft in line 14 of Algorithm1 with the block soft-thresholding operator [38]. However,we empirically found that Algorithm 1 works better than itsisotropic variant in terms of the signal-to-noise ratio, perhapsbecause separately treating the directional difference terms leadsto sparser Gabor representation, which can be recovered moreefficiently by the thresholding. Therefore, we left the sparsity-inducing regularizer as l1-norm instead of the mixed norm.

Finally, it is also well-known that considering some higher-order difference operators instead of only the first-order differ-ence D can improve the smoothness of a solution. However,just replacing D with a second-order difference performedworse in our case; this should be caused by the same reasonrelated to sparsity as explained in the previous paragraph. Thereare several methods for incorporating higher-order differencesusing infimal convolution, and the proposed method can beimproved by them [39]. In this paper, we chose only thefirst-order difference because of cheaper computational cost.For the same reason, although there are no limitations on usingmore than two Gabor frames, such possible extension will notbe examined here.

4. EXPERIMENTAL RESULTS

In this paper, all experiments were performed on a MATLAB2015b with an Intel core i7 3.0 GHz processor. Every calcu-lation of Gabor frame representation is done by the large

time-frequency analysis toolbox (LTFAT), which is openlyavailable software for frame-based signal processing [40,41].

A. Gabor Frame WFFBefore testing the proposed method, performance of the Gaborframe based WFF in Eq. (14) was investigated to show howredundancy is related to computational cost and denoisingability.

Figure 1 shows the computational time of Gabor frame rep-resentation for several redundancies obtained by varying thenumber of channels (frequency components) and sliding widthof a window. Note that redundancy can take only some discretevalues because the parameters (channel number and shiftingwidth) are both integers. Since discrete WFT can be regardedas a fully redundant Gabor frame representation, it is obviousthat replacing WFT with a Gabor frame can easily reduce thecomputational cost of any WFT-based method.

The denoising ability of Gabor frame-based WFF inEq. (14) with both hard- and soft-thresholding is illustratedin Fig. 2. The noisy wrapped phase map is obtained by addingcomplex-valued Gaussian noise of variance 2σ2 (the real andimaginary part are independent Gaussian with variance σ2)to EPF of a Gaussian function,

14πe−n2∕200−m2∕450; (23)

as in [14], where n; m ∈ Z are horizontal and vertical indicesrepresenting the position of a pixel in an image whose originis at the center of the image. The denoising performanceswere measured by the improvement in the signal-to-noise ratio(ISNR),

ISNR � 10 log10

264∥eiφ − eiφtrue∥22

∥eiφ̃ − eiφtrue∥22

375; (24)

where φ is a noisy wrapped-phase map, φ̃ is its denoisedversion, and φtrue is the noiseless one. Since the performance

101 102 103 104

Redundancy

10-3

10-2

10-1

100

101

102

Com

puta

tiona

l Tim

e [s

]

Directly implemented WFF (wft2) [20]

Faster WFFusing FFT(wft2f) [42]

Fig. 1. Measured computational time of Gabor frame representa-tion for an image of 64 × 64 pixels. It includes both forward andinverse transform as in Eq. (13). For reference, computational timeof two WFT functions implemented by Kemao (wft2 [20] and wft2f[42]) are also shown as red dots.


depends on both redundancy and threshold λ, every combina-tion of them are tested. The second row is the result for hard-thresholding, while the third row is for soft-thresholding.

From Figs. 2(f ) and 2(i), it can be confirmed that the per-formances of hard- and soft-thresholders are almost the samewhen they are combined with the fully redundant Gabor frame,or WFT. On the other hand, their performance greatly differswhen redundancy is cut down: soft-thresholding is more stablein decreasing redundancy than hard-thresholding. This resultindicates that WFF with a Gabor frame and soft-thresholdingoperator can be accelerated by reducing its redundancy withoutmuch loss on the denoising performance. It also justifies the

effectiveness of using the soft-thresholding operator insteadof hard-thresholding in the proposed method. Note that it alsohas advantages from the optimization point of view: the cor-responding cost function, l1-norm, is convex, which is animportant property, as explained in Section 3.B.

Based on these results, for comparison to the proposedmethod, WFF is performed with the soft-thresholding operatorin the next experiment.

B. Proposed MethodDenoising performance of the proposed method is comparedwith WFF and PEARLS (phase estimation using adaptive

Fig. 2. Comparison of denoising ability of hard- and soft-thresholding operators combined with Gabor frames as in Eq. (14). Noisy 100 × 100image of the Gaussian function in Eq. (23) �noise l evel σ � 0.5� in (a) is used as test data. The channel number [how many frequency componentsare used for each direction (vertical and horizontal)] and shifting step (how many pixels are shifted to compute adjacent window) of a Gabor frameare adjusted to control redundancy shown in (b). Their computational times are depicted in (c). Second row is the result for hard-thresholding, whilethird row is for soft-thresholding. The results for lower left part in (b) are omitted because their redundancies are less than or equal to 1, which meansthey are not a frame [a forward and inverse transform pair cannot reconstruct a function as in Eq. (13)]. (d) and (g) are ISNR obtained by varying thethreshold λ in Eqs. (5) and (9), where each line corresponds to each pixel in (b). The maximum values of each line in (d) and (g), which are the bestachievable ISNR for (a), are illustrated in (e), (f ), and (h), (i), respectively.


regularization based on local smoothing), which is one of thestate-of-the-art methods for wrapped phase denoising usingwindow-sized adaptation [14]. Testing data are the sameGaussian function in Eq. (23) except for 1/4 region wherethe Gaussian is replaced by zero. This is chosen for checkingthe ability of preserving the discontinuity of phase.

The two window functions g t1 and g t2 used in the Gaborframes of the proposed method are shown in Fig. 3(a) and 3(b),respectively. g t1 is the canonical tight window for a 15-channelGabor system, with 5-pixel shifting constructed from theHamming window, whose support is 15 × 15 pixels, whileg t2 is for 4-channel with 2-pixel shifting constructed fromthe iterated sine window of 3 × 3 pixels [43]. Their redundan-cies are R1 � 9 and R2 � 4. These windows are chosen intui-tively, with the hope that the Gabor system corresponding tog t1 is adapted to the smooth part, while g t2 is adapted to thedetailed part of the phase. There should be a better pair of win-dows for the testing data because, in general, the best windowfunction depends on its application and data; we do not con-sider such adaptations of windows here.

Figure 4 summarizes the denoising performances for noiselevel σ � 0.25, 0.5, 0.75, 1 that are approximately SNR � 9,3, −0.5, −3 dB, respectively. WFF was performed by soft-thresholding with threshold λ on a fully redundant Gabor

system whose redundancy is 10,000. Its window functionwas a Gaussian function, shown in Fig. 3(c), which is chosento have similar shape as g t1, Fig. 3(a). The filtering step ofPEARLS was tested by using the MATLAB code providedin [44]. It adaptively changes window size so that the smoothpart is treated by a large window while the rapidly changing partis treated by a smaller one. Γ is a parameter to control suchadaptation; see [14] for detail. The proposed method wasperformed by 100 iterations (K � 100) with α � 0.5, ρ � 1,and the initial value of x was set to F �1�

invTλαsoftF

�1�d .Figures 5 and 6 show visual examples of the results. The

parameters for obtaining them are listed in Table 1; they werechosen based on Fig. 4 so that nearly the best ISNR wasachieved for each situation. The corresponding unwrappedresults were obtained by the phase-unwrapping max-flow(PUMA) algorithm [4].

Let us point out a few things about the proposed method:

• Comparison to WFF. Since the proposed method is anextended version of WFF, their experimental results are similarin terms of ISNR. However, there are notable differences in thepresence of artifacts. The results for WFF contain some low-frequency patterns that deformed the flat part of the originalphase, as in Figs. 5 and 6. This is caused by violation ofthe sparsity assumption owing to artificial low-frequency

Fig. 3. Window functions used for the proposed method. The canonical tight windows g t1 and g t2 are illustrated in (a) and (b), respectively. Forcomparison, the Gaussian window used for calculating WFF is also depicted in (c); only part of it is shown in (c) for visual convenience since itssupport is not compact. These functions are symmetric (rotating the 25 × 25 pixel plane 90 deg does not change their appearance).

Fig. 4. ISNR for several noise levels. (a) is the results for WFF using soft-thresholding. (b) is for PEARLS, whose tuning parameter Γ is describedin [14], and (c) is for the proposed method. Each line corresponds to testing data with different noise levels: σ � 0.25 (blue), 0.5 (red), 0.75 (yellow),and 1 (green). Some illustrative examples of these data are shown in Figs. 5 and 6.


components created by windowing. Such components can beignored if a very large window is used, but that causes anotherproblem: oversmoothing. On the other hand, the results forthe proposed method contain fewer artifacts thanks to thehigh-pass filter D in the penalty term. In addition, the usageof multiple windows preserved discontinuity, or edge, moreclearly than WFF.

• Comparison to PEARLS. Since the core feature ofPEARLS is to choose the window size adaptively dependingon the smoothness of the underlying phase, estimation of win-dow size is the most important step in achieving a good result.Indeed, it works quite well when the noise level is moderate.However, window size cannot be estimated reasonably forseverely contaminated data because PEARLS tries to preservenoise pattern as detail in such a case. Although this phenome-non can be eliminated by choosing larger windows more fre-quently, this strategy cannot preserve discontinuity and detail,

which makes PEARLS less attractive. In contrast, the proposedmethod is more robust in a highly noisy situation. This shouldbe because it is formulated as a convex optimization problem:not changing window size directly but combining multiplewindows in the form of infimal convolution.

• Computational time. Table 2 shows the computationaltime of each method. Although the proposed method iteratedthe loop 100 times, which requires computation of forward andinverse Gabor frame transformations more than 400 times, theproposed method can be performed faster than WFF. This isbecause the redundancies of the Gabor frames, R1 � 9 andR2 � 4, are much smaller than that of WFF (10,000). Notethat the balance of computational cost and denoising perfor-mance of the proposed method can be adjusted by changingthe redundancies and number of iterations K . Also, it is pos-sible to increase its performance for fixed K by tuningthe parameters α and ρ. Moreover, implementation in faster

Fig. 5. Some examples of the results for mildly contaminated data. The test data are obtained by replacing a quarter part of Eq. (23) with zero.Each column shows (from left to right) noisy data to be denoised, results for WFF, results for PEARLS, and results for the proposed method. Eachrow shows (from top to bottom) wrapped-phase maps, error of wrapped phase, and corresponding unwrapped phase obtained by PUMA algorithm[4] for noise levels σ � 0.25, 0.5. The near side of 3D graphs of error and unwrapped phases corresponds to the upper left corner of the wrapped-phase maps. The parameters used for each method are listed in Table 1.


languages, such as C language, can decrease the execution timeof the proposed method that is implemented in MATLAB here.

5. CONCLUSIONS

In this paper, a convex optimization-based method for denois-ing a wrapped-phase map is proposed. It can be regarded as anextension of the well-known WFF in three senses: (1) replacingWFT with a Gabor frame decreases computational cost; (2) for-mulation based on infimal convolution allows simultaneous useof multiple windows; and (3) combination of a difference op-erator enhances sparsity and increases qualitative performance.The experimental results showed that the proposed method canobtain better results, which contain fewer artifacts than theordinary WFF and PEARLS.

Fig. 6. Some examples of the results for severely contaminated data. The test data is obtained by replacing a quarter part of Eq. (23) with zero.Each column shows (from left to right) noisy data to be denoised, results for WFF, results for PEARLS, and results for the proposed method. Eachrow shows (from top to bottom) wrapped phase maps, error of wrapped phase, and corresponding unwrapped phase obtained by PUMA algorithm[4] for noise levels σ � 0.75, 1. The near side of 3D graphs of error and unwrapped phases corresponds to the upper left corner of the wrapped-phasemaps. The parameters used for each method are listed in Table 1.

Table 1. Parameters for Obtaining Figs. 5 and 6

σ � 0.25 σ � 0.5 σ � 0.75 σ � 1

WFF λ � 0.006 λ � 0.015 λ � 0.02 λ � 0.03PEARLS Γ � 0.8 Γ � 2 Γ � 3 Γ � 15Proposed λ � 0.3 λ � 0.6 λ � 0.9 λ � 1.1

Table 2. Computational Time for Executing EachMethod Once

WFF PEARLS Proposed

10.1 [s] 0.04 [s] 3.9 [s]


We should note that automatic estimation of a good com-bination of regularization parameters is a highly desirablefeature for practical use. There is much research on suchautomatic tuning, and that topic should be investigated inthe future for the proposed method as well.

Funding. Japan Society for the Promotion of Science(JSPS) (15J08043).

REFERENCES1. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21,

2470 (1982).2. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping:

Theory, Algorithms and Software (Wiley, 1998).3. J. M. Bioucas-Dias and J. M. N. Leitao, “The F020F05AπM algorithm:

a method for interferometric image reconstruction in SAR/SAS,” IEEETrans. Image Process. 11, 408–422 (2002).

4. J. M. Bioucas-Dias and G. Valadao, “Phase unwrapping via graphcuts,” IEEE Trans. Image Process. 16, 698–709 (2007).

5. R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,”IEEE Trans. Geosci. Remote Sens. 45, 3240–3251 (2007).

6. S. Tomioka and S. Nishiyama, “Phase unwrapping for noisy phasemap using localized compensator,” Appl. Opt. 51, 4984-4994(2012).

7. D. Kitahara, M. Yamagishi, and I. Yamada, “A virtual resamplingtechnique for algebraic two-dimensional phase unwrapping,” inProceedings of the IEEE International Conference on Acoustics,Speech Signal Processing (ICASSP) (IEEE, 2015), pp. 3871–3875.

8. D. Kitahara and I. Yamada, “Algebraic phase unwrapping based ontwo-dimensional spline smoothing over triangles,” IEEE Trans.Signal Process. 64, 2103–2118 (2016).

9. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phaseunwrapping,” Opt. Eng. 36, 2466–2472 (1997).

10. H. A. Aebischer and S. Waldner, “A simple and effective method forfiltering speckle-interferometric phase fringe patterns,” Opt. Commun.162, 205–210 (1999).

11. K. Qian, S. H. Soon, and A. Asundi, “A simple phase unwrappingapproach based on filtering by windowed Fourier transform,” Opt.Laser Technol. 37, 458–462 (2005).

12. Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier-filteredand quality-guided phase-unwrapping algorithm,” Appl. Opt. 47,5420-5428 (2008).

13. V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation(phasela) technique for phase unwrap from noisy data,” IEEE Trans.Image Process. 17, 833–846 (2008).

14. J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolutephase estimation: adaptive local denoising and global unwrapping,”Appl. Opt. 47, 5358–5369 (2008).

15. M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas,“Fast two-dimensional simultaneous phase unwrapping and low-passfiltering,” Opt. Express 20, 2556–2561 (2012).

16. J.-F. Weng and Y.-L. Lo, “Integration of robust filters and phaseunwrapping algorithms for image reconstruction of objects containingheight discontinuities,” Opt. Express 20, 10896–10920 (2012).

17. H. Y. H. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G.Barbastathis, “Path-independent phase unwrapping using phasegradient and total-variation (TV) denoising,” Opt. Express 20,14075–14089 (2012).

18. X. Xie and Y. Li, “Enhanced phase unwrapping algorithm based onunscented Kalman filter, enhanced phase gradient estimator, andpath-following strategy,” Appl. Opt. 53, 4049–4060 (2014).

19. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,”Appl. Opt. 43, 2695–2702 (2004).

20. Q. Kemao, “Two-dimensional windowed Fourier transform for fringepattern analysis: principles, applications and implementations,” Opt.Lasers Eng. 45, 304–317 (2007).

21. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparativeanalysis on some filters for wrapped phase maps,” Appl. Opt. 46,7412–7418 (2007).

22. Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transformfor fringe pattern analysis: theoretical analyses,” Appl. Opt. 47,5408–5419 (2008).

23. C.Quan,H.Niu, andC. Tay, “An improvedwindowedFourier transformfor fringe demodulation,” Opt. Laser Technol. 42, 126–131 (2010).

24. Q. Kemao, “Applications of windowed Fourier fringe analysis in opticalmeasurement: a review,” Opt. Lasers Eng. 66, 67–73 (2015).

25. Q. Kemao, “On window size selection in the windowed Fourier ridgesalgorithm,” Opt. Lasers Eng. 45, 1186–1192 (2007).

26. Q. Kemao, “On window size selection in the windowed Fourier ridgesalgorithm: addendum,” Opt. Lasers Eng. 45, 1193–1195 (2007).

27. R. Chartrand, “Shrinkage mappings and their induced penalty func-tions,” in Proceedings of the IEEE International Conference onAcoustics, Speech and Signal Processing (ICASSP) (IEEE, 2014),pp. 1026–1029.

28. O. Christensen, Frames and Bases: An Introductory Course(Springer, 2008).

29. F. Bach, R. Jenatton, J. Mairal, and G. Obozinski, “Optimization withsparsity-inducing penalties,” Found. Trends Mach. Learn. 4, 1–106(2011).

30. M. Elad, P. Milanfar, and R. Rubinstein, “Analysis versus synthesis insignal priors,” Inv. Probl. 23, 947–968 (2007).

31. H. H. Bauschke and P. L. Combettes, Convex Analysis and MonotoneOperator Theory in Hilbert Spaces (Springer, 2011).

32. P. L. Combettes and J.-C. Pesquet, “Proximal splitting methods in sig-nal processing,” in Fixed-Point Algorithms for Inverse Problems inScience and Engineering, H. H. Bauschke, R. Burachik, P. L.Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, eds.(Springer, 2011), pp. 185–212.

33. N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim.1, 127–239 (2014).

34. Y. Ouyang, Y. Chen, G. Lan, and J. Eduardo Pasiliao, “An acceleratedlinearized alternating direction method of multipliers,” SIAM J. ImagingSci. 8, 644–681 (2015).

35. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributedoptimization and statistical learning via the alternating directionmethod of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).

36. A. Chambolle, “An algorithm for total variation minimization and appli-cations,” J. Math. Imaging Vis. 20, 73–87 (2004).

37. H. Zou, “The adaptive lasso and its oracle properties,” J. Am. Stat.Assoc. 101, 1418–1429 (2006).

38. S. Ono and I. Yamada, “Signal recovery with certain involved convexdata-fidelity constraints,” IEEE Trans. Signal Process. 63, 6149–6163(2015).

39. S. Setzer, G. Steidl, and T. Teuber, “Infimal convolution regulariza-tions with discrete l1-type functionals,” Commun. Math. Sci. 9,797–827 (2011).

40. P. L. Søndergaard, B. Torrésani, and P. Balazs, “The linear time fre-quency analysis toolbox,” Int. J. Wavelets Multiresolut. Inf. Process.10, 1250032 (2012).

41. Z. Průša, P. L. Søndergaard, N. Holighaus, C. Wiesmeyr, and P.Balazs, “The large time-frequency analysis toolbox 2.0,” in Sound,Music, and Motion, M. Aramaki, O. Derrien, R. Kronland-Martinet,and S. Ystad, eds. (Springer, 2014), pp. 419–442.

42. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis”(2009) [Online]. Available: http://www.mathworks.com/matlabcentral/fileexchange/24852.

43. E. Wesfreid and M. V. Wickerhauser, “Adapted local trigonometrictransforms and speech processing,” IEEE Trans. Signal Process.41, 3596–3600 (1993).

44. J. Bioucas-Dias, “Code” (2012) [Online] Available: http://www.lx.it.pt/~bioucas/code.htm.


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