On the application of the spectral projected gradient method in image segmentation

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1 Noname manusript No. will be inserted by the editor) On the appliation of the spetral projeted gradient method in image segmentation Laura Antonelli Valentina De Simone Daniela di Serafino May 22, 2015 Abstrat We investigate the appliation of the nonmonotone spetral projeted gradient SPG) method to a region-based variational model for image segmentation. We onsider a disretize-then-optimize approah and solve the resulting nonlinear optimization problem by an alternating minimization proedure that exploits the SPG2 algorithm by Birgin, Martínez and Raydan SIAM J. Optim., 104), 2000). We provide a onvergene analysis and perform numerial experiments on several images, showing the effetiveness of this proedure. Keywords Image segmentation Region-based variational model Spetral Projeted Gradient Mathematis Subjet Classifiation 2010) 68U10 65K05 90C30 Wor partially supported by INdAM-GNCS 2014 Projet First-order optimization methods for image restoration and analysis; 2015 Projet Numerial Methods for Nononvex/Nonsmooth Optimization and Appliations) and by MIUR FIRB 2010 Projet n. RBFR106S1Z002). L. Antonelli Institute for High-Performane Computing and Networing ICAR), CNR, via P. Castellino 111, I Naples, Italy laura.antonelli@na.iar.nr.it D. di Serafino Department of Mathematis and Physis, Seond University of Naples, viale A. Linoln 5, I Caserta, Italy daniela.diserafino@unina2.it V. De Simone Department of Mathematis and Physis, Seond University of Naples, viale A. Linoln 5, I Caserta, Italy valentina.desimone@unina2.it 1 Introdution: baground and motivations Image segmentation is the proess of partitioning an image into regions that are homogeneous aording to some feature, suh as intensity, texture or olour. This proess is generally aimed at identifying objets in the image, thus it plays a fundamental role in omputer vision, e.g., for objet detetion, reognition, measurement and traing. Many suessful methods for image segmentation are based on variational models where the regions of the desired partition, or their edges, are obtained by minimizing suitable ost funtionals see, e.g., [14, 9, 23, 26] and the referenes therein). In this wor we onsider a region-based variational model by Chan, Esedoḡlu and Niolova [13], heneforth referred to as CEN model, whih has been obtained by applying a suitable relaxation to the Chan-Vese CV) model [15] with the aim of overoming minimization diffiulties assoiated with its non-onvexity. The CV model is atually the Mumford-Shah one [27] restrited to the ase of pieewise onstant two-phase segmentation; its level-set formulation reads as follows: min E CV 1, 2, φ), 1) 1, 2,φ where E CV 1, 2, φ) = Hφx)) dx Ω + λ Hφx)) 1 ūx)) 2 dx Ω ) 2) + 1 Hφx))) 2 ūx)) 2 dx. Ω Here Ω R 2 is an open bounded set with Lipshitz boundary generally a retangle), ūx) : Ω R represents the image to be segmented, H is the Heaviside funtion, φ : Ω R is a Lipshitz-ontinuous funtion whose zero level set represents the boundary Σ of a set

2 2 Laura Antonelli et al. Σ Ω, 1, 2 R, and λ > 0 is a suitable regularization parameter. Solving 1) means finding the best approximation to ūx) among all the funtions that tae only two values; 1 and 2 represent these values, while Σ and Ω\Σ are the sets where they are taen, whih provide a two-phase partition of Ω. The first term in the right-hand side of 2) is a regularization term, whih penalizes the size of Σ. For any fixed φ, the values of 1 and 2 that minimize E CV are given by 1 = 2 = Ω ūx)hφx))dx Ω Hφx))dx, Ω ūx)1 Hφx)))dx Ω 1 Hφx)))dx, 3) i.e., by the mean values of ūx) in the regions Σ and Ω\Σ. Therefore, a natural approah to solve problem 1) is to alternate between the omputation of 1 and 2 through 3) and the minimization of E CV 1, 2, φ) with respet to φ. This minimization is usually performed by using a gradient-desent sheme, where the first variation of E CV 1, 2, φ) is omputed by using a smooth regularization of the Heaviside funtion [15]. However, beause of the non-onvexity of the funtional E CV with respet to φ, this method may get stu into loal minima, thus providing poor segmentations. In order to overome the previous problem, a onvex relaxation approah is used in the CEN model, whih is based on the idea of removing the onstraint that ūx) be approximated by funtions taing only two values. Speifially, in [13, Theorem 2] it is proved that, for any given 1, 2 ), a global minimizer for the pieewise onstant two-phase Mumford-Shah model an be obtained by solving the following onvex problem: min E CEN 1, 2, u), u s.t. 0 u 1, where E CEN 1, 2, u) = u dx Ω + λ ux) 1 ūx)) 2 dx Ω ) + 1 ux)) 2 ūx)) 2 dx, and by taing Ω 4) 5) Σ = {x Ω : ux) µ}, 6) for almost any µ 0, 1). Different methods have been proposed to solve problem 4). In [13] the onstraint 0 u 1 is enfored by using a penalty term and then applying the gradientdesent tehnique to the resulting unonstrained problem. Regularized versions of both u and the penalty term are used to deal with their non-differentiability. A penalty approah, ombined with a suitable splitting of the resulting energy funtional, is also used in [9]. In this ase, the minimization is performed by exploiting the dual formulation of the TV norm and adding a regularization term that allows the appliation of an alternating solution proedure, based on a semi-impliit gradient-desent algorithm and an expliit minimization formula [1]. As observed in [23], the previous methods may introdue too muh regularization, possibly yielding the elimination of fine segmentation details. Furthermore, when expliit disretizations of the gradient flow equation are used, the resulting numerial methods are generally slow, beause of the time step limitations imposed by stability onditions. An approah that avoids regularization is based on the appliation of the Split Bregman SB) method [23]. In this ase, problem 4) is splitted into easier subproblems that an be effiiently solved within an alternating minimization sheme. The onvergene of the SB method has been proved when the subproblems are solved exatly [24, 29]; however, it has been observed that approximate solutions an be used in pratie without deteriorating onvergene speed [23]. Furthermore, in [23] it has been shown that this approah outperforms the duality-based algorithm desribed in [9]. Another approah that exploits the dual formulation of the TV norm without introduing regularization terms has been proposed in [31]. In that wor problem 4) is reformulated as a saddle-point one and solved by the first-order primal-dual method desribed in [12], whih alternates minimizations over the primal and dual variables. We observe that alternating minimization methods for problem 4), or for variants of it, an be derived in the general ontext of Lagrangian methods see, e.g., [30, 26]). There are indeed strong onnetions between the alternating diretion method of multipliers and the SB method, as shown in [20,29]. We also note that a relaxation of the CV model whih is onvex with respet to all its variables has been proposed in [10]. More preisely, the CV model has been approximated by a sequene of onvex problems obtained by using a suitable reformulation of the original problem and a tehnique for embedding the resulting problem into a higher-dimensional spae. The onvex problems an be written as saddle-point ones and solved by using the aforementioned primal-dual method of [12]. However, we do not onsider this approah, sine the results reported in [10] show that its omputational ost an be very high. An interesting disussion on onvex relaxations of non-onvex funtionals used in image proessing an be found in [28]. In this artile we investigate the appliation of the nonmonotone spetral projeted gradient SPG) method,

3 On the appliation of the spetral projeted gradient method in image segmentation 3 within an alternating minimization proedure, to the CEN model. The interest for the SPG method is due to its faster onvergene with respet to lassial gradient projetion methods, whih results from the ombination of the spetral properties of the Barzilai and Borwein BB) steplength [2] with the nonmonotone linesearh tehnique by Grippo, Lampariello and Luidi GLL) [25]. We note that the idea of using steplengths related somehow to the spetrum of the Hessian of the objetive funtion, in order to speedup the onvergene of gradient methods, has gained widespread aeptane in the last years; therefore, an inreasing amount of wor has been devoted to designing and applying new gradient methods that employ suh steplengths see, e.g., [6,8,17 19,21,22,33]). In the ontext of image proessing, the BB steplength has been used, e.g., in [32] to aelerate Chambolle s gradient projetion method for total variation image restoration [11]. In this ase, the nonmonotone line-searh tehnique by Dai and Flether [16] has been applied to ensure global onvergene. Notie that the use of the BB steplength together with a nonmonotone line-searh tehnique allows to overome the limitations on the steplength required by lassial expliit gradient-desent algorithms. On the other hand, the appliation of the SPG method requires some regularization of the TV term; however, we did not experiene any drawbas with using a small amount of regularization in our experiments see Setion 4). We onsider a disretize-then-optimize approah, i.e., we first onsider a disretization of the CEN model and then solve the resulting optimization problem. We fous of the SPG2 algorithm proposed in [4]. The -th iteration of our minimization proedure onsists of two steps: omputation of an approximation u to u by applying an SPG2 step, with a suitable GLL-type line searh, to the disretized CEN funtional where 1 and 2 are equal to the previously omputed values 1 1 and 1 2 ; omputation of 1 and 2 by exat minimization, by using a disrete version of 3) where u is set equal to u. This method is proved to be globally onvergent and its effetiveness is illustrated through numerial experiments on several test problems. This paper is organized as follows. In Setion 2 we provide a disrete formulation of the CEN model. In Setion 3 we desribe our SPG-based minimization algorithm and prove its onvergene. In Setion 4 we disuss the results of numerial experiments performed by applying our algorithm to several images. We also mae a omparison with the SPG2 algorithm and with the SB tehnique presented in [23]. Finally, in Setion 5, we draw some onlusions. 2 Disrete formulation The image domain Ω is disretized by using an m n grid of pixels Γ = {i, j) : 0 i m 1, 0 j n 1}. We identify eah pixel with its enter and denote by v i,j the value in i, j) of any funtion v defined in Ω. Now we introdue some finite-differene operators that are useful to present the disretization of the funtional 5) over the grid Γ. For any v i,j with i, j) Γ, we set δ x +v i,j = v i+1,j v i,j, δ y +v i,j = v i,j+1 v i,j, δ x v i,j = v i,j v i 1,j, δ y v i,j = v i,j v i,j 1, where we assume v i 1,j = v i,j for i = 0, v i,j 1 = v i,j for j = 0, v i+1,j = v i,j for i = m 1, v i,j+1 = v i,j for j = n 1, i.e., we define by repliation the values of v with indies outside Γ. To disretize the total variation term in 5) we proeed as in [11]; however, in order to deal with non-differentiability, we use a lassial regularization of u, i.e., we set u i,j = δ x +u i,j ) 2 + δ y +u i,j ) 2 + ε, where ε is a small positive parameter. Then we onsider the following disretization of the funtional E CEN : E 1, 2, u) = u i,j i,j + λ ) 1 ū i,j ) 2 u i,j + 2 ū i,j ) 2 1 u i,j ), i,j where the indies i and j in the sum run from 0 to m 1 and from 0 to n 1, respetively. Obviously, the funtion E 1, 2, u) is ontinuously differentiable. Its derivatives tae the following form: E = 2λ 1 i,j E = 2λ 2 i,j 1 ū i,j ) u i,j, 2 ū i,j ) 1 u i,j ), E = a 0 u i,j u i,j a 1 u i+1,j a 2 u i 1,j a 3 u i,j+1 a 4 u i,j 1 + λ 1 ū i,j ) 2 2 ū i,j ) 2), 7)

4 4 Laura Antonelli et al. where δ ) x 2 a 1 = + u i,j + δ y ) ) u i,j + ε, δ ) x 2 a 2 = u i,j + δ y ) ) u i 1,j + ε, δ ) x 2 a 4 = + u i,j 1 + δ y ) ) u i,j + ε, a 3 = a 1, a 0 = a 1 + a 2 + a 3 + a 4. 8) By imposing E/ 1 = E/ 2 = 0 we trivially obtain the values of 1 and 2 that minimize E for any fixed u: 1 = 2 = i,j ūi,ju i,j i,j u i,j i,j ūi,j1 u i,j ) i,j 1 u. i,j), From 9) and 0 u i,j 1 it follows that u min 1, 2 u max, 9) where u min = min i,j ū i,j and u max = max i,j ū i,j. Note also that 9) provides a disretization of 3) where Hφx)) has been replaed by ux). In the following we fous on the solution of the problem min E 1, 2, u), s.t. 0 u 1, u min 1, 2 u max, 10) where 0 u 1 is intended omponentwise. Heneforth, to simplify the notation, we denote by G the vetor E, by G the vetor with omponents E/ 1 and E/ 2, and by G u the vetor with omponents E/ u i,j we impliitly assume some ordering in the grid of pixels). For any vetors v and w, we use the notation v, w) to denote v T, w T ) T ; furthermore, we denote by, the Eulidean inner produt and by the indued vetor norm. 3 An SPG-based alternating method for the disretized CEN model We ompute a solution to problem 10) by exploiting the SPG2 spetral projeted gradient method proposed in [4] within an alternating minimization framewor. Therefore, we first provide a short desription of SPG2. Given a losed onvex set X R n and a funtion f : X R that is ontinuously differentiable in an open set ontaining X, the SPG2 method attempts to ompute a minimizer of f in X by building a sequene of approximate solutions as follows: x +1 = x + θ d. Here the searh diretion d is defined as d = P X x α fx ) ) x, where P X denotes the orthogonal projetion on X, α is obtained through the BB formula α BB = s 1, s 1 s 1, y 1, 11) with s 1 = x x 1 and y 1 = fx ) fx 1 ), and the steplength θ is suh that the Grippo-Lampariello- Luidi GLL) ondition [25] holds, i.e., fx + θ d ) max 0 j min{,ν 1} fx j ) + γ θ fx ), d 12) with ν N and γ 0, 1). More preisely, given a small parameter α min > 0 and a large parameter α max > 0, the value of α is set as { { min α αmax, max { }} α = min, αbb, if α BB > 0, 13) α max, otherwise. A steplength θ satisfying 12) is omputed by a line searh proedure, starting from the trial value θ0 = 1. That is, if θj 1 is suh that 12) does not hold, then a new andidate steplength θj is omputed by using a lassial one-dimensional quadrati interpolation tehnique; furthermore, given the safeguard parameters σ 1 and σ 2, with 0 < σ 1 < σ 2 < 1, if θj does not belong to [σ 1, σ 2 θj 1 ], then θ j = θ j 1 /2. It an be proved that the SPG2 algorithm is onvergent { } in the sense that any limit point ˆx of the sequene x generated by SPG2 is a stationary point of f in X, or equivalently P X ˆx fˆx)) ˆx = 0. Our alternating minimization method based on SPG2 has the following struture: given u, 1, 2) suh that 9) holds, we first ompute u +1 by applying an SPG step to Eu, 1, 2) with 1, 2) fixed, and then ompute +1 1 and +1 2 by 9), using u +1. However, in the omputation of u +1 = u + θ d u, where d u is the SPG searh diretion, we require that E 1, 2, u + θ d ) u max E j 1, j 2, u j) 14) 0 j min{,ν 1} + γ θ G u 1, 2, u ), d u, i.e., we use a GLL ondition that involves values of 1 and 2 different from the fixed ones 1 and 2 unless ν = 1). On the other hand, sine G 1, 2, u ) = 0, eah iteration of the alternating minimization proedure an be regarded as the appliation of a projeted gradient step to the funtion E with respet to all variables, followed by an improvement of the approximate solution resulting from that step. Furthermore,

5 On the appliation of the spetral projeted gradient method in image segmentation 5 Algorithm 3.1 SPGASeg) given tol > 0, α max > α min > 0, α 0 [α min, α max ], 0 < σ 1 < σ 2 < 1, ν N, γ 0, 1) hoose u 0 and ompute 0 1 and 0 2 as in 9), using u 0 set = 0 while PW u G u 1, 2, u )) u > tol do ompute d u = P W u α G u 1, 2, u )) u ompute θ that satisfies 14) by a line searh along u + θd u and set u +1 = u + θ d u ompute +1 1 and +1 2 as in 9), using u +1 ompute α +1 aording to 13) and 11), where s = u +1 u and y = G u +1 1, +1 2, u +1 ) G u 1, 2, u ) set = + 1 end while letting d = d u, , ), the GLL ondition holds with respet to all variables, i.e., E +1 1, +1 2, u +1) max E j 1, j 2, u j) 15) 0 j min{,ν 1} + γ θ G 1, 2, u ), d, although +1 1, +1 2, u +1) is not obtained through a line searh along d. The SPG-based Alternating proedure desribed so far, heneforth referred to as SPGASeg SPG-based Alternating proedure for Segmentation), is summarized in Algorithm 3.1, where W = [0, 1] mn. Note that, sine G 1, 2, u ) = 0 for all, the stopping ondition reported in the algorithm is equivalent to PV 1, 2, u ) G 1, 2, u )) 1, 2, u ) > tol, where V = [u min, u max ] 2 [0, 1] mn. Furthermore, Algorithm SPGASeg differs from the inexat blo oordinate desent methods presented in [7], beause it does not satisfy onditions 3.5) in [7] and uses a nonmonotone line-searh tehnique. Now we prove that Algorithm SPGASeg preserves the onvergene properties of SPG2. To this end, we first observe that for all 1, 2, u) V and t 0, α max ], G 1, 2, u), G t 1, 2, u) where the notation 1 t Gt 1, 2, u) 2 1 α max G t 1, 2, u) 2, G t 1, 2, u) = P V 1, 2, u) t G 1, 2, u)) 1, 2, u) 16) has been used for simpliity. This is a nown result see [4, Lemma 2.1]), whih follows straightforwardly from the properties of the projetion on a onvex set see, e.g., [3, Proposition 2.1.3, b)]). In order to shorten the notation, we also define G t u 1, 2, u) = P W u t G u 1, 2, u)) u. The onvergene of Algorithm SPGASeg is stated in the following theorem. Theorem 1 Algorithm SPGASeg is well defined and any limit point of the sequene { 1, 2, u )} that it generates is a stationary point of E in V. Proof This proof losely follows the proof of Theorem 2.4 in [4], however we outline it for the sae of ompleteness. If 1, 2, u ) is not a stationary point for E in V, then, by 16), for all t 0, α max ] we have G 1, 2, u ), G t 1, 2, u ) 1 t It follows that Gu 1, 2, u ), d u G t 1, 2, u ) 2 1 α max G t 1, 2, u ) 2 < 0. = G u 1, 2, u ), G t u 1, 2, u ) < 0, beause G 1, 2, u ) = 0 for all. Therefore, d u is a desent diretion for E 1, 2, u ) in u, and a stepsize θ suh that 14) holds an be omputed with a finite number of trials. Thus, Algorithm SPGASeg is well defined. Now we suppose that ĉ 1, ĉ 2, û) V is a limit point of the sequene generated by Algorithm SPGASeg. Without loss of generality, we an assume that the sequene

6 6 Laura Antonelli et al. { 1, 2, u )} onverges to ĉ 1, ĉ 2, û). We distinguish two ases. Case 1: inf θ = 0. By ontradition we suppose that ĉ 1, ĉ 2, û) is not a stationary point, i.e., G α ĉ 1, ĉ 2, û) 0 for all α > 0. From G 1, 2, u ) = 0 it follows that G ĉ 1, ĉ 2, û) = 0 and hene G α uĉ 1, ĉ 2, û) 0. Then, by reasoning as in the proof of Theorem 2.4 in [4], we find that there exists δ > 0 suh that for all suffiiently large, G u 1, 2, u ) G α u, 1, 2, u ) G α u 1, 2, u) < δ 2, 17) for all α [α min, α max ]. Sine inf θ = 0, there exists K N suh that lim K θ = 0. Then, moving from the way θ is hosen to satisfy the GLL ondition 14) and proeeding again as in [4], we find that for all suffiiently large K, there exists ρ, with 0 < σ 1 ρ σ 2, and t [0, θ /ρ ] suh that Gu 1, 2, u + t d ) u, d u > γ Gu 1, 2, u ), d u. Hene, by taing a subsequene { } d u K 1 K suh that d u/ d u is onvergent with limit ˆd u, we get G u ĉ 1, ĉ 2, û), ˆd u = 0. From d u = G α u 1, 2, u ) it follows, by ontinuity, that for all suffiiently large K 1, G u 1, 2, u ) G α u, 1, 2, u ) G α u 1, 2, u) > δ 2, whih ontradits 17). Case 2: inf θ ρ > 0. By ontradition, we suppose that ĉ 1, ĉ 2, û) is not a onstrained stationary point. Then G α ĉ 1, ĉ 2, û) > 0 for all α 0, α max ] and hene there exists δ > 0 suh that G α ĉ 1, ĉ 2, û) δ for all α [ρ, α max ]. Let l) be an integer suh that min{, ν 1} l) and E l) 1, l) 2, u l)) = max E j 1, j 2, u j). 0 j min{,ν 1} By reasoning{ as in the proof of)} Theorem 2.4 in [4], we find that E l) 1, l) 2, u l) is a noninreasing sequene; furthermore, for suffiiently large, we have G α l) 1, l) 2, u l)) δ/2. Then, by 15) and 16), we get E l) 1, l) 2, u l)) E l) 1 1, l) 1 2, u l) 1) γρ G l) 1 α l) 1 1, l) 1 2, u l) 1) 2 α max E l) 1 1, l) 1 2, u l) 1) γδ2 ρ. 4α max It follows that lim E l) 1, l) 2, u l)) =, whih is a ontradition beause, by ontinuity, lim E l) 1, l) 2, u l)) = E ĉ 1, ĉ 2, û). This ompletes the proof. It is worth noting that Algorithm SPGASeg an be applied in a more general ontext than the one onsidered in this wor. More preisely, let X Y R p R q be a losed onvex set and F x, y) a real-valued funtion that is ontinuously differentiable in an open set ontaining X Y. If we assume that for all y Y argmin x X F x, y) exists and an be exatly omputed, then we an apply Algorithm SPGASeg to F x, y), with x playing the role of 1, 2 ) and y the role of u. Of ourse, the onvergene of the algorithm still holds. 4 Computational experiments We performed numerial experiments to evaluate the effetiveness of our approah. To this aim, we applied the Algorithm SPGASeg to several gray-sale images with different sizes and features, whih are widely used in the literature of image segmentation. For the sae of spae, we report only the results onerning six images, showing either real senes or syntheti pitures, whih are representative of the general behaviour of the proposed method see the first olumn in Figures 1-2). Algorithm SPGASeg was implemented in C, in single preision, exploiting the implementation of Algorithm SPG2 [5] available as a part of the TANGO projet egbirgin/tango/). It was run under Matlab R2011b, v. 7.13), through a MEXinterfae, using the Image Proessing Toolbox to read and display images. The C ode was ompiled by using g v ) and the experiments were performed on an Intel Core i5 proessor with lo frequeny of 2.7 GHz and 8 GB of RAM.

7 On the appliation of the spetral projeted gradient method in image segmentation 7 ameraman SPGASeg SB arplate SPGASeg SB zebras SPGASeg SB Fig. 1 Real senes and orresponding segmentations by SPGASeg and SB. The number of pixels of eah image is speified after the name. The images have been resized using different salings to fit the figure layout. The parameters in the objetive funtion E 1, 2, u) were set as λ = 0.5 and ε = 10 6, and the threshold parameter in 6) as µ = 0.5. Following the implementation of SPG2, the l norm was used instead of the l 2 norm in the SPGASeg stopping riterion, with tol = A maximum number of iterations equal to 1000 and a maximum number of funtion evaluations equal to were also onsidered. The values ν = 10 and γ = 10 4 were used in the GLL ondition 14). The parameters α min, α max, σ 1 and σ 2 were left equal to their default values in SPG2. The initial value of α was also hosen as in SPG2, i.e., α 0 = min{α max, max{α min, 1/ G 1 u 0 1, 0 2, u 0 }}. For eah test problem we used three hoies of the initial funtion u 0 : ū [0,1], the image to be segmented, with intensity values saled between 0 and 1; u ws, a small white square p q pixels, with p m and q n) on bla baground; u bs, a small bla square p q pixels, with p m and q n) on white baground. Sine the three funtions led to the same segmentations, we report here the results obtained by hoosing u 0 = ū [0,1]. For omparison purpose, we also applied to the CEN model the SPG2 algorithm and the SB-based algorithm desribed in [23], for all the test problems. The former was hosen to evaluate the behaviour of our approah with respet to a straightforward appliation of the spetral projeted gradient algorithm; the latter is

8 8 Laura Antonelli et al. irles SPGASeg SB ninetyeight SPGASeg SB spiral SPGASeg SB Fig. 2 Syntheti pitures and orresponding segmentations by SPGASeg and SB. The number of pixels of eah image is speified after the name. The images have been resized using different salings to fit the figure layout. nown to be very effiient and therefore it was taen as a referene algorithm for our analysis. The SB approah omputes an approximation u +1 to u by performing the following steps: u +1, d +1 ) = argmin 0 u 1; d Ω d + λur + η 2 d u b 2) dx, 18) b +1 = b + u d, 19) where r = 1 ū) 2 2 ū) 2. Problem 18) is approximately solved by an alternating minimization proedure, where u +1 is obtained by applying the Gauss- Seidel method to the Euler-Lagrange equations with d = d ) and projeting the omputed solution in [0, 1], while d +1 is obtained by applying the so-alled shrin operator see [23] for more details). A single preision version of the aforementioned implementation of SPG2 was used in the experiments, with the same values of the parameters as in the SP- GASeg ode. Conerning the SB approah, the implementation available from was applied. This is a single-preision C ode with a MEX interfae, allowing to run it under Matlab. Its native stopping riterion for the outer iterations was substituted by the stopping riterion used in SPGASeg and SPG2, in order to mae a fair omparison. The parameter η in 18) was set equal to 1; the stopping riterion for the Gauss-Seidel iterations remained unhanged. Details about the behaviour of SPGASeg, SPG2 and SB are given in Table 1. For all the algorithms we show the number of iterations, nit, performed to satisfy the stopping riterion. For SPGASeg and SPG2 we also show the number, nf, of objetive funtion evaluations, eah requiring Omn) floating-point operations.

9 On the appliation of the spetral projeted gradient method in image segmentation 9 IMAGE SPGASeg SPG2 SB nit nf nit nf nit n GS ameraman arplate zebras irles ninetyeight spiral noisy ameraman noisy ninetyeight Table 1 Details about the exeution of SPGASeg, SPG2 and SB on the seleted test problems indiates that the required auray has not been ahieved within the maximum number of iterations). The number of gradient evaluations, whih osts Omn) floating-point operations plus the omputation of the square roots in 8), is not reported sine it is equal to the number of iterations plus 1. For the SB algorithm we also show the number, n GS, of Gauss-Seidel iterations, eah performing Omn) floating-point operations. The number of updates of d +1 and b +1 is equal to n GS eah update is again Omn)). We see that SPGASeg is muh more effiient than SPG2. SPGASeg generally ahieves the required auray in at most 7 iterations, with at most 15 objetive funtion evaluations, while SPG2 requires a muh larger number of iterations and objetive funtion evaluations. This behaviour is more evident for images with a partiular texture, suh as zebras or ninetyeight, or orrupted by a large amount of noise, suh as arplate and spiral. The large inrease in the number of SPG2 iterations beause of noise is onfirmed by further numerial experiments, performed adding noise to some images. For example, in the last two rows of Table 1 we report the results obtained after orrupting the real image ameraman and the syntheti one ninetyeight with Gaussian noise having zero mean and standard deviation values of 15 and 25, respetively the orrupted images are shown in the left olumn of Figure 3). We see that the noise strongly deteriorates the performane of SPG2, while it slightly affets the behaviour of SP- GASeg note that SPG2 does not ahieve the required auray within the maximum number of iterations on the noisy ninetyeight image). Conversely, the SPGASeg and SB algorithms perform about the same number of iterations and show similar time omplexity for all the test images. In order to evaluate more ompletely the effiieny of our approah, we also ompare the exeution time of the SPGASeg ode with that of the SB one. These times, obtained as average values over 20 runs, are reported in Table 2. Note that the time measurement of SB does not IMAGE SPGASeg SB ameraman arplate zebras irles ninetyeight spiral noisy ameraman noisy ninetyeight Table 2 Exeution times seonds) of SPGASeg and SB on the seleted test problems. inlude the gradient evaluations needed by the modified stopping riterion introdued in that ode for omparison purpose. We see that for arplate, spiral and noisy ameraman the times of SPGASeg and SB are omparable, while for the other images SB is slightly faster. This is a satisfatory result, sine the SB method is nown to be very effiient on the segmentation model onsidered in this paper. Furthermore, we thin that our implementation of SPGAseg an be improved to enhane its performane; however, this issue is beyond the sope of this artile. Finally, we highlight that the segmentations obtained using SPGASeg and SB are the same, for both the images of the original test set and those obtained by adding Gaussian noise see the seond and third olumn of Figures 1-3). In partiular, the small amount of regularization introdued in the TV term does not affet the results. The quality of the segmentations is generally good, sine the meaningful regions of eah image are identified. 5 Conlusions We presented an alternating minimization proedure alled SPGASeg, whih exploits the nonmonotone SPG method for solving the formulation of segmentation problems provided by the CEN model. The hoie of the SPG method was motivated by its steplength seletion strategy, allowing faster onvergene than lassial gradient methods. We proved that Algorithm SPGASeg is globally onvergent to a onstrained stationary point of the disretized CEN model. Numerial experiments on images widely used in the image segmentation literature showed that SPGASeg is able to solve problem 10) with a small number of iterations, produing segmentations equal to those obtained with the SB algorithm presented in [23], whih is very effetive on the model onsidered in this wor. Furthermore, the effiieny of

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