CAPE: combinatorial absolute phase estimation

Technical report, IT/IST Communications theory

and Pattern Recognition Group

Goncalo Valadao and Jose Bioucas-Dias

Instituto de Telecomunicacoes and Instituto Superior Tecnico,

Av. Rovisco Pais, Torre Norte, Piso 10, 1049-001 Lisboa, Portugal

email:{gvaladao,bioucas}@lx.it.pt.

February, 2009

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Abstract

This paper introduces an absolute phase estimation algorithm for in-terferometric applications. The approach is Bayesian. Besides coping withthe 2π-periodic sinusoidal non-linearity in the observations, the proposedmethodology assumes a first order Markov random field (MRF) prior anda maximum a posteriori probability (MAP) viewpoint. For computing theMAP solution, we provide a combinatorial suboptimal algorithm that in-volves a multi-precision sequence. In the coarser precision, it unwraps thephase by using essentially the, previously introduced, PUMA algorithm[J. Bioucas-Dias and G. Valadao, IEEE Trans. Image Proc., vol. 16, no.3, pp. 698-709 (2007)], which blindly detects discontinuities and yieldsa piecewise smooth unwrapped phase. In the subsequent increasing pre-cision iterations, the proposed algorithm denoises each piecewise smoothregion, thanks to the previously detected location of the discontinuities.For each precision, we map the problem into a sequence of binary opti-mizations, which we tackle by computing min-cuts on appropriate graphs.This unified rationale for both phase unwrapping and denoising inheritsthe fast performance of graph min-cuts algorithms. In a set of experimen-tal results, we illustrate the effectiveness of the proposed approach.

1 Introduction

There are nowadays many applications based on phase images, e.g., inter-ferometric synthetic aperture radar (InSAR) [1], magnetic resonance imaging(MRI) [2], adaptive optics [3, 4], vibration and deformation measurements [5],and diffraction tomography [6]. InSAR is being successfully applied, e.g., to thegeneration of digital elevation models (DEM), and in the monitoring of landsubsidence; among the plethora of MRI applications, we emphasize venography(angiography as well) [7] and tissues elastography [8]; concerning adaptive op-tics, we point up applications in medicine and industry [9]; interferometry basedvibration and deformation measurements is widespread among metrology tech-niques; diffraction tomography finds application in, e.g., geophysical subsurfaceprospection and 3D microscopic imaging.

In all of these imaging systems, the acquisition sensors read only the sineand cosine of the absolute phase; that is, we have only access to the phasemodulo-2π, the so-called interferogram. Besides this sinusoidal nonlinearity,the observed data is corrupted by some type of noise. Due to these degradationmechanisms, absolute phase estimation is known to be a very difficult problem.In particular, if the magnitude of absolute phase variation between neighboringpixels is larger than π, i.e., the so-called Itoh condition [10] is violated, then theinference of the absolute phase is an ill-posed problem [11]. These violationsmay be due to spatial undersampling, original phase discontinuities, or noise.The goal of this paper is to present an algorithm to solve this nonlinear inverseproblem: the so-called absolute phase estimation problem.

The structure of the observation models relating the noisy wrapped phasewith the true phase depends on the system under consideration (see, e.g., [12–14]

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for an account of observation models in different coherent imaging systems). Theessential of most of these observation mechanisms is, however, captured by therelation

z = A exp(jφ) + n, A > 0

= |z| exp[j(φ+ φn)],

where φ is the true phase value (the so-called absolute phase value), n = nI+jnQis complex-valued zero-mean circular white additive noise with variance σ2 (i.e.,nI and nQ are zero-mean independent Gaussian random variables with varianceσ2/2), and φn is the phase due to n.

Let us define W , the wrapping operator, such that

ψ = angle(z) = W(φ+ φn), ψ ∈ [−π, π). (1)

Operator W maps the noisy phase φ + φn into the principal phase interval[−π, π). For φn = 0, there is an obvious link between the wrapped phase ψ andthe non-wrapped absolute phase φ, φ = ψ+ 2πk, ψ ∈ [−π, π), where k is an in-teger. The basic unwrapping problem is to reconstruct φ(x, y), (x, y) ∈ X ⊂ Z2,from the observations ψ(x, y). There is, of course, no one-to-one relation be-tween the wrapped and the absolute phases. It is, however, straightforward toshow that, in the absence of noise, if the phase difference ∆φ between any neigh-boring pixels is smaller than π in magnitude (i.e., Itoh condition is verified),then the correspondent wrapped phase difference is the same, i.e.,

∆φ = W(∆ψ). (2)

We are, then, able to compute (unwrap) the absolute phase for every other pixel,up to a constant, by integrating the wrapped differences W(∆ψ). This is thebasis of the path following phase unwrapping (PU) algorithms [15]. It happens,quite often, that phase images have discontinuities, areas of high phase rate,and noise. In this case, the Itoh condition may fail and, therefore, differentintegration paths may lead to different unwrapped phase values.

To cope with these difficulties, there are in the literature two main classesof algorithms: basic phase unwrapping type, which solve only the 2π multi-ples problem [15–18]; and estimation type (or regularization type) in which theabsolute phase is estimated [13, 19–22], and thus have a broader scope.

1.1 Contributions

The main contribution of this paper is twofold:

1. A new approach to absolute phase estimation. Contrarily to the major-ity of absolute phase methods, we first apply unwrapping and only thendenoising. By using a discontinuity preserving prior, the unwrapping al-gorithm not only infers the 2π multiples of the absolute phase, but also,implicitly, detects the discontinuity locations. This is a crucial informationfor the effectiveness of the phase denoising that would not be available, ifthis step was applied prior to unwrapping.

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2. A unified rationale for making both phase unwrapping and denoising. Thisis achieved through the proposal of a (state-of-the-art competitive) multi-precision combinatorial optimization algorithm based on graph cuts. Themulti-precision technique improves the performance by decreasing the al-gorithm’s complexity.

1.2 Related work

In the field of phase unwrapping, we refer to [12] and references therein. [23] givesa phase unwrapping algorithm with discontinuity preserving capabilities. Fur-thermore, to our knowledge, PUMA [24] is a state-of-the-art phase unwrappingtype algorithm, while ZπM [13] and PEARLS [25] are state-of-the-art absolutephase estimation type algorithms. Regarding ZπM, the present work extendsit by allowing a wider family of MRFs, namely nonconvex pairwise potentials,which brings discontinuity preservation ability. PEARLS algorithm [25] em-ploys PUMA to make phase unwrapping and differs from ours essentially in thedenoising technique: namely it applies denoising before unwrapping. This is asensitive issue, since the effectiveness of the denoising step depends on the knowl-edge of the discontinuities location. PEARLS implements a filtering techniquebased on local polynomial approximation with varying adaptive neighborhoodused in reconstruction (see [25] for details). The adaptive mechanism trades biaswith variance in such a way that the window size stretches in areas where theunderlying true phase is smooth and shrinks otherwise, namely in the presenceof discontinuities.

In this paper, we circumvent the need of filtering the interferogram (noisywrapped phase image) by basically first using PUMA to unwrap the interfer-ogram; only then, denoising is applied after PUMA has implicitly located thediscontinuities. Compared with PUMA [24], the present work is also an exten-sion by both allowing denoising and improving the performance of each binarystep in the optimization sequence. With respect to this optimization, we dealwith a labeling problem on first-order Markovian random field made of unary(one variable dependence) and pairwise on difference (difference between twovariables dependence) terms. If the unary and pairwise terms are convex, thisproblem is known as the dual of the convex cost network flow problem which isextensively discussed in the literature (see, e.g., [26–29]). Works [13], [30], [24]have attacked particular instances of the dual of the convex cost network flowproblem by solving a sequence of descent binary optimization problems, whichmay be solved by computing the max-flow/min-cuts on appropriate graphs.This is a relevant practical advantage, as there are max-flow/min-cut algorithmsspecifically tuned to image processing and computer vision problems [31].

Works [32–34] and [28] have introduced and characterized generalizations ofthe descent algorithms [13], [30], [24] to dual of the convex cost network flowproblems containing pairwise and unary terms. An even more general classof functions, the so-called L-convex and L\-convex functions, has been consid-ered in [35]. The works [32, 33], in addition to introducing new results, give anexcellent account of the connections between the dual of the convex cost net-

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work flow problems and of L-convex and L\-convex based problems. ConcerningMarkovian first-order objective functions (having unary and pairwise terms) westill should refer [36] (pairwise term depending on the difference of the twovalues involved) which, although being theoretically very appealing, seems asnot very practical for most of the computer vision problems; we should alsorefer to the work [37], which deals with general first-order submodular objec-tive functions. A binary pairwise interaction function E(·, ·) is submodular iffE(0, 0) + E(1, 1) ≤ E(1, 0) + E(0, 1). Concerning the submodularity property,in general, we refer to [35] and references therein.

Discontinuity preserving pairwise interaction terms are, quite often, non-convex. In this case, the underlying optimization problem is NP-hard [38] andnone of the referred to algorithms applies. The source of difficulties is twofold:a) A sequence of descent optimizations leads to a local minimum and b) Eachsubproblem cannot be anymore mapped onto a max-flow/min-cut. In the veinof [24] we insist, however, in the descent approach, which, in the case of phaseunwrapping, leads systematically to high quality results. Concerning each bi-nary subproblem, we replace the objective function with a convex majorizer,thus adopting the majorization-minimization framework [39]. In an experimen-tal comparison with the powerful roof duality based methods [40–42] aimedat computing “good” suboptimal non-submodular binary-labeling solutions, wegive evidence that our approach is much faster and yet it achieves local minimawith the same quality.

1.3 Paper organization

In the next section, we present the core ideas and concepts of the proposed ap-proach. Namely Section 2.1 characterizes the posterior density of the proposedBayesian framework; then Section 2.2 concerns about the roles of unwrappingand denoising in the optimization process to compute the MAP, and subse-quently Section 2.3 establishes and discuss the optimization technique in somedetail. Finally, Section 3 presents a series of experimental results and, then,Section 4 draws some concluding remarks.

2 Proposed Approach

Let G = (V , E) be an undirected graph associated to a first order Markov randomfield (MRF), where the set of nodes V represents image pixels and the set ofedges E represents pairs of neighboring pixels. We assume that if (i, j) ∈ E then(j, i) /∈ E .

In this paper, we consider first order MRFs and, therefore, the set of edgesE represents the set of horizontal and vertical neighbors. Nevertheless, all theconcepts and results next presented are valid for any set of pairwise interactions.

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2.1 Posterior density

We follow the Bayesian framework. Accordingly, we need to build the posteriordensity p(φ|z) of the phase image φ ∈ R|V| given the observed complex image z ∈C

|V| (C denotes the complex field). Invoking the Bayes law, we have p(φ|z) ∝p(z|φ)p(φ), where p(z|φ) is the likelihood function, measuring the data fit andp(φ) is the prior density encoding a priori knowledge about the phase image φ.

Let us assume conditional independence in the observation mechanism, i.e.,p(z|φ) =

∏i∈V p(zi|φi). Furthermore, let us consider priors such that log p(φ) =

−µ∑

(i,j) Vi,j(φi − φj) + c, where c is an irrelevant constant, µ > 0 is a scale

parameter often termed the regularization parameter, and Vi,j(·) is the so-calledpotential associated with edge (i, j). In these circumstances, computing theMAP estimate is equivalent to minimize the negative logarithm of the posteriordensity E : R

|V| → R ∪ {+∞} given by

E(φ) ≡∑

i∈V

Di(φi)

︸ ︷︷ ︸Data fidelity term

+µ∑

(i,j)∈E

Vi,j (φi − φj)

︸ ︷︷ ︸Prior term

, (3)

where Di(φi) ≡ − log p(φi|zi).Given the observation mechanism introduced in (1), we have (see e.g. [13])

Di(zi) = −λi cos(φi − ψi), for i ∈ V ,

with λi ≡ A|zi|/(2σ2) and ψi ≡ angle(zi), i.e., the loglikelihood function is

proportional to a shifted cosine. The MAP absolute phase estimate is thenobtained by minimizing the negative of the logposterior function given by

E(φ) ≡∑

i∈V

−λi cos(φi − ψi) + µ∑

(i,j)∈E

V (φi − φj) . (4)

Notice that µ, the regularization parameter, sets the relative weight betweenthe data fidelity and the prior terms.

Aiming at simultaneously noise filtering and discontinuity preservation, weuse in this paper half-quadratic [43, 44] type potentials given by

V (x) =

{|x|2 if |x| ≤ π

π2 − πp + |x|p if |x| > π ,(5)

with 0 < p < 1 and thus nonconvex. Fig. 1 plots this function for p = 0.4.These potentials are quadratic near the origin, thus modeling Gaussian noise,and with a flatness trend for phase magnitudes larger than π, in order to preservediscontinuities [45]. In case we do not have discontinuities to deal with, wesimply employ quadratic (or near quadratic) potentials. Anyway, the conceptsand ideas next introduced are valid for a wide class of potential functions.

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Figure 1: Graph of a half quadratic potential employed in this paper (p = 0.4).

2.2 Optimization

Our goal is to computeφ∗ = arg min

φ[E(φ)] , (6)

where E(φ) is given by (4). This is a hard problem because both data fidelityand prior terms are nonconvex.

2.2.1 First unwrap then denoise

Very often, in interferometric applications, we are just interested in computingthe 2π multiple in the representation φi = ψi + 2πki, the so-called phase un-wrapping problem. Since the data fidelity term, Di(φi = ψi + 2πki) = −λi,does not depend on ki, then the unwrapping optimization problem consists inminimizing the prior term of (4) with respect to φ ∈

∏i∈V{ψi+2πki : ki ∈ Z}.

In our previous works [24, 30], we have introduced a descent method that, de-pending on the potentials, yields exact (in the case of convex potentials) orapproximated (in the case of nonconvex potentials) solutions. Each step of thismethod solves a binary problem by computing the min-cut of an appropriategraph.

We note that the basic phase unwrapping step can be viewed as a discretiza-tion of the original domain, using a sampling interval of 2π. A consequence ofthis discretization is that the resulting objective function is easier to deal withsince, for φ ∈

∏i∈V{ψi + 2πki : ki ∈ Z}, it does not depend on the nonconvex

data terms −λi cos(φi − ψi).

After the unwrapping step, we get absolute phase estimates given by φi =ψi + 2πki. Even if the integer image estimate is exact, we still have error inφi due to the noise present in ψi [please see expression (1)]. In order to filterout the noise, we compute a sequence of binary descent optimizations usinga multi-precision schedule. The precision q ∈ {0, 1, . . . , N}, corresponds to a

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sampling interval of ∆ = 2π/2q. Thus, the coarser precision implements phaseunwrapping and the following denoising.

We highlight the following qualitative characteristics of the approach justdescribed:

1. The PUMA algorithm [24], used for phase unwrapping, is able to dealwith discontinuities and implicitly locate them.

2. The PUMA solution yields an error much smaller than π in magnitude inmost of the pixels.

3. Given that for precisions q > 0, we have, for most i ∈ V , |φi − ψi| � π,then most of the unary terms −λi cos(φi−ψi) behave as convex functions,rendering a much easier optimization problem.

One expectable advantage of coarse-to-fine multi-precision schedule is com-putation time. However, in our scenario this is not, perhaps, the most importantfeature: owing to the nonconvexity of the problem, the algorithm would have,very likely, got stuck in a local minimum if we had begun with the highestprecision.

2.3 Algorithm

Algorithm 1 shows the pseudo-code for the optimization, where we use thefollowing sets:

MU (φ′,∆) ≡{φ ∈ R

|V| : φi = φ′i + δi∆}

MD(φ′,∆) ≡{φ ∈ R

|V| : φi = φ′i − δi∆},

where δi ∈ {0, 1}, ∆ ∈ R.Our algorithm engages on a greedy succession of up and down binary op-

timizations. The precision interval ∆ starts with the value 2π and ends withthe value 2π/(2N), where N is a depth of precision. In order to characterizethe algorithm, we start assuming E(φ) = E(φ) and the terms Di and Vi,j inexpression (3) to be convex. Then, for each precision interval ∆, the algorithm1 finds, in finite time, a minimizer of E in a grid of size ∆. More precisely, wehave

Theorem 1 If the unary and pairwise terms of E, defined in (3), respectively,Di(·), for i ∈ V, and Vij(·), for (i, j) ∈ E, are convex, then, at a given precisioninterval ∆ = 2π/2q, for q ∈ {0, . . . , N}, the following holds:

1. The output of the loop “while” is a minimizer of E(φ) in the set{φ ∈ R

|V| : φi = ψi + zi∆, zi ∈ Z}.

2. The number of “while” iterations at each precision q ∈ {0, . . . , N} isbounded by Kq + 1, where Kq = K02

q is the range of the 2π multiplesvariables at precision q.

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Algorithm 1 Combinatorial absolute phase estimation.

Initialization: φ = ψ {Interferogram}, successup = false, successdown = false1: for ∆ = 2π ×

{20, 2−1, . . . , 2−N

}do

2: while (successup = false OR successdown = false) do

3: if successup = false then

4: φ = arg minφ∈MU (φ,∆) E(φ)

5: if E(φ) < E(φ) then

6: φ = φ7: else

8: successup = true9: end if

10: end if

11: if successdown = false then

12: φ = arg minφ∈MD(φ,∆) E(φ)

13: if E(φ) < E(φ) then

14: φ = φ15: else

16: successdown = true17: end if

18: end if

19: end while

20: end for

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Proof: For the precision interval ∆, the pseudo code embraced by the“while” loop (lines between 2 and 19) finds a minimizer ofE(φ) in

{φ ∈ R|V| : φi = φ′ + zi∆, zi ∈ Z

},

where φ′ is the minimizer obtained in the previous precision. Proofs are givenin [24], in the absence of unary terms, and in [32], [34], for the general case, i.e.,when E contains unary and pairwise terms.

Since the successive precisions are powers of (1/2), we can write φ′i = ψi +li∆, where φ′i and ψi are the ith components of φ′ and ψ, respectively, and liis an integer. Therefore, it follows that

{φ ∈ R|V| : φi = φ′ + zi∆, zi ∈ Z

}={

φ ∈ R|V| : φi = ψi + zi∆, zi ∈ Z}. �

As already pointed out, we solve the binary minimizations displayed in lines 5and 14 of Algorithm 1 by computing max-flows/min-cuts on appropriate graphs.Denoting by T (n,m) the time complexity of the used max-flow/min-cut algo-rithm, where n is the number of graph nodes and m the number of edges, thenAlgorithm 1 takes the pseudo-polynomial time O (KqT (n,m)) to find a mini-mum of E at the precision q.

The rationale underlying the multi-precision minimization is that of a min-imum length search for a minimizer of E. Still considering the convex sce-nario, given a minimizer at a precision q, say φq, there exists a minimizer atthe precision q + 1 such that ‖φq − φq+1‖∞ < n [35, Theorem 7.18], where‖x‖∞ ≡ maxi xi is the l∞-norm of x. Therefore, the algorithm takes at mostn iterations to find a minimizer at resolution q + 1 [24, 33, 34]. Consequently,the number of iterations to find a minimizer of E is bounded by O(n logKN ).In practice, we have observed systematically ‖φq − φq+1‖∞ � n, and very of-ten we have ‖φq − φq+1‖∞ < 2, making the algorithm highly efficient fromthe time complexity point of view. This behavior is illustrated in Fig. 4 (c),where we show the evolution of Algorithm 1 in a convex scenario, both with andwithout multi-precision. Notice that in the former case the number of “up anddown”iterations (lines 6 and 14 of Algorithm 1) to find a minimum, in a givenprecision, is 2 × (Kq + 1).

To solve the binary optimizations shown in lines 4 and 12, we use graph cutstechniques. The main idea, which was introduced into computer vision in [46],and popularized in [36, 38, 47, 48], is a one-to-one mapping of a submodularbinary function onto the set of cuts of a certain source-terminal graph. Wefollow the mapping proposed in [38]. As referred at the very beginning of thissection, to each pixel of the image to infer there corresponds a node. Each nodeis linked, through edges, with each neighbor (either vertical or horizontal in ourcase) and, furthermore, to the source (s) and/or to the sink (t) (the edges alwayshave either positive or null weights, the latter corresponding to non existenceof the edge). An s− t cut C = (S, T ) is a partition of the set of vertices V intotwo disjoint sets S and T , such that s ∈ S and t ∈ T . The cost of the cut is thesum of costs of all edges between S and T . A min-cut is a cut of minimum cost.After computing an s− t min-cut, each node v ends having the binary value 1if v ∈ S or binary value 0 otherwise.

Fig. 2 illustrates the building of such a graph. Given that the binary func-

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Source

Sink

v

v'

v v'

s

t

E(0,1)+E(1,0)-E(0,0)-E(1,1)

E(1,0)-E(0,0)

E(1,0)-E(1,1)

Cut

s

t

(a) (b)

Source

Sink

S

T

E(1)-E(0)

Figure 2: (a) Elementary graph representing a unary and a pairwise interactionenergy terms. Nodes s and t represent source and sink, respectively, and v andv′ represent the two pixels involved in the interaction; pixel v also has a unaryterm attached to it. In this case E(1) − E(0) > 0, E(1, 0) − E(0, 0) > 0 andE(1, 0) − E(1, 1) > 0. We note that the two edges linking pixels s and t areequivalent to a single edge with weight E(1) − E(0) + E(1, 0) − E(0, 0) (b)Illustration of the graph obtained at the end: it results from adding elementarygraphs.

tion is submodular, the exact global minimum of the function is attained [38].According to the Ford-Fulkerson theorem [49], the min-cut is equal to the max-flow. As there are plenty of efficient low-order polynomial complexity algorithmsto compute min cuts/max flows, it turns out that each binary problem is effi-ciently solved.

In our case, neither the unary nor the pairwise energy (3) interaction terms,Di(φi) and Vi,j(φi − φj), respectively, are convex. First of all, this means thatwe cannot map the binary optimizations discussed above into min-cuts onto ap-propriate graphs, because the resulting binary energy is non-submodular [38];moreover, we do not meet the conditions in which Theorem 1 applies. We ad-dress this problem by approximating energy E(φ) with E(φ). This approxima-tion consists in replacing the non-submodular pairwise terms Vi,j by submodularmajorizers. Then we apply the majorize-minimize (MM) principle [39]. Fig. 3illustrates the process of finding the majorizer.

We note that although we employ the approximation E(φ) to compute theoptimizations, the evaluation of descending energy in lines 5 and 13 of Algorithm1 is made through E(φ), the true energy.

We now emphasize two facts:

• For the first precision (∆ = 2π) there is no unary term in the energy (3)E(φ). This means that with E we are in a convex scenario.

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Figure 3: Replacing nonsubmodular energy terms by submodular ones; we end-up with an approximate energy. One of the possible approximations is to in-crease Eij(0, 1).

• For the subsequent precisions the nonconvex sinusoidal unary term in Eis not null; however, for these small precisions, we are likely to be in aconvex attraction basin of E(φ) because, as already referred in Section2.2.1, |φi − ψi| � π for most of the unary terms.

Fig. 4 illustrates the virtues of employing a multi-precision approach. In thefirst example [(a), (b), (c)], we intend to illustrate the pure effect of algorithm’sspeed enhancing. For that, we consider a convex energy (3) for which we setDi(zi) = |φi−ψi|

2, instead of the sinusoidal nonconvex observation data model,and V (x) = |x| for the pairwise interaction, discontinuity preserving, potential.In this example, ψ plays the role of a noisy observation of φ in a Gaussianadditive model. Fig. 4 (a) shows an image that corresponds to a discretizedpyramid with additive Gaussian noise (σ = 1). Fig. 4 (b) shows the imagein (a) denoised by applying algorithm 1; the result is very good. Fig. 4 (c)illustrates the energy decreasing versus iterations. The curve with marks on itrepresents the energy evolution by using multi-precision; each mark correspondsto a change of precision. The other curve corresponds to a performance ofthe algorithm in the finest precision from the beginning. It can be seen thatboth ways we end up with the same energy (and in fact the same denoisedimage), as expected by using a convex energy function; however, multi-precisionturns the algorithm much faster. In the second example [(d), (e), (f), (g)], weillustrate the absolute phase estimation using energy (3) as almost elsewhere inthe paper. Fig. 4 (d) shows the wrapped phase corresponding to a true phasegiven by a clipped Gaussian with 14π rad height. (e) shows a perfect unwrappingobtained by algorithm 1 using the multi-precision approach. (f) displays the

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completely failed unwrapping that one obtains by running the algorithm withthe finest precision from the very beginning. Finally, Fig. 4 (g) displays theenergy evolutions either with and without multi-precision; the curve with markson it corresponds to the multi-precision run (each mark represents a precisionchange). These plots illustrate that multi-precision avoids poor energy localminima as the one obtained without multi-precision. Again multi-precisionenhances the speed of the algorithm.

In the next section, we present a series of experiments confirming the effec-tiveness of the proposed approach in absolute phase estimation.

3 Experimental Results

In this section, we present four experiments illustrating the performance ofour algorithm: the first three concern synthetic data; the fourth deals withInSAR data distributed with [12], which is a commonly used benchmark toscore absolute phase estimation algorithms. In all the experiments, we haveemployed a depth of precision (see Algorithm 1) N = 8, which gives a minimumprecision interval of 2π/28 ' 2.5 × 10−3. The regularization parameter µ [see(4)] and the exponent p [see (5)] of the half quadratic discontinuity preservingpotential were hand tuned for the best performance. We employ the definitionof improvement in signal to noise ratio introduced in [25] as

ISNR = 10 log 10

∥∥ejφ − ejψ∥∥2

∥∥∥ejφ − ejψ∥∥∥

2 ,

where φ is the true absolute phase, ψ the wrapped phase and φ the estimatedabsolute phase.

We note that in all the experiments, due to discontinuities (original phasedifferences with magnitude greater than π), noise or aliasing (due, e.g., to sub-sampling), there is a huge number of neighboring pixels having phase differencesgreater than π in magnitude, making the absolute phase estimation a very dif-ficult problem.

3.1 Synthetic data

3.1.1 Sheared ramp

For this experiment, we set the prior parameter µ = 0.4 and the potential withexponent p = 0.4. Fig. 5 (a) displays an image (150×100 pixels) correspondingto an absolute phase surface formed by two equal sized planes with slopes,respectively, of 0 and 1 rad/pixel (maximum height is 99 rad); (b) displaysthe image shown in (a) wrapped, and with a noise having root mean squareerror RMSE = 0.7324 rad; (c) shows the image in (b) unwrapped and denoised:RMSE = 0.2219 rad. For this experience we have ISNR = 11.0 dB. By usingPUMA to do phase unwrapping it can be observed that with the evolution of

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Figure 4: Fig. 4 (a) Discretized pyramid with additive Gaussian noise (σ = 1).(b) Image in (a) denoised by applying algorithm 1 with convex energy. (c) En-ergy decreasing versus iterations: marks correspond to increase of precision; thecurve without marks corresponds to use the finest precision from the beginning.(d) Wrapped phase corresponding to a true phase given by a clipped Gaussianwith 14π rad height. (e) Perfect unwrapping obtained by algorithm 1 using themulti-precision approach. (f) Completely failed unwrapping obtained by run-ning the algorithm with the finest precision from the very beginning. (g) Energyevolutions either with and without multi-precision; the curve with marks on itcorresponds to the multi-precision run.

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the algorithm (phase unwrapping steps) the discontinuities (original neighboringpixels having phase differences greater than π rad in magnitude) progressivelyappear. Obviously this is not an error free process in the sense that, e.g.,noise or high phase rate may be wrongly signaled as discontinuities. Fig. 5 (h)displays the discontinuities that the unwrapping step is able to blindly detect, asmentioned in Section 1.1. Fig. 5 (d) shows the re-wrapped image shown in (c).We emphasize that the shear discontinuity plus the noise turns this into a hardtask; it is noticeable the good unwrapping and denoising. (e) is a 3D renderingof image shown in (c); (f) is a corresponding 3D rendering after unwrappingand before denoising. Fig. 5 (g) shows the descending objective function alongthe iterations of the minimization algorithm.

3.1.2 Partial Gaussian

For this experiment, we used prior parameter µ = 0.4 and the potential withexponent p = 0.5. The Fig. 6 is analogous to Fig. 5 but now the originalabsolute phase surface is a 150 × 100 pixels sized Gaussian elevation with aheight of 14π rad and standard deviations σ = 15 pixels (horizontally) andσ = 10 pixels (vertically); additionally, in a quarter of the plane the Gaussianhas zero height introducing surface discontinuities. We stress that this is ahard absolute phase estimation problem given these discontinuities plus thenoise. The noise introduced in (b) corresponds to RMSE = 0.5005 rad, and (c)exhibits an almost perfect unwrapping and RMSE = 0.1753 rad. This denoisingcorresponds to ISNR = 9.1 dB. We emphasize that the quarter of the Gaussianhaving zero height, plus the noise, introduces a lot of discontinuities, which theunwrapping algorithm is able to detect (the algorithm also marks other pixelsas discontinuities due to noise plus high phase rate effects) as shown in (h). Thedenoising effect is quite evident in (d). The 3D rendering of Figs. 6 (e) and 6(f)illustrates the unwrapping and unwrapping plus denoising effects respectively;in (g) is shown the evolution of the objective function along with the iterations.

We again mention that binary optimizations such as those employed in lines4 and 12 of Algorithm 1 are demanding not only because they handle objectivefunctions with a huge (typical of usual phase unwrapping problems) numberof discrete variables, but also because those functions have, generally, non-submodular terms (see section 1.1). Roof duality [40] is a technique speciallydeveloped to handle such functions. It yielded, recently, promising results incomputer vision, with the so-called quadratic pseudo boolean optimization al-gorithms as described in e.g. [50]. For non-submodular functions these are sub-optimal algorithms as the problem is NP-hard. Being so, we have also appliedsuch algorithms to handle the binary minimizations. In a series of experimentsour technique was similar regarding accuracy, however always (significantly)faster. To illustrate this, Table 1 shows numerical results regarding the unwrap-ping of an image as the one shown in Fig. 6 (b), by both our algorithm andQPBOP [50].

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Figure 5: (a) Sheared ramp image (99 rad height) (b) Wrapped and noisyimage shown in (a) (RMSE = 0.7324 rad). (c) Image in (b) unwrapped anddenoised by our algorithm (RMSE = 0.2219 rad). (d) Image in (c) re-wrapped.(e) 3D rendering of image in (c). (f) 3D rendering of the image in (c) beforedenoising. (g) objective function evolution along the iterations of the algorithm.(h) discontinuities blindly detected by the algorithm.

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Figure 6: (a) Gaussian image with 14π maximum height and with a quarterset to zero. (b) Wrapped and noisy image shown in (a) (RMSE = 0.5005 rad).(c) Image in (b) unwrapped and denoised by our algorithm (RMSE = 0.1753rad). (d) Image in (c) re-wrapped. (e) 3D rendering of image in (c). (f) 3Drendering of the image in (c) before denoising. (g) objective function evolutionalong the iterations of the algorithm. (h) discontinuities blindly detected by thealgorithm.

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Table 1: Comparative performance of the proposed algorithm and QPBOP.no. of non-submodular terms

Iteration Our algorithm QPBOP1 590 5902 410 3263 271 2634 179 1545 141 1236 117 947 91 888 57 57

Time 1s 120s

3.1.3 Gaussian

For this experiment we used prior parameter µ = 0.4 and the potential withexponent p = 1.7. We found out better results with such a relatively high ex-ponent p (when comparing with the p used in the last presented results); thiscan be explained by the fact that the original surface does not have disconti-nuities, although it has high phase rates that may create problems when noiseis added). Fig. 7 is as Fig. 6 but now the original absolute phase surface is a,100×100 pixels sized, Gaussian elevation with a height of 14π rad and standarddeviations σ = 15 pixels (horizontally) and σ = 10 pixels (vertically). The noiseintroduced in (b) corresponds to RMSE = 0.6187 rad, and (c) exhibits an al-most perfect unwrapping and RMSE = 0.2149 rad. This denoising correspondsto ISNR = 8.3dB and is quite evident in (d) where we show the rewrapped de-noised image. The 3D rendering of Figs. 7 (e) and (f) illustrate the unwrapping[(f)] and unwrapping plus denoising effects [(e)]. In (g) we show the evolutionof the objective function along with the iterations. We emphasize that this ab-solute phase estimation experiment is presented as the first experiment in [25](there employed with a there called σ = 0.5 noise). It can be seen that ouralgorithm is able to make denoising similar to the state-of-the-art, ZπM andPEARLS algorithms.

3.2 Real data

Finally, we illustrate the performance of the algorithm on a (152 × 458 pixels)InSAR image. We have employed a prior parameter µ = 0.2, and a quadraticpotential p = 2. Fig. 8 (a) displays an image corresponding to an absolutephase surface generated by a (simulated) InSAR acquisition for a real steep-relief area1, thus inducing many discontinuities and posing a very hard absolutephase estimation problem. Fig. 8 (b) displays a corresponding wrapped and

1Long’s Peak, Colorado, USA. Data distributed with book [12].

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Figure 7: (a) Gaussian image with 14π maximum height. (b) Wrapped andnoisy image shown in (a) [RMSE = 0.6187 rad]. (c) Image in (b) unwrappedand denoised by our algorithm (RMSE = 0.2149 rad). (d) Image in (c) re-wrapped. (e) 3D rendering of image in (c). (f) 3D rendering of the image in(c) before denoising. (g) objective function evolution along the iterations of thealgorithm.

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noisy image. In some areas the characteristic fringes are destroyed due to typicalphenomena as shadowing and layover (see, e.g., [12]) The noise is given byRMSE = 0.28 rad. Fig. 8 (e) shows a quality map that is an input to thealgorithm: white color corresponds to pixels whose phase value is meaninglessand gray corresponds to the rest of the pixels. This information accounts forjamming phenomena such as, the above mentioned, layover and shadowing. Ouralgorithm is able to screen those white pixels from the absolute phase estimationprocess (therefore those pixels do not “contaminate” the results for the rest ofthe image), and this is the reason why we employ a non-discontinuity preservingquadratic potential. Obviously the error values here presented refer only to grayimage areas on the quality map. Fig. 8 (c) shows the unwrapped resulting image,with RMSE = 0.18 rad corresponding to ISNR = 3.8dB. Again comparing withthe results mentioned in [25] our algorithm outperforms ZπM (reported to haveRMSE = 0.20 rad in this case) and gets very close to PEARLS (which is reportedto have RMSE = 0.17 rad in this case). Fig. 8 (d) displays the image in (c)rewrapped. Comparing it with image shown in (b), it is apparent the denoisingeffect; this is quite clear comparing the zoomed patches in (b) and (d). Fig. 8(f) displays a 3D rendering of the image in (c), and (g) displays the energy (4)evolution along the iterations within the algorithm performance.

Finally we remark that all the illustrated experiments have been run in a2.2 GHz Intel dual core processor, in a maximum of few dozens of seconds2.

4 Concluding Remarks

We have introduced a (discontinuity preserving) denoising Bayesian approachto absolute phase estimation in interferometric applications. Among the sci-entific community there is still an alive debate of whether denoising should bedone after phase unwrapping, before phase unwrapping, or any other solutionin-between. In this paper we have chosen the first option in order to avoidthe denoising step to corrupt the phase unwrapping process. The - graph cutsbased - proposed algorithm first makes phase unwrapping, with discontinuitypreserving capabilities; then, basically using the same rationale, but with amulti-precision scheduling technique, it achieves denoising. In the experimen-tal results the simplicity of the algorithm translates into fast execution and,furthermore, high accuracy. We emphasize that a short running time can beessential for some applications (e.g., some medical applications in magnetic res-onance imaging). A further step to speed up the algorithm may be obtained inthe future by using dynamic graph cuts [51].

The obtained experimental results for absolute phase estimation have, toour knowledge, at least almost state-of-the-art accuracy; the running time istypically of few seconds in a 2.2 GHz Intel dual core for images of size 150×100,and experimentally found out to grow linearly with image size. Theoretically,the worst-case complexity is O(kn2m) (see [24]) where n = |V| and m = |E|,

2We should note that, even so, the code is a mix of matlab and C++, therefore is notoptimized.

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Figure 8: (a) Absolute phase gray level image generated by a (simulated) InSARacquisition for a real steep-relief area. (b) Wrapped and noisy image shown in(a). (c) Image in (b) unwrapped and denoised by our algorithm. (d) Imagein (c) re-wrapped. (e) Quality map; white pixels phase is meaningless andare screened by the algorithm. (f) Energy along the iterative evolution of thealgorithm.

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respectively the number of nodes and edges of the graph that our algorithmdeals with, and k is the excursion of the 2π labels of the nodes in the absolutephase. We emphasize that our algorithm, in a set of representative experiments,turns out to be much more faster than recent QPBOP [25] algorithm in dealingwith non-submodular terms.

In the future, we intend to explore diversity techniques (e.g., chinese remain-der theorem) applied to absolute phase estimation, in order to extend the setof the phase images that we can unwrap (namely images for which the abso-lute phase exhibits high rates); we should note that dealing with noise in sucha scenario turns out to be quite more challenging. We also intend to employhigher order Markov random fields; this is an effort to avoid the metricationartifacts well known to be associated with graph-cuts based techniques [52]. Inaddition we intend to develop learning schemes for the selection of the best priorparameter µ, given some absolute phase estimation problem.

Acknowledgments

This work was supported by Fundacao para a Ciencia e Tecnologia under grantSFRH / BD / 25514 / 2005.

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