Key words. Convexity, deblurring, multiplicative noise, primal-dual algorithm, total variation regularization, variational model.

Transcription

1 A CONVEX VARIATIONAL MODEL FOR RESTORING BLURRED IMAGES WITH MULTIPLICATIVE NOISE YIQIU DONG AND TIEYONG ZENG Abstract. In this paper, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primal-dual algorithm is proposed and its convergence is studied. The paper ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well. Key words. Convexity, deblurring, multiplicative noise, primal-dual algorithm, total variation regularization, variational model. AMS subject classifications. 5A4, 5K0, 5K5, 0C30, 0C47. Introduction. In real applications, degradation effects are unavoidable during image acquisition and transmission. For instance, the photos produced by astronomical telescopes are often blurred by atmospheric turbulence. In order to benefit further image processing tasks, image deblurring and denoising continue to attract the attentions in the applied mathematics community. Based on the imaging systems, various kinds of noise were considered, such as additive Gaussian noise, impulse noise, and Poisson noise, etc. We refer the reader to [7,,35,40,44,45] and references therein for an overview of those noise models and the restoration methods. However, multiplicative noise is a different noise model, and it commonly appears in synthetic aperture radar (SAR), ultrasound imaging, laser images, and so on [8,, 3, 43]. For a mathematical description of such degradations, suppose that an image û is a real function defined on, a connected bounded open subset of R with compact Lipschitz boundary, i.e., û : R. The degraded image f is given by: f = (Aû)η, (.) where A L(L ()) is a known linear and continuous blurring operator, and η L () represents multiplicative noise with mean. Here, f is obtained from û, which is blurred by the blurring operator A, and then is corrupted by the multiplicative noise η. Usually we assume that f > 0. In this paper, we concentrate on the assumption that η follows a Gamma distribution, which commonly occurs in SAR. The deblurring process under noise is well-known to be an ill-posed problem in the sense of Hadamard [30]. Since the degraded image only provides partial restrictions on the original data, there exist various solutions which can match the observed degraded image under the given blurring operator. Hence, in order to utilize variational methods, the main challenge in image restoration is to design a reasonable and easily solved optimization problem based on the degradation model and the prior information on the original image. Until the past decade, a few variational methods have been proposed to handle the restoration problem with the multiplicative noise. Given the statistical properties Department of Applied Mathematics and Computer Science, Technical University of Denmark, 800 Kgs. Lyngby, Denmark. (yido@dtu.dk) Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. (zeng@hkbu.edu.hk)

2 of the multiplicative noise η, in [3] the recovery of the image û was based on solving the following constrained optimization problem: inf u S() Du f subject to: Au dx = (.) ( f Au ) dx = θ, where θ denotes the variance of η, S() = {v BV () : v > 0}, BV () is the space of functions of bounded variation (see [] or below), and the total variation (TV) of u is utilized as the objective function in order to preserve significant edges in images. In (.), only basic statistical properties, the mean and the variance, of the noise η are considered, which somehow limits the restored results. For this reason, based on the Bayes rule and Gamma distribution with mean, by using a maximum a posteriori (MAP) estimator Aubert and Aujol [5] introduced a variational model as follows: inf u S() ( log(au) + f Au ) dx + λ Du, (.3) where the TV of u is utilized as the regularization term, and λ > 0 is the regularization parameter which controls the trade-off between a good fit of f and a smoothness requirement due to the TV regularization. Below we refer (.3) as AA model. Since both (.) and (.3) are non-convex, the gradient projection algorithms proposed in [3] and [5] may stick at some local minimizers, and the restored results strongly rely on the initialization and the numerical schemes. To overcome this problem and provide a convex model, in [3] Huang et al. introduced an auxiliary variable z = log u in (.3), and in [4] Shi et al. modified (.3) by adding a quadratic term. With convex models, these two methods both provide better restored results than the method proposed in [5], and they are independent of the initial estimations. In addition, Steidl et al. combined the I-divergence as the data fidelity term with the TV regularization or the nonlocal means to remove the multiplicative Gamma noise [4]. A general patch-based denoising filter is proposed in [0]. In [], the denoising problem was handled by using L fidelity term on frame coefficients. In [3], the approach with spatially varying regularization parameters in the AA model (.3) was considered in order to restore more texture details of the denoised image. In [3], the multiplicative noise removal was addressed via a learned dictionary and extensive experimental results illustrated the leading performance of this approach. However, all of the above methods only work on the multiplicative noise removal issue, and it is still an open question to extend them to the deblurring case. In this paper, we focus on the restoration of images that are simultaneously blurred and corrupted by multiplicative noise. Since the non-convexity of the model (.3) proposed in [5] causes uniqueness problem and the issue of convergence of the numerical algorithm, we introduce a new convex model by adding a quadratic penalty term based on the statistical properties of the multiplicative Gamma noise. Furthermore, we study the existence and uniqueness of a solution to the new model on the continuous, i.e. functional space, level. Here, we still use the TV regularization in order to preserve edges during the reconstruction. Evidently, it can be readily extended to some other modern regularization terms such as non-local TV [] or framelet approach []. The minimization problem in our method is solved by the primal-dual algorithm proposed in [5,4] instead of the gradient projection method in [5,3]. The numerical results in this paper show that our method has the potential to outperform the other approaches in multiplicative noise removal with deblurring simultaneously.

3 The rest of the paper is organized as follows. In Section, we briefly review the total variation regularization and provides its main properties. In Section 3, based on the statistical properties of the multiplicative Gamma noise we propose a new convex model for denoising, and study its existence and uniqueness of a solution with several other properties. Then in Section 4 we extend the model and those properties to the case of denoising and deblurring simultaneously. Section 5 gives the primal-dual algorithm for solving our restoration model based on the work proposed in [5]. The numerical results shown in Section demonstrate the efficiency of the new method. Finally, conclusions are drawn in Section 7.. Review of Total Variation Regularization. In order to preserve significant edges in images, in their seminal work [40] Rudin et al. introduced total variation regularization into image restoration. In this approach, they recover the image in BV (), which denotes the space of functions of bounded variation, i.e. u BV () iff u L () and the BV-seminorm: { Du := sup u div(ξ(x))dx } ξ C0 (, R ), ξ L (,R ), (.) is finite. The space BV () endowed with the norm u BV = u L + Du is a Banach space. If u BV (), the distributional derivative Du is a bounded Radon measure and the above quantity defined in (.) corresponds to the total variation (TV). Based on the compactness of BV (), in two-dimensional case we have the embedding BV () L p () for p which is compact for p <. See [3,, ] for more details. 3. A Convex Multiplicative Denoising Model. To propose a convex multiplicative denoising model, we start from the multiplicative Gamma noise. Suppose that the random variable η follows Gamma distribution, i.e., its probability density function (PDF) is: p η (x; θ, K) = θ K Γ(K) xk e x θ for x 0, (3.) where Γ is the usual Gamma-function, θ and K denote the scale and shape parameters in the Gamma distribution, respectively. Furthermore, the mean of η is Kθ, and the variance of η is Kθ, see []. As multiplicative noise, in general we assume that the mean of η equals, then we have that Kθ = and its variance is K. Now, set a random variable Y = η, whose PDF is: p Y (y) = θ K Γ(K) y K e θy for y 0. (3.) We can prove the following properties of Y. Proposition 3.. Suppose that the random variable η follows Gamma distribution with mean. Set Y = η. Then the means of Y and Y are: E(Y ) = KΓ(K /) Γ(K) respectively. Furthermore, we have: E(Y n ) = K n Γ(K n ) Γ(K) 3 and E(Y ) = K K,

4 for any n N. Proof. Based on the PDF of η shown in (3.) and Kθ =, we obtain: E(Y ) = Similarly, we have: = + 0 x θ K Γ(K) xk e x θ dx Γ(K /) + θγ(k) θ K / Γ(K /) xk 3/ e x θ 0 Γ(K /) + = p η (x; θ, K /) dx θγ(k) KΓ(K /) =. Γ(K) E(Y ) = Γ(K ) = θγ(k) = K K. x θ K Γ(K) xk e x θ dx + 0 θ K Γ(K ) xk e x θ dx Further, for any n N we can calculate E(Y n ) similarly, and obtain E(Y n ) = K n Γ(K n ). Γ(K) dx According to the properties of Gamma function [4] and Proposition 3., if K is large, the mean of Y is close to. Furthermore, we have: Proposition 3.. Suppose that the random variable η follows Gamma distribution with mean. Set Y = η. Then: lim E((Y K + ) ) = 0. (3.3) Proof. With Proposition 3., readily we have: E((Y ) ) = K K KΓ(K /) +. Γ(K) Based on the property of the Gamma function, see [4], lim K + taking α =, immediately we can obtain: Γ(K + α) =, α R, Γ(K)Kα lim E((Y K + ) ) = 0. 4

5 Proposition 3. ensures that the value of E((Y ) ) is small with a large K. On the other hand, from Table 3. we can see that even with a small K its value is still rather small. Remark 3.3. Since p Y (y) defined in (3.) is monotonic on either side of the single maximal point y = θ, it is a strictly unimodal function. In statistics, Gaussian distribution is commonly used to approximate the unimodal or other distributions, see [3, 38]. To further understand the relationship between Y and Gaussian distribution, we need the concept of the Kullback-Leibler (KL) divergence. Suppose that P and Q are continuous random variables with the PDFs p(x) and q(x), respectively. The KL-divergence of P and Q is defined as ( ) p(x) D KL (P Q) := log p(x) dx, q(x) which is always non-negative and is zero if and only if P = Q almost everywhere, see [7]. Based on the PDF of the random variable Y, we can prove that the KLdivergence of Y with the Gaussian distribution N (µ K, σk ) tends to 0 when K goes to infinity, where µ K and σk are the mean and variance of Y, respectively, i.e., µ K := E(Y ) and σ K := E(Y ) (E(Y )). (3.4) Proposition 3.4. Suppose that the random variable η follows Gamma distribution with mean. Set Y = η. Then, (i) + p 0 Y (y) log p Y (y)dy = log log( KΓ(K)) + K+ ψ(k) K, where d log Γ(K) ψ(k) := dk is the digamma function (see []); (ii) + p 0 Y (y) log p N (µk,σk ) (y)dy = log(πeσ K ), where p N (µ K,σK ) (y) denotes the PDF of the Gaussian distribution N (µ K, σk ); (iii) lim D KL(Y N (µ K, σk )) = 0. K + Proof. See Appendix I. Furthermore, let us consider a re-scaled version of Y, the random variable Z := Y µ K σ K. When K goes to infinity, Y degenerates to, however, Z tends to follow the Gaussian distribution N (0, ). Further, based on Proposition 3.4, we have: lim D KL(Z N (0, )) = lim D KL(Y N (µ K, σk)) = 0. K + K + Using Proposition 3.4 and the definition of the KL-divergence, in Table 3. we list the KL-divergence between Y and the Gaussian distribution N (µ K, σk ) with respect to different values of K. Obviously, even with small K, D KL (Y N (µ K, σk )) is already very small. In addition, in Figure 3. we show the PDFs of Y and N (µ K, σk ) with K =, K = 0 and K = 30. We can see that the PDF of Y decreases quickly and is close to symmetry as K increases, which are the essential properties of Gaussian distribution. In the denoising case, that is, A is the identity operator, from the degradation model (.), we obtain that Y = η = u f. Inspired by (3.3), we introduce a quadratic penalty term into the AA model (.3), which turns out to be: ( inf E(u) := log u + f ) u S() u ( u dx + α f 5 ) dx + λ Du (3.5)

6 K E((Y ) ) D KL (Y N (µ K, σk )) e-04 Table 3. The values of E((Y ) ) and the KL-divergence of Y with the Gaussian distribution N (µ K, σk ) for different values of K..8 p Y (y) p N(µ,σ ) (y).5 p Y (y) p N(µ,σ ) (y) p Y (y) p N(µ,σ ) (y) p(y) p(y) p(y) y y y (a) (b) (c) Fig. 3.. The comparisons of the PDFs of Y and N (µ K, σk ) with different K. (a) K =, (b) K = 0, (c) K = 30. with the penalty parameter α > 0. In addition, we set: S() := {v BV () : v 0}, which is closed and convex, and we define log 0 = and 0 = +. Note that as f > 0, we need not define Existence and uniqueness of a solution. For the existence and uniqueness of a solution to (3.5), we start with discussing the convexity of the model. Since the quadratic penalty term provides extra convexity, we prove that E(u) in (3.5) is convex, if the parameter α satisfies certain condition. Proposition 3.5. If α, then the model (3.5) is strictly convex. Proof. With t R + and a fixed α, we define a function g as: g(t) := log t + t + α( t ). Easily, we have that the second order derivative of g satisfies: g (t) = t + t 3 + α t 3. A direct computation shows that the function t 3 g (t) reaches its unique minimum, α 8, at t =. Hence, if α, we have g (t) 0, i.e., g is convex. Furthermore, since the function g has only one minimizer, g is strictly convex when α. Setting t = u(x) f(x) for each x, we obtain the strict convexity of the first two terms in (3.5). Based on the convexity of the TV regularization, we deduce that E(u) in (3.5) is strictly convex, if α. Since the feasible set S() is convex, the assertion is an immediate consequence. Based on Proposition 3.5, we see that with a suitable α, (3.5) is a convex approximation of the non-convex model (.3) with A as the identity operator. Now, we

7 argue the existence and uniqueness of a solution to (3.5) and the minimum-maximum principle. Theorem 3.. Let f be in L () with inf f > 0, then the model (3.5) has a solution u in BV () satisfying: 0 < inf f u sup f. Moreover, if α, the solution of (3.5) is unique. Proof. Set c := inf f, c := sup f, and let: ( E 0 (u) := log u + f ) ( ) u dx + α u f dx. For each fixed x, easily we have: log t+ f(x) t +log f(x) with t R + {0}. Noting that E(u) is defined in (3.5), we thus have: E(u) ( log u + f ) u dx ( + log f) dx. In other words, E(u) in (3.5) is bounded from below, and we can choose a minimizing sequence {u n S() : n =,, }. Since for each fixed x, the real function on R + {0}: g(t) := log t + f(x) t ( t + α f(x) is decreasing if t [0, f(x)) and increasing if t (f(x), + ), one always has g(min(t, M)) g(t) with M f(x). Hence, we obtain that: E 0 (inf(u, c )) E 0 (u). Combining with D inf(u, c ) Du, see Lemma in Section 4.3 of [34], we have E(inf(u, c )) E(u). In the same way we are able to get E(sup(u, c )) E(u). Hence, we can assume that 0 < c u n c, which implies that u n is bounded in L (). As {u n } is a minimizing sequence, we know that E(u n ) is bounded. Furthermore, Du n is bounded, and {u n } is bounded in BV (). Therefore, there exists a subsequence {u nk } which converges strongly in L () to some u BV (), and {Du nk } converges weakly as a measure to Du []. Since S() is closed and convex, by the lower semi-continuity of the total variation and Fatou s lemma, we get that u is a solution of the model (3.5), and necessarily 0 < c u c. Moreover, if α, uniqueness follows directly from the strict convexity of the function E. Similar as Theorem 4. in [5], here we also need the assumption inf f > 0, which ensures the lower boundedness of E(u) defined in (3.5). In the numerical practice, this assumption is always kept by using max(f, ϵ) as the observed image with a very small positive value ϵ. In [5] a comparison principle was given concerning the model (.3). With the α-term in (3.5), it is satisfied with certain condition on α as well. Proposition 3.7. Let f and f be in L () with a > 0 and a > 0, where a = inf f and a = inf f. Further, set b = sup f and b = sup f. Assume 7 )

8 f < f. Suppose u (resp. u ) is a solution of (3.5) with f = f (resp. f = f ). Then when α < aa b b a a, we have u u. Proof. Referring to [5], we define u v = inf(u, v) and u v = sup(u, v). Since u i is a minimizer of E(u) with f = f i, which is defined in (3.5), with respect to i =,, we have: E(u u ) + E(u u ) E(u ) + E(u ). Based on the result D(u u ) + D(u u ) Du + Du in [3,7], we get: ( ) log(u u ) + f u u + α u dx u f ( ) + log(u u ) + f u u + α u dx u f ( ) log u + f u u + α dx f ( ) + log u + f u u + α dx. f Writing = {u > u } {u u }, we easily deduce that: [ ] (f f )(u {u >u } u ) + α α f f f f ( f + f )( u + u ) 0. u u Based on Theorem 3., we have 0 < a u b and 0 < a u b, which imply and: u u b b f f ( f + f )( u + u ) f f a a ( a + a ). b b a a b b Hence, we find that: u u + α α f f f f ( f + f )( u + u ) > 0, a as soon as α < a b b a a. Taking account of f < f, in this case we deduce that {u > u } has a zero Lebesgue measure, i.e. u u a.e. in. 4. The Extension to Simultaneous Deblurring and Denoising. The model (3.5) is based on the statistical properties of Gamma distribution, and is specifically devoted to the multiplicative Gamma noise removal. In this section, we extend it to the simultaneous deblurring and denoising case, i.e., to restore the image û in (.) with the blurring operator A. The restoration is processed by solving the optimization problem: ( inf E A (u) := log Au + f ) u S() Au ( ) Au dx + α f 8 dx + λ Du, (4.)

9 where A L(L ()). As a blurring operator, we assume that A is nonnegative, i.e., A 0 in short. Then we have Au 0 with u S(). Similar as Proposition 3., since A is linear, we can readily establish the following convexity result. Proposition 4.. If α, then the model (4.) is convex. 4.. Existence of a solution. Based on the properties of total variation and the space of bounded variation functions, we prove the existence and uniqueness of a solution to (4.). Theorem 4.. Recall that R is a connected bounded set with compact Lipschitz boundary. Suppose that A L(L ()) is nonnegative, and it does not annihilate constant functions, i.e., A 0. Let f be in L () with inf f > 0, then the model (4.) admits a solution u. Moreover, if α and A is injective, then the solution is unique. Proof. Similar to the proof of Theorem 3., readily, E A is bounded from below. We can choose a minimizing sequence {u n } S() for (4.). So { Du n } with n =,, is bounded. Using the Poincaré inequality (see Remark 3.50, [3]), we obtain: u n m (u n ) C D(u n m (u n )) = C Du n, (4.) where m (u n ) = u n dx and denotes the measure of. Further, C is a constant. Recalling that is bounded, it follows that u n m (u n ) is bounded for each n. Since A L(L ()) is continuous, {A(u n m (u n ))} must be bounded in L () and in L (). ( On the other hand, according to the boundedness of E A (u n ), for each n, Aun f is bounded in L (), which implies that Aun is bounded, then we obtain that Au n is bounded. Moreover, we have: m (u n ) A = A(u n m (u n )) Au n A(u n m (u n )) + Au n. Hence, m (u n ) A is bounded. Thanks to A 0, we obtain that m (u n ) is uniformly bounded. Together with the boundedness of {u n m (u n )}, it leads to the boundedness of {u n } in L () and thus in L (). Since S() is closed and convex, {u n } is bounded in S() as well. Therefore, there exists a subsequence {u nk } which converges weakly in L () to some u L (), and {Du nk } converges weakly as a measure to Du. Due to the continuity of the linear operator A, one must have that {Au nk } converges weakly to Au in L (). Then based on the lower semi-continuity of the total variation and Fatou s lemma, we obtain that u is a solution of the model (4.). Based on Proposition 4., when α, the model (4.) is convex. Furthermore, if A is injective, (4.) is strictly convex, then its minimizer has to be unique. Same as in Theorem 3., the assumption that inf f > 0 basically guarantees the well-posedness of the model. Moreover, in the discrete settings, adding a few more technicalities, the assumption that A is injective may be weakened to ker(a) ker( ) = {0}, where denotes the discrete gradient operator. Remark 4.3. In the proof of Theorem 4., because of the α-term, we obtain that the sequence {Au n } is bounded in L (). Thus, when α = 0, that is, in the case of the model (.3), it is difficult to get the same result, and in [5] the existence of a solution to (.3) is still an open question. f )

10 According to the constraint in (4.), we find that its minimizer is nonnegative. Further, we have the following result. Proposition 4.4. Suppose that u is the solution of (4.). Then there exists a constant C, such that: ( ) {x : (Au ϵ )(x) ϵf(x)} C + log f dx + ϵ log ϵ for any 0 < ϵ <. Proof. Suppose that C is the minimal value of (4.), which is independent of ϵ. Set w = Au f, then we have: ( + log f dx log w + ) w dx C + log f dx, where we have used the fact that for each t > 0, log t + t. Moreover, if t ϵ <, then log t + t log ϵ + ϵ, and we get that: ( ) ϵ {x : w(x) ϵ} C + log f dx. + ϵ log ϵ Based on w = Au f, we obtain the assertion. As a consequence, {x : (Au )(x) = 0} = 0, i.e., Au is positive almost everywhere. Especially, in the discrete situation, Au is strictly positive. 4.. Bias correction. In [], a variational model was proposed in the logimage domain for multiplicative noise removal. In order to ensure that the mean of the restored image equals to that of the observed image, the method ends up an exponential transform along with a bias correction. In addition, we recall that through the theoretical analysis in [4, ] the classical ROF model in [40]: inf u BV () (Au f) dx + λ Du, (4.3) is proved to preserve the mean, i.e., m (Au ) = m (f) with u as a solution. However, (4.3) is proposed to remove the additive white Gaussian noise. In this section, we consider our new model (4.) through the similar theoretical analysis as in [4], and try to find the relation between the observed image f and the solution u. Proposition 4.5. Suppose that A =. Let u be a solution of (4.) and assume that we have inf Au > 0, then the following properties hold true: (i) [ ( )] f (Au ) α f f Au dx = Au (ii) If there exists a solution in the case of α = 0, then we have, f dx Au dx. dx. Proof. 0

11 (i) We define a function with single variable t ( inf Au, + ): ( ) e(t) := log A(u f + t) + A(u + t) + λ D(u + t). Concerning A =, we necessarily have: e(t) = ( ) A(u dx + α + t) f ( ) ( ) log(au f Au + t) + Au dx+α + t dx+λ Du. + t f Since t = 0 is a (local) minimizer of e(t), we have e (0) = 0, which leads to: [ Au f ( )] (Au ) + α f dx = 0. f Au (ii) With α = 0 the result in (i) becomes: f (Au ) dx = Au dx. Moreover, according to Hölder s inequality and the nonnegativity of Au and f, we obtain: Combining both, we have: ( ) f (Au ) dx f dx Au dx. f dx Au dx. dx Note that in the discrete situation the assumption that inf Au > 0 is always satisfied, based on the conclusion of Proposition 4.4. From Proposition 4.5 we cannot obtain the similar theoretical result as in [4, ], that is, the mean of the observed image is automatically preserved, for the model (4.). In order to reduce the influence from the bias and keep the restored image in the same scale as f, we improve the model (4.) as: ( inf log Au + f ) {u S():m (u)=m (f)} Au ( ) Au dx + α f dx + λ Du. (4.4) It is straightforward to show that the feasible set {u S() : m (u) = m (f)} is closed and convex, then the existence and uniqueness of a solution to (4.4) are easily obtained by extending Theorem 4.. Note that the bias-variance trade-off in statistics does not always advocates for unbiased estimators. It is possible to obtain better PSNR results with a small (but different from zero) bias. However, in our numerical simulations, we do observe the improvement of PSNR with our bias correction step.

12 In (4.4), we implicitly suppose that: m (u) m (Au), m ((Au)η) m (Au). Under some independence conditions, the above assumptions are theoretically rooted in statistics. Moreover, in the practical simulations, we find that these two assumptions provide rather reasonable results. 5. Primal-Dual Algorithm. Since the model (4.4) is convex, there are many methods that can be extended to solve the minimization problem in (4.4). For example, the alternating direction method [,5], which is convergent and is well-suited to large-scale convex problems, and its variant, the split-bregman algorithm [8], which is widely used to solve the L regularization problems such as the TV regularization. In this section, we apply the primal-dual algorithm proposed in [5, 4, 37] to solve the minimization problem in (4.4). This algorithm can be used for solving a large family of non-smooth convex optimization problems, see the applications in [5]. It is simple, and comes with the convergence guarantees. Now, we focus on the discrete version of (4.4). For the sake of simplicity, we keep the same notations from the continuous context. Then the discrete model reads as follows: f min E A(u) := log Au, + u X Au, Au + α f + λ u, (5.) where X = {v R n : v i 0 for i =,, n, and n i= v i = n i= f i}, n is the number of pixels in the images, f X is obtained from a two-dimensional pixel-array by concatenation in the usual columnwise fashion, and A R n n. Moreover, the vector inner product u, v = n i= u iv i is used, and denotes the l -vector-norm. The discrete gradient operator R n n is defined by: [ ] x v v =, y v for v R n with x, y R n n corresponding to the discrete derivative in the x- direction and y-direction, respectively. In our numerics, x and y are obtained by applying finite difference approximations for the derivatives with symmetric boundary conditions in the respective coordinate directions. In addition, v denotes the discrete total variation of v, which is defined as: v = n i= ( x v) i + ( yv) i. Define the function G : X R as: f G(u) := log Au, + Au, Au + α f. Based on the definition of total variation in Section, we give the primal-dual formulation of (5.): max min p Y u X G(u) λ u, div p, (5.)

13 where Y = {q R n : q }, q = max i {,,n} qi + q i+n denotes the l -vector-norm, p is the dual variable, and the divergence operator div =. This is a generic saddle-point problem, and we can apply the primal-dual algorithm proposed in [5] to solve the above optimization task. The algorithm is summarized as follows. Algorithm for solving the model (5.) : Fixed σ, τ. Initialize u 0 = f, ū 0 = f and p 0 = (0,, 0) R n. : Calculate p k+ and u k+ from: p k+ = arg max p Y λ ūk, div p σ p pk, (5.3) u k+ = arg min u X G(u) λ u, div pk+ + τ u uk, (5.4) ū k+ =u k+ u k. (5.5) 3: Stop; or set k := k + and go to step. In order to apply the algorithm to (5.), the main questions are how to solve the dual problem in (5.3) and the primal problem in (5.4). For (5.3), the solution can be easily given by: p k+ i = π ( λσ( ū k ) i + p k i ), for i =,, n, (5.) where π is the projector onto the l -normed unit ball, i.e., π (q i ) = q i max(, q i ) and π (q n+i ) = q n+i, for i =,, n, max(, q i ) with q i = qi + q i+n. Since the minimization problem in (5.4) is strictly convex and its objective function has the second derivative, it can be solved efficiently by the Newton method following with one projection step, u k i := n j= f j n j= max(uk j, 0) max(uk i, 0), for i =,, n, (5.7) to ensure that u k is nonnegative and preserves the mean of f. This projection is inspired by Prop.. of [8] or Prop. of []. Based on Theorem in [5], we end this section by the convergence properties of our algorithm. The proof refers to [5]. Proposition 5.. The iterates (u k, p k ) of our algorithm converge to a saddle point of (5.) provided that στλ <. According to the result 8 with the unit spacing size between pixels in [3], we only need στλ < 8 in order to keep the convergent condition. In our numerical practice, λ is tuned empirically (and usually it is around 0., see next Section for details) and we simply set σ = 3 and τ = 3, which in most cases ensures the convergence of the algorithm. 3

14 (a) (b) (c) Fig... Original images. (a) Phantom, (b) Cameraman, (c) Parrot.. Numerical Results. In this section we provide numerical results to study the behavior of our method with respect to its image restoration capabilities and CPUtime consumption. Here, we compare our method with the one proposed in [5] (AA method) by solving (.3) and the one in [3] (RLO method) by solving (.), and both of them are able to remove the multiplicative noise and deblurring simultaneously. In the denoising case, we also provide numerical results from [0] where a probabilistic patch-based method was proposed for several kind of noise, including multiplicative noise. We abbreviate this method by the PPB filter. Since in the AA method and the RLO method the preservation of the mean of the observed image is not guaranteed, in order to compare fairly, we add the same projection step as in (5.7) before outputting the results. Indeed, assume that u o is the result of the AA method or the RLO method, then we revise it as: n u new j= i := f j n j= max(uo j, 0) max(uo i, 0), for i =,, n. (.) Note that usually this will improve the result by around 0. to 0.3 db, depending on images and noise levels. For the PPB filter, we numerically find that this correction is not necessary. For illustrations, the results for the 5-by-5 gray level images Phantom, Cameraman and Parrot, are presented, see the original test images in Figure.. The quality of the restoration results is compared quantitatively by means of the peak signal-to-noise ration (PSNR) [0], which is a widely used image quality assessment measure. In addition, all simulations listed here are run in Matlab 7. (R0a) on a PC equipped with.40ghz CPU and 4G RAM memory... Image denoising. Although our method is proposed for the simultaneous deblurring and denoising of images subject to multiplicative noise, here we show that it also provides good results for noise removal only. However, since our model is rather basic, the more advanced approaches such as patch-based method [0] or dictionary learning method [3] for multiplicative noise removal could have better results. In our example, the test images are corrupted by multiplicative noise with K = 0 and K =, respectively. The results are shown in Figure.-.7. For the AA method and the RLO method, we use the time-marching algorithm to solve the minimization models as proposed in [5, 3]. We set the step size as 0. in order to obtain a stable iterative procedure. The algorithms are stopped when the maximum 4

15 (a) (b) (c) (d) (e) Fig... Results of different methods when removing the multiplicative noise with K = 0. Row 3 : the plots of the objective versus. (a) Noisy Phantom, (b) PPB filter, (c) AA method (λ = 0.), (d) RLO method (λ = 0.), (e) our method (λ = 0. and α = 8). number of is reached. In addition, after many experiments with different λ-values in the AA model, RLO model and ours, the ones that provide the best PSNRs are presented here. In the PPB filter [0], we use the codes provided by the authors in [], which are written as C++ mex-functions. Since the codes were written for removing multiplicative noise with Nakagami-Rayleigh distribution, a square root transform was applied in order to eliminate the gap between Gamma distribution and Nakagami-Rayleigh distribution. We use the suggested parameter values for the PBB filter except for Phantom where the search window size has to be enlarged to get better result, which costs more time (see Table.). In our method, we stop the iterative procedure as soon as the value of the objective function has no big relative decrease, i.e., E(u k ) E(u k+ ) E(u k ) < ε. (.) In denoising case, we set ε = Notice that the computing time of the PPB filter in [0] is not comparable with the remaining three TV-based methods, since the 5

16 (a) (b) (c) (d) (e) Fig..3. Results of different methods when removing the multiplicative noise with K = 0. Row 3 : the plots of the objective versus. (a) Noisy Cameraman, (b) PPB filter, (c) AA method (λ = 0.4), (d) RLO method (λ = 0.4), (e) our method (λ = 0. and α = ). programming languages are different. Comparing the results by the three TV-based methods, i.e., the AA, RLO and our method in Figure.-.5, we see that all of their objective are monotonically decreasing, and our method performs best visually with the least. Note that in the restored results by the AA method and the RLO method, much more noise remains comparing with the ones by our method; see, e.g., white boundary of phantom and the background in Cameraman. Moreover, the contrasts of the details by the AA method and the RLO method are noticeably reduced because of oversmoothing during noise removal, however, our method preserves more details. In this respect observe the tripod and the trousers in Cameraman, especially when recovering the images corrupted by high-level noise. In order to compare the capability of recovering details, in Figure. and Figure.7 we show the results for denoising the image Parrot which includes more details. Comparing the textures surrounding the eye and the background of the parrot, we can clearly see that our method suppresses noise successfully while preserving significantly more details.

17 (a) (b) (c) (d) (e) Fig..4. Results of different methods when removing the multiplicative noise with K =. Row 3 : the plots of the objective versus. (a) Noisy Phantom, (b) PPB filter, (c) AA method (λ = 0.3), (d) RLO method ( λ = 0.3), (e) our method (λ = 0. and α = 4). Since the PPB filter in [0] is a patch-based method, which has a different framework with the TV-based methods, it can remove the multiplicative noise more effectively and preserve more details. But we still find that some unpleasant artifacts are noticeable in the results obtained by the PPB filter; see, e.g., the homogeneous regions in Cameraman, Phantom and Parrot. Furthermore, some tiny structures in the images are completely missing; see, e.g., the spots in Phantom and the buildings in Cameraman. Since we utilize TV regularization in our method, those spurious artifacts are absent from our results, but at the same time there might be some undesirable stair-casing effects; see the beak and neck of the parrot. For the comparison of the performance quantitatively and the computational efficiency, in Table. we list the PSNR values of the restored results, the number of the and the CPU-times. In the three TV-based methods, we observe that the PSNR values from our method are more than 0.5 db higher than others. Due to a large step size, with much less our method reaches the stopping rule and also spends much less CPU-times. However, in order to obtain a stable iterative procedure, the AA method and the RLO method have to use a small step size, and 7

18 (a) (b) (c) (d) (e) Fig..5. Results of different methods when removing the multiplicative noise with K =. Row 3 : the plots of the objective versus. (a) Noisy Cameraman, (b) PPB filter, (c) AA method (λ = 0.), (d) RLO method ( λ = 0.), (e) our method (λ = 0. and α = ). then need more than 0 times more to provide the results with the best PSNRs... Image deblurring and denoising. In this section, we consider the restoration of the noisy blurred images. In our experiments, we test two blurring operators, which are motion blur with length 5 and angle 30, and Gaussian blur with a window size 7 7 and a standard deviation of. Further, after blurred, the test images are corrupted by multiplicative noise with K = 0. In the blurring case, we set ε = 0 4 in (.) as the stopping rule for our method. In Figure.8-.0, we show the degraded images and the restored results by all three TV-based methods, and Table. lists the PSNR values, the number of, and the CPU-times. In contrast to the results by the AA method and the RLO method, our method also performs best both visually and quantitatively, and it preserves more details; see, e.g., the tripod in Cameraman and the texture near the eye in Parrot. Due to the blurring, more are needed in all three methods, but our method still provides the best results in much less with 8

19 (a) (b) (c) (d) (e) Fig... Restored images by different methods for restoring the image Parrot. (a) Noisy image with K = 0, (b) PPB filter, (c) AA method (λ = 0.4), (d) RLO method (λ = 0.4), (e) our method (λ = 0. and α = ). the least CPU-times. In a conclusion, our method turns out to be more efficient and outperforms the other methods which are able to deblur while removing multiplicative noise simultaneously. 7. Conclusion. In this paper, we propose a new variational model for restoring blurred images subject to multiplicative Gamma noise. In order to obtain the convexity, we add a quadratic penalty term according to the statistical properties of the multiplicative Gamma noise in the model proposed in [5], which combines a MAP estimator with the total variation regularization. The existence and uniqueness of a solution to the new model are obtained. Furthermore, some other properties are studied, such as the minimum-maximum principle, the bias correction, and so on. Due to the convexity, we are allowed to employ the primal-dual algorithm proposed in [5] to solve the corresponding minimization problem in the new model, and its convergence is guaranteed. Compared to other recently proposed methods, our method appears to be very competitive with respect to image restoration capabilities and CPU-time consumption. Acknowledgement. This work is supported by RGC 70, RGC, NSFC 704 and the FRGs of Hong Kong Baptist University. The authors would like to thank the reviewers for the careful reading of the manuscript and the insightful and constructive comments. 8. Appendix I: Proof of Proposition 3.4. In order to prove Proposition 3.4, we need the following lemma.

20 (a) (b) (c) (d) (e) Fig..7. Restored images by different methods for restoring the image Parrot. (a) Noisy image with K =, (b) PPB filter, (c) AA method (λ = 0.8), (d) RLO method (λ = 0.8), (e) our method (λ = 0. and α = 8). Lemma 8.. Let B(K) denote the Beta function B(K, K) with K, i.e., B(K) := Γ (K) Γ(K). Then, we have B(K ) B(K) = + ( ) K + O K. (8.) Proof. Based on a property of Gamma function, we readily obtain: ( B(K log ) ) B(K) Γ(x + ) Γ(x) = x for x > 0, (8.) ( = log K ) + (log Γ(K ) ) log Γ(K). Then using the expansion of log Γ(x) with x > 0 in []: log Γ(x) =x log x x log x π + x + O( x ) (8.3) 0

21 (a) (b) (c) (d) (e) Fig..8. Results of different methods when restoring the degraded images corrupted by motion blur and then multiplicative noise with K = 0. Row : degraded images. Row and 4: restored images with different methods. Row 3 and 5: the plots of the objective versus. (a) Degraded Phantom, (b) degraded Cameraman, (c) AA method (row : λ = 0.05; row 4: λ = 0.0), (d) RLO method (row : λ = 0.05; row 4: λ = 0.0), (e) our method (row : λ = 0.0 and α = ; row 4: λ = 0.0 and α = ).

22 (a) (b) (c) (d) (e) Fig... Results of different methods when restoring the degraded images corrupted by Gaussian blur and then multiplicative noise with K = 0. Row : degraded images. Row and 4: restored images with different methods. Row 3 and 5: the plots of the objective versus. (a) Degraded Phantom, (b) degraded Cameraman, (c) AA method (row : λ = 0.03; row 4: λ = 0.05), (d) RLO method (row : λ = 0.03; row 4: λ = 0.05), (e) our method (row : λ = 0.07 and α = ; row 4: λ = 0.07 and α = ).

23 K = 0 K = Images Methods PSNR(dB) Iter Time(s) PSNR(dB) Iter Time(s) AA Phantom RLO PPB Ours AA Cameraman RLO PPB Ours AA Parrot RLO PPB Ours Table. The comparisons of PSNR values, the number of and CPU-time in seconds by different methods for denoising case. Motion Blur Gaussian Blur Images Methods PSNR(dB) Iter Time(s) PSNR(dB) Iter Time(s) AA Phantom RLO Ours AA Cameraman RLO Ours AA Parrot RLO Ours Table. The comparisons of PSNR values, the number of and CPU-time in seconds by different methods for deblurring with denoising. and the Taylor expansion of log( + x) for x (, ), we get: ( B(K log ) ) B(K) ( = + (K ) log K ( = log + ) + O( K K ). ) + Further, in view of the Taylor expansion of e x, we have B(K ) B(K) K(K ) + O( K ) = ( + K ) exp ( O( K )) = + K + O( K ). Now we are ready to prove Proposition 3.4: Proof. 3

24 (a) (b) (c) (d) (e) Fig..0. Results by different methods for restoring the image Parrot blurred by different kernels and then corrupted by multiplicative noise with K = 0 (row : by motion blur; row 3: by Gaussian blur). (a) Degraded image by motion blur, (b) degraded image by Gaussian blur, (c) AA method (row : λ = 0.05; row 3: λ = 0.04), (d) RLO method (row : λ = 0.05; row 3: λ = 0.04), (e) our method (row : λ = 0.08 and α = ; row 3: λ = 0.07 and α = ). (i) Based on (3.) and θ = K, we have: + p Y (y) log p Y (y)dy = + 0 ( p Y (y) log θ K Γ(K) (K + ) log y K ) y dy. 4

25 Let x = y. Note that the mean of η equals, then we get: + + ( p Y (y) log p Y (y)dy = p η (x) log θ K Γ(K) + K + ) log x Kx dx, = log θ K Γ(K) + K + (ψ(k) log K) K, = log log( KΓ(K)) + K + ψ(k) K, where we use a property of Gamma distribution, E η (log x) = ψ(k) log K d log Γ(K) (Theorem. in [33]), and recall that ψ(k) := dk is the digamma function (see []). (ii) Based on the PDFs of Y and Gaussian distribution N (µ K, σk ), we have: + ( ) + p Y (y) log p N (µk,σk ) (y)dy = p Y (y) log e (y µ K ) σ K dy πσk = + = log( πσ K ) = log(πeσ K), p Y (y) (log( πσ K ) + (y µ K) + where we have utilized the definition of the variance, i.e., p Y (y)(y µ K ) dy = σ K. p Y (y)dy σ K σk + (iii) In view of the results in (i) and (ii), using the approximation of ψ(x) in []: we have: ψ(x) = log x x + O( x ), D KL (Y N (µ K, σk)) = log log( KΓ(K)) + K + = log log( KΓ(K)) + K + ) dy ψ(k) K + log(πeσ K) ( log K K p Y (y)(y µ K ) dy ) K + log(πeσ K) + O( K ) = log + log Kσ K + O( ). (8.4) K Here, we obtain (8.4) by simplifying D KL (Y N (µ K, σk )) with (8.3). Moreover, based on the results in Proposition 3., we have: σ K = E(Y ) E (Y ) = KΓ(K ) Γ(K) = K K = K K KΓ (K ) Γ (K) K K K K 5 Γ (K ) Γ(K) Γ(K ) Γ (K) B(K ). (8.5) B(K)

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