Artifact-free Wavelet Denoising: Non-convex Sparse Regularization, Convex Optimization

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1 IEEE SIGNAL PROCESSING LETTERS. 22(9): , SEPTEMBER 215. (PREPRINT) 1 Artifact-free Wavelet Denoising: Non-convex Sparse Regularization, Convex Optimization Yin Ding and Ivan W. Selesnick Abstract Algorithms for signal denoising that combine avelet-domain sparsity and total variation (TV) regularization are relatively free of artifacts, such as pseudo-gibbs oscillations, normally introduced by pure avelet thresholding. This paper formulates avelet-tv (WATV) denoising as a unified problem. To strongly induce avelet sparsity, the proposed approach uses non-convex penalty functions. At the same time, in order to dra on the advantages of convex optimization (unique minimum, reliable algorithms, simplified regularization parameter selection), the non-convex penalties are chosen so as to ensure the convexity of the total objective function. A computationally efficient, fast converging algorithm is derived. I. INTRODUCTION Simple thresholding of avelet coefficients is a reasonably effective procedure for noise reduction hen the signal of interest possesses a sparse avelet representation. Hoever, avelet thresholding often introduces artifacts such as spurious noise spikes and pseudo-gibbs oscillations around discontinuities [1]. In contrast, total variation (TV) denoising [4] does not introduce such artifacts; but it often produces undesirable staircase artifacts. Roughly stated, noise spikes resulting from avelet thresholding are due to noisy avelet coefficients exceeding the threshold, hereas pseudo-gibbs artifacts are due to nonzero coefficients being erroneously set to zero. An approach to alleviate pseudo-gibbs artifacts is to estimate thresholded coefficients by variational means [5], [14]. The use of TV minimization as a ay to recover thresholded coefficients is particularly effective [1], [11], [18], [19]. In this approach, thresholding is performed first. Those coefficients that survive thresholding (i.e., significant coefficients) are maintained. Those coefficients that do not survive (i.e., insignificant coefficients) are then optimized to minimize a TV objective function. TV regularization as subsequently used ith avelet packets [26], curvelets [7], complex avelets [25], shearlets [21], tetrolets [2], [7], and generalized in [2]. Poerful proximal algorithms have been developed for signal restoration ith general hybrid regularization (including avelet-tv) alloing constraints and non-gaussian noise [2]. In this letter, e propose a unified avelet-tv (WATV) approach that estimates all avelet coefficients (significant and insignificant) simultaneously via the minimization of a Copyright (c) 215 IEEE. Personal use of this material is permitted. Hoever, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. The authors are ith the Department of Electrical and Computer Engineering, School of Engineering, Ne York University, 6 Metrotech Center, Brooklyn, NY yd72@nyu.edu and selesi@nyu.edu. MATLAB softare available at single objective function. To induce avelet-domain sparsity, e employ non-convex penalties due to their strong sparsityinducing properties [29] [1]. In general practice, hen nonconvex penalties are used, the convexity of the objective function is usually sacrificed. Hoever, in our approach, e restrict the non-convex penalty so as to ensure the strict convexity of the objective function; then, the minimizer is unique and can be reliably obtained through convex optimization. The proposed WATV denoising method is relatively resistant to pseudo-gibbs oscillations and spurious noise spikes. We derive a computationally efficient, fast converging optimization algorithm for the proposed objective function. The formulation of convex optimization problems for linear inverse problems, here a non-convex regularizer is designed in such a ay that the total objective function is convex, as proposed by Blake and Zimmerman [4] and Nikolova [27], [28], [1]. The use of semidefinite programming (SDP) to attain such a regularizer has been considered [5]. Recently, the concept has been applied to group-sparse denoising [12], TV denoising [6], and non-convex fused-lasso []. II. PROBLEM FORMULATION We consider the estimation of a signal x R N observed in additive hite Gaussian noise (AWGN), y n = x n + v n, n =, 1,..., N 1. (1) We denote the avelet transform by W and the avelet coefficients of signal x as = W x. We index the coefficients as j,k here j and k are the scale and time indices, respectively. In this ork, e use the translational-invariant (i.e., undecimated) avelet transform [1], [24], hich satisfies the Parseval frame condition, W T W = I. (2) But the proposed WATV denoising algorithm can be used ith any transform W satisfying (2). The total variation (TV) of signal x R N is defined as TV(x) := Dx 1 here D is the first-order difference matrix, D =......, () and x 1 denotes the l 1 norm of x, i.e., x 1 = n x n. We also denote x 2 2 := n x n 2. For a set of doubly-indexed avelet coefficients, e denote 2 2 := j,k j,k 2.

2 IEEE SIGNAL PROCESSING LETTERS. 22(9): , SEPTEMBER 215. (PREPRINT) 2 The proposed WATV denoising method finds avelet coefficients by solving the optimization problem, ŵ F () = 1 2 W y 2 2 (a) x φ(x) 2 (b) + j,k λ j φ( j,k ; a j ) + β DW T 1. (4) 2 1 The estimate of the signal x is then given by the inverse avelet transform of ŵ, i.e., ˆx = W T ŵ. The penalty term DW T 1 is the total variation of the signal estimate ˆx. The regularization parameters are λ j > and β >. We allo the avelet regularization and penalty parameters (λ j and a j ) to vary ith scale j. A. Parameterized Penalty Function In the proposed approach, the function φ( ; a): R R is a non-convex sparsity-inducing penalty function ith parameter a. We assume φ satisfies the folloing conditions: 1) φ is continuous on R 2) φ is tice continuously differentiable, increasing, and concave on R + ) φ(x; ) = x 4) φ(; a) = 5) φ( x; a) = φ(x; a) 6) φ ( + ; a) = 1 7) φ (x; a) a for all x The folloing proposition indicates a suitable range of values for the parameter a. This proposition combines Propositions 1 and 2 of [12]. Proposition 1. Suppose the parameterized penalty function φ satisfies the above-listed properties and λ >. If then the function f : R R, a < 1 λ, (5) f(x) = 1 2 (y x)2 + λ φ(x; a), (6) is strictly convex and the function θ : R R, θ(y; λ, a) x R 1 2 (y x)2 + λ φ(x; a), (7) is a continuous nonlinear threshold function ith threshold value λ. That is, θ(y; λ, a) = for all y < λ. Note that (7) is the definition of the proximity operator, or proximal mapping, of φ [2], [15]. Several common penalties satisfy the above-listed properties, for example, the logarithmic and arctangent penalties (hen suitably normalized), both of hich have been advocated for sparse regularization [8], [29]. We use the arctangent penalty as it induces sparsity more strongly [12], [5], ( ( ) ) 2 a φ(x; a) = atan 1+2a x π 6, a > (8) x, a =. Figure 1 illustrates this penalty and its corresponding threshold function θ, hich is given by the solution to a cubic polynomial; see Eqn. (21) in [5] θ(y) soft(y) 1 2 Fig. 1. (a) Non-convex penalty. (b) Threshold function (λ = 1, a =.95). The parameter a (ith a < 1/λ) controls the nonconvexity of the penalty φ; specifically, φ ( + ; a) = a. This parameter also controls the shape of the threshold function θ; namely, the right-sided derivative at the threshold λ, is θ (λ + ) = 1/(1 aλ). If a =, then θ (λ + ) = 1 and θ is the soft-threshold function. If a 1/λ (and a < 1/λ), then θ is a continuous approximation of the hard-threshold function. But, if a > 1/λ, then f in (6) is not convex, the associated threshold function is not continuous, and the threshold value is not λ. As discussed in [5], the value a = 1/λ is the critical value that maximizes the sparsity-inducing behavior of the penalty hile ensuring convexity of f. B. Convexity of the Objective Function In the WATV denoising problem (4), e use the non-convex penalty φ to strongly promote sparsity, but e seek to ensure the strict convexity of the objective function F. Proposition 2. Suppose the parameterized penalty function φ satisfies the properties listed in Sec. II-A. The objective function (4) is strictly convex if for each scale j. a j < 1 λ j (9) Proof. We rite the objective function in (4) as F () = 1 2 ([W y] j,k j,k ) 2 + λ j φ( j,k ; a j ) j,k + β DW T 1. (1) By Proposition 1, condition (9) ensures each term in the curly braces is strictly convex. Also, the l 1 norm term is convex. It follos that F, being the sum of strictly convex and convex functions, is strictly convex. When β = in (4) and a j < 1/λ j for all j, then ŵ j,k = θ([w y] j,k ; λ j, a j ). That is, the solution is obtained by aveletdomain thresholding. C. Parameters The proposed formulation (4) requires parameters λ j, a j, and β. We suggest using a j = 1/λ j or slightly less (e.g.,

3 IEEE SIGNAL PROCESSING LETTERS. 22(9): , SEPTEMBER 215. (PREPRINT).95/λ j ) to maximally induce sparsity hile keeping F convex. To set λ j and β, e consider separately the special cases of (i) β = and (ii) λ j = for all j. In case (i), problem (4) reduces to pure avelet denoising, for hich e presume suitable parameters, denoted λ W j, are knon. In case (ii), problem (4) reduces to pure TV denoising, for hich e presume a suitable parameter, denoted β TV, is knon. We suggest setting λ j = η λ W j, β = (1 η) β TV, (11) here < η < 1 controls the relative eight of avelet and TV regularization. We further suggest the smaller range of.9 < η < 1., so that TV regularization makes a relatively minor adjustment of the avelet solution, to reduce spurious noise spikes and pseudo-gibbs phenomena. In this ork, e set the avelet thresholds λ W j = 2.5σ j, here σj 2 denotes the noise variance in avelet scale j. For an undecimated avelet transform satisfying (2), e have σ j = σ/2 j/2 here σ 2 is the noise variance in the signal domain. For pure TV denoising, e use β TV = Nσ/4 in accordance ith [17]. Therefore, e suggest using λ j = 2.5 η σ/2 j/2, β = (1 η) Nσ/4, (12) ith a nominal value of η =.95. Additionally, e omit regularization of lo-pass avelet coefficients. III. OPTIMIZATION ALGORITHM To solve the WATV denoising problem (4) e use the split augmented Lagrangian shrinkage algorithm (SALSA) [1] hich in turn is based on the alternating direction method of multipliers (ADMM) [6], [22]. Proximal methods (e.g., Douglas-Rachford) could also be used [15]. We assume that a j satisfies (9) for each j, so that F is strictly convex. With variable splitting, problem (4) is expressed as a constrained problem, here arg min g 1() + g 2 (u) u, subject to u =, g 1 () = 1 2 W y j,k g 2 (u) = β DW T u 1. λ j φ( j,k ; a j ) (1) (14a) (14b) Both g 1 and g 2 are strictly convex, hence e may utilize the convergence theory of [22]. Even though φ is non-convex, the function g 1 is strictly convex because a j satisfies (9). The augmented Lagrangian is given by L(, u, µ) = g 1 () + g 2 (u) + µ 2 u d 2 2. (15) The solution of (1) can be found by iteratively minimizing ith respect to and u alternately, as proven in [22]. Thereby, e obtain an iterative algorithm to solve (4), here each iteration consists of three steps: g1 () + µ 2 u d 2 2, (16a) u g2 (u) + µ u 2 u d 2 2 d = d (u ),, (16b) (16c) for some µ >. We initialize u = W y and d =. Note that both (16a) and (16b) are strictly convex. (Sub-problem (16a) is strictly convex because a j satisfies (9).) Sub-problem (16a) can be solved exactly and explicitly. Combining the quadratic terms, (16a) may be expressed as j,k 1 2 (p j,k j,k ) 2 + λ j µ + 1 φ( j,k; a j ), here p = (W y + µ(u d))/(µ + 1). Since this problem is separable, it can be further ritten as 1 j,k v R 2 (p j,k v) 2 + λ j µ + 1 φ(v; a j), (17) for each j and k. This is the same form as (6). Consequently, the solution to sub-problem (16a) is given by ( ) λ j j,k = θ p j,k ; µ + 1, a j (18) here θ is the threshold function (7). Sub-problem (16b) can also be solved exactly. We use properties of proximity operators to express (16b) in terms of a TV denoising problem [15]. In turn, the TV denoising problem can be solved exactly by a fast finite-time algorithm [16]. Define function q as q(v) β DW T u 1 + µ u 2 u v 2 2 (19) ψ(w T u) + 1 u 2 u v 2 2 (2) here h(u) u u v 2 2 (21) = prox h (v) (22) = v + W ( prox ψ (W T v) W T v ) (2) ψ(x) = β µ Dx 1, h(u) = ψ(w T u), (24) and here e have used the semi-orthogonal linear transform property of proximity operators to obtain (2) from (22), hich depends on (2). See Table 1.1 and Example in [15], and see [9], [] for details. Note that prox ψ (W T v) x ψ(x) x W T v 2 2 x β µ Dx x W T v 2 2 (25) = tvd ( W T v, β/µ ) (26) is total variation denoising. We use the fast finite-time exact softare of [16]. Hence, (16b) can be implemented as v = d + (27) u = v + W ( tvd(w T v, β/µ) W T v ). (28) Hence, solving problem (4) by (16a)-(16c), consists of iteratively thresholding (18) [to solve (16a)] and performing total variation denoising (28) [to solve (16b)]. Table I summarizes the hole algorithm. Each iteration involves one avelet transform, one inverse avelet transform, and one total variation denoising problem. (The term W y in the algorithm

4 IEEE SIGNAL PROCESSING LETTERS. 22(9): , SEPTEMBER 215. (PREPRINT) 4 TABLE I ITERATIVE ALGORITHM FOR WATV DENOISING (4). 6 4 (a) Simulated data (σ = 4.) Input: y, λ j, β, µ Initialization: u = W y, d = a j = 1/λ j Repeat: p = (W y + µ(u d))/(µ + 1) j,k = θ(p j,k ; λ j /(µ + 1), a j ) for all j, k v = d + u = v + W ( tvd(w T v, β/µ) W T v ) d = d (u ) Until convergence Return: x = W T (b) Wavelet hard thresholding (RMSE = 1.68) TABLE II AVERAGE RMSE FOR DENOISING EXAMPLE 6 4 (c) To step WATV (RMSE = 1.58) σ Thresholding To-step WATV Proposed WATV may be precomputed.) Since both minimization problems, (16a) and (16b), are solved exactly, the algorithm is guaranteed to converge to the unique global minimizer according to the convergence theory of [22]. IV. EXAMPLE We illustrate the proposed WATV denoising algorithm using noisy data of length 124 (Fig. 2a), generated by Wavelab ( ith AWGN of σ = 4.. We use a 5-scale undecimated avelet transform ith to vanishing moments. Hard-thresholding (ith λ W j = 2.5σ j ) results in noise spikes due to avelet coefficients exceeding the threshold (Fig. 2b). To-step WATV denoising [19] (avelet hard-thresholding folloed by TV optimization of coefficients thresholded to zero) does not correct noise spikes due to noisy coefficients that exceed the threshold, so e use higher thresholds of σ j 2 log N to avoid noise spikes from occurring in the first step. The result (Fig. 2c) is relatively free of noise spikes and psuedo-gibbs oscillations. The proposed WATV algorithm [ith parameters (12), η =.95, and a j = 1/λ j ] is resistant to spurious noise spikes, preserves sharp discontinuities, and results in a loer rootmean-square-error (RMSE) (Fig. 2d). The proposed algorithm as run for 2 iterations in a time of.15 seconds using Matlab 7.14 on a MacBook Air (1. GHz CPU). Table II shos the RMSE for several values of the noise standard deviation (σ) for each method. Each RMSE value is obtained by averaging 2 independent realizations. The higher RMSE of to-step WATV appears to be due to distortion induced by the greater threshold value (d) Proposed WATV (RMSE = 1.41) Fig. 2. (a) Simulated data. (b) Wavelet denoising (hard-thresholding). (c) To-step WATV denoising [19]. (d) Proposed WATV denoising. V. CONCLUSION Inspired by [18], [19], this paper describes a avelet- TV (WATV) denoising method that is relatively free of artifacts (pseudo-gibbs oscillations and noise spikes) normally introduced by simple avelet thresholding. The method is formulated as an optimization problem incorporating both avelet sparsity and TV regularization. To strongly induce avelet sparsity, the approach uses parameterized non-convex penalty functions. Proposition 2 gives the critical parameter value to maximally promote avelet sparsity hile ensuring convexity of the objective function as a hole. A fast iterative implementation is derived. We expect the approach to be useful ith transforms other than the avelet transform, provided they satisfy the Parseval frame property (2). ACKNOWLEDGMENT The authors gratefully acknoledge constructive comments by anonymous revieers. In particular, a revieer noted that (16b) can be expressed explicitly in terms of total variation denoising, an improvement upon the inexact solution in the original manuscript.

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