DERIVATIVE COMPRESSIVE SAMPLING WITH APPLICATION TO PHASE UNWRAPPING

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1 17th European Signal Proessing Conferene (EUSIPCO 2009) Glasgow, Sotland, August 24-28, 2009 DERIVATIVE COMPRESSIVE SAMPLING WITH APPLICATION TO PHASE UNWRAPPING Mahdi S. Hosseini and Oleg V. Mihailovih Department of ECE, University of Waterloo 200 University Avenue West, Waterloo, N2L 3G1 Ontario, Canada s: {smhossei, ABSTRACT Reonstrution of multidimensional signals from the samples of their partial derivatives is known to be an important problem in imaging sienes, with its fields of appliation inluding optis, interferometry, omputer vision, and remote sensing, just to name a few. Due to the nature of the derivative operator, the above reonstrution problem is generally regarded as ill-posed, whih suggests the neessity of using some a priori onstraints to render its solution unique and stable. The ill-posed nature of the problem, however, beomes muh more onspiuous when the set of data derivatives ours to be inomplete. In this ase, a plausible solution to the problem seems to be provided by the theory of ompressive sampling, whih looks for solutions that fit the measurements on one hand, and have the sparsest possible representation in a predefined basis, on the other hand. One of the most important questions to be addressed in suh a ase would be of how muh inomplete the data is allowed to be for the reonstrution to remain useful. With this question in mind, the present note proposes a way to augment the standard onstraints of ompressive sampling by additional onstraints related to some natural properties of the partial derivatives. It is shown that the resulting sheme of derivative ompressive sampling (DCS) is apable of reliably reovering the signals of interest from muh fewer data samples as ompared to the standard CS. As an example appliation, the problem of phase unwrapping is disussed. 1. INTRODUCTION Numerous appliations are known in whih one is provided with the measurements of the gradient of a multidimensional signal, rather than of the signal itself. Central to suh appliations, therefore, appears the problem of reonstrution of signals from their partial derivatives subjet to some a priori onstraints (whih ould be either probabilisti or deterministi in nature) [1]. One of suh appliations, whih has been hosen to exemplify the major ontribution of this note, is the problem of phase unwrapping. Note that solving this problem is known to be a standard proedure in, e.g., optial and synthethi aperture radar (SAR) interferometry [2], stereo vision [1], blind deonvolution [3, 4], et. In order to speify the problem of phase unwrapping, let F(x, y) be an arbitrary ontinuously differentiable funtion defined over a losed subset of the real plane R 2. If F happens to be the phase of a omplex-valued funtion, it an only be measured in its wrapped form, i.e. modulo 2π. Formally, the proess of phase wrapping an be represented by its assoiated operator W : R 2 ( π,π]. In this notation, the wrapped prinipal phase R is given as R = W [F]. Speifially, the operator W adds to F a pieewise-onstant funtion K : R 2 {2π k} k Z resulting in R = W [F] = F +K that obeys [5]: π < W[F (x,y)] π, (x,y) R 2. (1) In omplex notation, the gradients of F and R an be defined as F = F x 1 i + F x 2 j K = K x 1 i + K x 2 j, (2) where i and j denote the unit vetors assoiated with the x- and y-axis, respetively. Consequently, omputing the gradient of the wrapped phase R using equations (2) yields R = W[F] = F + K. (3) Finally, applying the wrapping operator W one more time to both sides of (3) results in W[ W[F]] = W[ R] = F + K + K. (4) Due the property of operator W to produe the values in interval [ π,π], the term K + K vanishes as long as [2] π < F π. (5) Therefore, as long as the ondition (5) above holds, the gradient of the original phase F an be unambiguously reovered from the gradient of the orresponding prinipal phase R aording to F = W[ R]. (6) From the above onsiderations it follows that, if the ondition (5) was known to hold, given a measured R, an estimate ˆF of the original phase F ould be obtained as a solution to the following optimization problem ˆF = argmin F F W{ R} 2 dxdy, (7) whih amounts to solving a Poisson equation subjet to appropriate boundary onditions. Unfortunately, situations are rare in whih the ondition (5) an be a priori guaranteed. In this ase, the estimate of F as W[ R] is ontaminated by, so alled, residuals, whih ause the solution of (7) to be of little pratial value. In order to overome the limitations of phase unwrapping inflited by using the gradients estimated aording to (6), EURASIP,

2 a magnitude of different approahes has been hitherto proposed [2]. In the urrent note, we introdue a different solution to the problem whih is based on the onepts of the theory of ompressive sampling [5, 6, 7, 8, 9, 10]. In partiular, let Γ R 2 be a finite disrete subset over whih the values of F need to be reovered. Let further Γ 0 denote a subset of those points in Γ at whih the ondition (5) is known to hold, and hene at whih the gradient F estimated aording to (6) an be assumed to be errorless. (Note that the subset Γ 0 an be identified based on analysis of the field of residuals as detailed in [2]). Subsequently, we first reover the values of F over the whole Γ from its inomplete measurements over Γ 0, followed by estimating the original phase F using (6). Moreover, in addition to the standard onstraints of ompressive sampling, we propose to use the onstraints stemming from the nature of the gradient as a potential field, viz. F(x,y) x y = F(x,y) y x. (8) We will refer to the problem of reonstrution of F from { F(x,y} (x,y) Γ0 as the problem of derivative ompressive sampling (DCS), and show that using (8) allows onsiderably reduing the ardinality of Γ 0, while preserving a predefined error rate. The remainder of this paper is organized as follows. Setion 2 provides an overview of ompressive sampling, and shows how this theory an be used for solution of the problem of phase unwrapping. In Setion 3, some neessary tehnial details are speified. The performane of the method is analyzed in Setion 4, while Setion 5 finalizes the paper with a disussion and an outline of our future researh diretions. 2. DERIVATIVE COMPRESSIVE SAMPLING The theory of ompressive sampling addresses the problem of perfet reonstrution of signals of interest from their subritially sampled measurements [5, 6, 7, 8, 9, 10, 11]. In the ase when inomplete measurements of the derivatives of the signals are available, the resulting reonstrution problem is refereed to as derivative ompressive sampling. 2.1 Basis of Compressive Sampling The idea of ompressive sampling was first formulated by D. Donoho [6] in the form of the, so alled, generalized unertainty priniple. In this initial setup, a bandlimitted signal f (t) L 2 (R) in used for transmission over a hannel, in whih it loses its values on a subset T. Formally, one an define r (t) = (I P T ) f (t) + n(t), (9) where I denotes the identity operator, n(t) is observation noise, and P T denotes the spatial limiting operator of the form { f (t), t T P T f (t) = (10) 0, otherwise The seond operator used in [6] is a band-limiting operator defined as given by P Ω f (t) ˆf (ω)e 2πıωt dω, (11) where ˆf (ω) denotes the Fourier transform of f (t). Ω The main goal of ompressive sampling is to reonstrut the transmitted signal f from the noisy reieved signal r. The possibility of suh a reovery is assured by Theorems 2 & 4 in [6] asserting that if Ω T < 1 (with T being the omplement of T ) there exists a linear operator Q and a onstant p suh that f Q[r] p n, (12) ( where p 1 1. T Ω ) Speifially, the reonstrution operator Q is given by Q = (I P T P Ω ) 1 = k=0 (P T P Ω ) k. (13) Moreover, the resulting solution is unique, and it an be approximated by trunating the Neumann series in (13) at some finite k = N. An addition impetus to the theory of ompressive sampling has been given in [7, 8, 9] via introduing the onept of two orthonormal bases Φ and Ψ of R n, whih are used for sampling and signal representation, respetively. Moreover, entral to the modern theory of ompressive sampling are the notions of Sparsity, in the sense that signals an be represented by a relatively small number of non-zero oeffiients in a properly hosen Ψ, and Inoherene, whih represents the duality between the sampling Φ and representing Ψ domains, where the ohereny µ(φ,ψ) = n max Φ T Ψ (14) remains low. Note that here n is the number of vetors in the basis. In its typial setting, ompressive sampling refers to the ase when an n-dimensional signals f has to be reovered from its m measurements y k = f,φ k, k M {1,...,n} (15) where m = #M and m < n. Let Φ M be the n m matrix whose olumns are formed by those φ k for whih k M. Then, assuming that the signal f an be represented as f = Ψ, for some oeffiient vetor R n, the reonstrution is arried out via solving min 1 = n k=1 k, s.t. Φ T MΨ = y, (16) where y R m stands the vetor of m measurements in (15). The above problem an be solved by means of linear programming whih, among all solutions obeying the measurement onstraint Φ T MΨ = y, piks the one that has the sparsest representation in the domain of Ψ as measured by the l 1 -norm of. Moreover, Theorem 1.2 derived in [10] defines a bound on the number of measurements m m C µ 2 (Φ,Ψ)S logn (17) for whih perfet reovery is possible. Note that in (17), C is a onstant, while S denotes the number of non-zero elements in Ψ T f. 116

3 2.2 Reonstrution from Partial Derivatives In the ase when only partial derivatives of a signal of interest are available, the sampling operator of ompressive sampling beomes the kernel of a derivative operator. In partiular, in the 2-D ase, we are given the measurements of F x = F/ x and F y = F/ y. At this point, there are two possibilities to find F. The first would be to define Φ to be a disretized version of the 1st-order derivative operator. This hoie, however, ould result in relatively large values of the ohereny µ(φ,ψ) for the ase when Ψ is a wavelet orthobasis (whih is the hoie in the present study). This would, in turn, inrease the bound in (17), whih ould be unaeptable for pratial onsiderations. On the other hand, one an define Φ to be the Dira omb (i.e., Φ = I). In this ase, the partial derivatives an be reovered first, followed by integrating the latter using (7). To proeed with the seond of the above-mentioned possibilities, we turn to a disrete setup in whih F, F x and F y are onsidered to be n n matries. In this ase, the maximal possible number of measurements is equal to 2n 2, and hene M {1,2,...,2n 2 }. Speifially, we are interested in the ase when m = #M < n 2. In 2-D, the partial derivatives F x and F y an be approximated aording to F x FD F y D T F, (18) where D is 2-D differene matrix D = (19) (Note that the last olumn of D defines the boundary ondition.) For the sake of notational simpliity, let Φ = Φ Φ and Ψ = Ψ Ψ, where stands for the Kroneker matrix produt. Moreover, sine the sampling sets for the x- and y-derivatives may be in general different, we denote the orresponding sampling matries by Φ x and Φ y, respetively. Hene, assuming that there exist oeffiients x and y suh that ve(f x ) = Ψ ve( x ) and ve(f y ) = Ψ ve( y ) (with ve denoting the operation of matrix onatenation), the measurement onstraints of the DCS problem are defined as Φ x Ψ ve( x ) = G x Φ y Ψ ve( y ) = G y, (20) where G x and G y are the vetors of measured derivatives. In what follows, the onstraints in (20) will be referred to as primary. On the other hand, the ross-derivative (seondary) onstraints in (8) an now be expressed as { x Ψ y Ψ T } F y { = y Ψ x Ψ T }. (21) F x Figure 1: Cirular integration paths involving the (i, j) pixel. Using some standard rules of the matrix alulus [12], the above equation an be rewritten as [ ( D T Ψ ) ( Ψ Ψ D T Ψ ) ] [ ] ve(x ) = 0 (22) ve( B x B y y ) The matrix B = [B x, B y ] is a full (row) rank matrix, whose ondition number is approximately equal to 5n/4. In the DCS formulation, this matrix of seondary (rossderivative) onstraints is ombined with the primary onstraints to result in the following optimization problem subjet to min 1 = n 2 k x (k) + n 2 k Φ x Ψ 0 [ ] 0 Φ y Ψ Gx = Φ B x Φ G B y y y (k), (23) where Φ denotes a sub-sampling operator whih removes from the seondary onstraints (21) those whih are linearly dependent on the primary onstraints (20) (see below). Finally, having estimated the partial derivatives F x and F y as Ψ x and Ψ y, respetively, we reover the funtion (phase) F as a solution to (7). 3. RECOVERING PRIMARY CONSTRAINTS FROM SECONDARY CONSTRAITNS Sine the gradient F is a potential field, its integral over any losed path in R 2 should be equal to zero, namely F(s)ds = 0. (24) Moreover, in the disrete ase, the shortest of suh paths onnets eah 4-pixels neighborhood, as shown in Fig. 1. The shown integration paths result in the following set of equations q 1 : F y (i, j) = F y (i + 1, j) + F x (i, j) F x (i, j + 1) q 2 : F y (i, j) = F y (i 1, j) + F x (i 1, j) F x (i 1, j) q 1 : F x (i, j) = F x (i, j + 1) + F y (i, j) F y (i + 1, j) q 3 : F x (i, j) = F x (i, j 1) + F y (i + 1, j 1) F y (i, j 1). (25) The onstraint equations in (25) an be used to enlarge the set of primary onstraints through reovering some of unknown values of F x and F y from those of their known neighbors. This proedure an be implemented using Algorithm 1 provided below. 117

4 Algorithm 1 Optimal reovery of primary onstraint while A pixel an be reovered do for i, j = 1 to n do if all 3 elements in q 1 & q 2 are knwon F y (i, j) = (q 1 + q 2 )/2 else if all 3 elements in q 1 are knwon F y (i, j) = q 1 else if all 3 elements in q 2 are knwon F y (i, j) = q 2 if all 3 elements in q 1 & q 3 are knwon F x (i, j) = (q 1 + q 3 )/2 else if all 3 elements in q 1 are knwon F x (i, j) = q 1 else if all 3 elements in q 3 are knwon F x (i, j) = q 3 end for end while Algorithm 2 Elimination of dependent rows of B for i, j = 1 to n do if F y (i, j): known AND F x (i, j): known if F y (i + 1, j: known OR F x (i, j + 1: known else if F y (i, j): known OR F x (i, j): known if F y (i + 1, j): known AND F x (i, j + 1): known else if F y (i, j): unknown AND F x (i, j): unknown if F y (i + 1, j): known AND F x (i, j + 1): known end for (a) (b) Figure 3: (a) Original phase; (b) Wrapped phase. Figure 2: Reovering the indies of primary onstraint for test images. Performing Algorithm 1 is a ritial step as it maximizes the ardinality of the set of primary onstraints, thereby improving the overall probability of reovering the true gradient. Typially, five iterations of the algorithm are suffiient to omplete the task. Fig. 2 provides a quantitative harateristis of the algorithm whih have been averaged over a number of test images. Finally, in order to exlude any linear dependeny between the primary and seondary onstraints, the matrix Φ in (23) should be identified. However, in pratie, instead of finding the matrix, we simply remove the rows of B whih are linearly dependent on the primary onstraints. This an be done using Algorithm 2. It should be pointed out that while Algorithm 1 maximizes the number of primary onstraints, Algorithm 2 guarantees that the onstraint matrix in (23) is of full row rank. After the exeution of both algorithms, the problem of reovering the (sparse) representation oeffiients an be solved by linear programming. However, in the 2-D ase, suh a solution ould be rather impratial onsidering the size of signals under onsideration. To alleviate the omputational burden, in the urrent work, the sparse solutions have been found using the algorithm detailed in [13]. 4. RESULTS We demonstrate the performane of our algorithm using a fringe pattern from the fringe phase data [14]. The original phase and its wrapped version are shown in Figure (3(a)) and (3(b)), respetively. Throughout the experimental study, Ψ was defined to be an orthogonal basis matrix orresponding to the nearly symmetri wavelet of I. Daubehies having six vanishing moments. In our first numerial experiment, we ompared the performane of the DCS algorithm with that of the standard CS method. Note that the latter an be obtained from the former by simply disarding the ross-derivative onstraints (21). Through analyzing the field of residuals orresponding to the estimated phase gradient (see Fig. 4(b)), 23.76% of the total number of gradient samples were dismissed as unreliable. Fig. (4(a)) shows the unwrapped phase estimated using the DCS algorithm. The mean-squared error (MSE) of the estimation was found to be 0.116%. As a omparison, the same solution was omputed using the standard CS, whose MSE was found to be equal to 2.35%. Thus, one an see 118

5 (a) Figure 4: (a) Original phase estimated by the DCS method; (b) Residual points. that augmenting the primary onstraints of CS by the rossderivative onstraints results in substantial redution in the level of MSE. In our seond experiment, we fixed the MSE of 2.35% as obtained by the standard CS above, and further redued (via random exlusion) the number of available data points to find the perentage for whih the DCS method would result in the same error rate. This perentage was found to be equal to 60%. As a omparison, for the same number of exluded data points, the standard CS resulted in MSE of 8.45%. 5. CONCLUSION The main idea of derivative ompressive sampling is to augment the onstraints of the standard CS by some additional onstraints related to the properties of the gradient as a potential field. In this ase, the reonstrution algorithm resolves the ambiguity of too few samples not by only looking for a solution of maximal sparseness in the domain of Ψ, but also a solution that omplies with the properties of a gradient field. The proposed DCS algorithm is performed in two stages: first, the partial derivatives of a signal of interest are reovered from their sub-ritially sampled measurements, followed by integrating the estimated derivatives via solving a Poisson equation. It should be noted, however, that it is possible to get rid of the seond stage via defining the sampling system Φ to be a disretized version of the 1st-order derivative operator. In this ase, the resulting CS problem should be apable of diretly reovering the original signal. Unfortunately, this solution seems to be appliable only for reovering relatively smooth signals. This is beause the derivativebased Φ an be inoherent only with bases of smooth (slow varying) funtions, thereby ruling out the use of wavelets and the relatives thereof. The urrent researh results leave open a lot of exiting theoretial questions (e.g., as to what other onstraints ould be inorporated into the problem of CS). Moreover, it is still to be proved what bases ould be onsidered to be optimal for representing the gradients of natural senes [15]. Computation effiieny of DCS is another issue that should not be overseen when pratial appliations of DCS are of onern [16]. (b) [1] J. D. Jakson, A. J. Yezzi, and S. Soatto, Joint priors for variational shape and appearane modeling, in IEEE Conferene on Computer Vision and Pattern Reognition, CVPR 07, June , pp [2] D. C. Ghiglia, M. D. Pritt, Two-dimensional phase unwrapping: Theory, algorithms, and software. New York: Wiley-Intersiene, [3] O. Mihailovih and A. Tannenbaum, A fast approximation of smooth funtions from samples of partial derivatives with appliation to phase unwrapping, Signal Proessing, Vol. 88, pp , [4] O. Mihailovih and D. Adam, Phase unwrapping for 2-D blind deonvolution of ultrasound images, IEEE Trans. Med. Imag., vol. 23, No. 1, pp. 7 25, Jan [5] A. V. Oppenheim, R. W. Shafer, Disrete time signal proessing. London: Prentie Hall, [6] D. L. Donoho and P. B. Stark, Unertainty priniples and signal reovery, SIAM J. Appl. Math, vol. 49, no. 3, pp , June [7] E. Candes, J. Romberg and T. Tao, Robust unertainty priniples: Exat signal reonstrution from highly inomplete frequeny information, IEEE Trans. Infrom. Theory, vol. 52, no. 2, pp , Feb [8] E. Candes and T. Tao, Near optimal signal reovery from random projetions: Universal enoding strategies?, IEEE Trans. Infrom. Theory, vol. 52, no. 12, pp , De [9] D. L. Donoho, Compressed sensing, IEEE Trans. Infrom. Theory, vol. 52, no. 4, Apr [10] E. Candes and J. Romberg, Sparsity and inoherene in ompressive sampling, Inverse Problems, vol. 23, no. 3, pp , [11] E. Candes and T. Tao, Deoding by linear programming, IEEE Trans. Inform. Theory, vol. 51, no. 12, De [12] J. W. Brewer, Kroneker produts and matrix alulus in system theory, IEEE Trans. Ciruits and Sys., vol. 25, no. 9, Sep [13] E. V. D. Berg and M. P. Frielander, Probing the pareto frontier for basis pursuit solution, Tehnial Report TR , Department of CS, Univ. of British Columbia, May [14] J. A. Quiroga, D. Crespo and J. A. Gomez-Pedrero XtremeFringe R : state-of-the-art software for automati proessing of fringe patterns, Optial Measurment Sys. for Industrial Inspetion V, Pro. of SPIE, vol Y-1, [15] M. Aharon, M. Elad, and A.M. Brukstein, The K- SVD: An algorithm for designing of overomplete ditionaries for sparse representation, IEEE Trans. On Signal Proessing, Vol. 54, no. 11, pp , Nov [16] D. L. Donoho, Y. Tsaig, I. Drori and J. l. Stark, Sparse solution of underdetermined linear equations by stagewise orthogonal mathing pursuit, Tehnial Report, Univ. of Stanford, REFERENCES 119