Semi-explicit Solution and Fast Minimization Scheme for an Energy with l 1 -Fitting and Tikhonov-Like Regularization

Transcription

1 DOI /s y Sem-explt Soluton and Fast Mnmzaton Sheme for an Energy wth l 1 -Fttng and Tkhonov-Lke Regularzaton Mla Nkolova Sprnger Sene+Busness Meda, LLC 2008 Abstrat Regularzed energes wth l 1 -fttng have attrated a onsderable nterest n the reent years and numerous aspets of the problem have been studed, manly to solve varous problems arsng n mage proessng. In ths paper we fous on a rather smple form where the regularzaton term s a quadrat funtonal appled on the frst-order dfferenes between neghborng pxels. We derve a semexplt expresson for the mnmzers of ths energy whh shows that the soluton s an affne funton n the neghborhood of eah data set. We then desrbe the volumes of data for whh the same system of affne equatons leads to the mnmum of the relevant energy. Our analyss nvolves an ntermedate result on random matres onstruted from trunated neghborhood sets. We also put n evdene some drawbaks due to the l 1 -fttng. A fast, smple and exat optmzaton method s proposed. By way of applaton, we separate mpulse nose from Gaussan nose n a degraded mage. Keywords Non-smooth analyss l 1 data-fttng Image denosng Sgnal denosng Random matres Tkhonov regularzaton Impulsve nose Nonsmooth optmzaton Numeral methods 1 Introduton For any postve nteger p>0 e.g. the number of the pxels n an mage we denote I def ={1,...,p}. M. Nkolova CMLA, ENS Cahan, CNRS, PRES UnverSud, 61 Av. Presdent Wlson, Cahan, Frane e-mal: nkolova@mla.ens-ahan.fr Wth every ndex I we assoate a subset N I suh that I and j I,wehave { / N, 1 j N N j. Typally, N represents the set of the neghbors of. Thus we all N I the neghborhood system on I. We onsder the followng mnmzaton problem: for any y R p, fnd ˆx suh that F ˆx,y = mn Fx, y, x Rp p where Fx, y = x y + α p x x j 2, 2 2 =1 =1 where α>0 s a parameter. In ths paper we study how the soluton ˆx depends on y. The pratal ontext we have n mnd s sgnal or mage proessng where y s the data and ˆx s the sought-after sgnal or mage. In the latter ase we assume that all pxels of the mage are arranged n a p-length vetor and that N I s an usual neghborhood system orrespondng to the 4 or the 8 adjaent pxels. E.g., for a m n-sze pture whose olumns are onatenated n a p = mn-length vetor, the 4 adjaent neghbors of a pxel n the nteror of the mage are N ={ m, 1, + m, + 1}. Any other onfguraton fttng 2 an also be onsdered. The ontnuous verson of the energy n 2, where α orresponds to a Lagrange multpler for an underlyng onstraned optmzaton problem, was onsdered n [19] to denose smooth H0 1 -regular mages. The authors propose and analyze an atve-set method for solvng the onstraned non-smooth optmzaton problem. Regularzed energes wth l 1 -fttng were onsdered by Allney n [1, 2] n the ontext of one-dmensonal flters.

2 Energes nvolvng general non-smooth data-fttng terms were analyzed n [21] where t was shown that the orrespondng mnmzers ˆx satsfy ˆx = y for numerous ndexes. 3 Ths result ntally suggested applatons for mpulse nose proessng. The geometr propertes of mages restored usng l 1 -fttng and total-varaton TV regularzaton were studed by Chan and Esedoglu n [9] and appled to the restoraton of bnary mages n [10]. Deblurrng under mpulse nose usng l 1 -fttng and dfferent regularzaton terms was explored n [3 5]. Fast optmzaton methods for ths knd of problems were proposed by [13, 14, 16]. The frst two papers, wrtten by Darbon and Sgelle, are based on graph-uts and suppose that mages are quantzed. A lot of papers usng l 1 -fttng wth regularzaton were publshed n the last years and t s hard to evoke all of them. In ths paper we fous on l 1 -fttng wth a smple Tkhonov-lke regularzaton for two man reasons: n ths ontext t s possble to obtan a lot of explt results on the propertes of the soluton and we an provde a fast and smple numeral sheme. Our Contrbuton Our man ontrbuton s to show that F, as defned n 2, s mnmzed by an ˆx whh s a loally affne funton of the data y. We exhbt an affne formula gvng the result and determne a subset of data where the same formula leads to the mnmum of the relevant energy. Suh a set s seen to be a polyhedron of R p whh s unbounded. Sne many omponents of the soluton satsfy ˆx = y, we deompose the remanng samples ˆx y nto onneted omponents. We show that eah onneted omponent s estmated usng amatrxwthnon-negatve entres appled to the adjaent pxels satsfyng ˆx = y, plus a fxed vetor dependent only on the sgn of all y ˆx n the onneted omponent. An nterestng ntermedate result s to prove the nvertblty and the postvty of ths matrx whh s random, sne t orresponds to randomly trunated neghborhoods. We also gve a very fast and smple mnmzaton sheme whh easly reovers the most dffult ponts those where the energy s non-dfferentable, namely ˆx = y. It s fully explt there s no lne-searh and nvolves only sums and omparsons to a fxed threshold. Numeral results on data ontamnated wth mpulse nose are provded n order to llustrate the propertes of the energy and the mnmzaton sheme. Other applatons an ertanly be envsaged. 1.1 Notatons and Defntons We use the symbol to denote the l 2 -norm of vetors, x = x,x. Next, we denote by N the postve ntegers. For a subset ζ I, the symbol #ζ stands for ts ardnalty whle ζ s the omplement of ζ n I. Defnton 1 A subset ζ I s sad to be a onneted omponent wth respet to N I f ether ζ s a sngleton, say ζ ={} and ζ N =, or f the followng hold:, j ζ 2, n N, k n k=1 suh that { 1 = and n = j 4 k N k +1 ζ, k = 1,...,n 1, def N ζ = N \ ζ ζ. 5 ζ By extendng 1, we all N ζ n 5 the neghborhood of ζ. The ondton n 4 means that any and j n ζ are onneted by a sequene of neghbors w.r.t. N I that belong to ζ. The requrement n 5 means that ζ s maxmal there are no other elements n I \ ζ that an satsfy 4. Guded by the property n 3, we systematally use the set-valued applaton J from R p R p to the famly of all subsets of I defned by Jx,y def = { I : x y }. 6 Note that F., y, as defned n 2, s nondfferentable n the lassal sense; t has ontnuous partal dervatves Fx,y x for Jx,y but not for J x, y. We reall some fats about nondfferentable funtons see e.g. [18]. Defnton 2 The rght-sde dervatve of F., y at x n the dreton of u s defned by δfx, yu def Fx + εu,y Fx, y = lm, ε 0 ε whenever ths lmt exsts. Ths one-sded dervatve always exsts for the energy n 2 sne t s onvex and ontnuous. Let us remnd that the relevant left-sde dervatve s δfx, y u. If F s nonsmooth at x along u, wehaveδfx, y u δfx, yu and by the onvexty of F the followng nequalty holds: δfx, y u < δfx, yu. More detals an be found e.g. n [18]. Otherwse, f F s smooth at x along u, wejusthave δfx, y u = δfx, yu. Fnally, f F., y s dfferentable at x n the lassal sense, we have δfx, yu = x Fx, y, u whh s lnear n u. The omponents of a matrx A are denoted by A, j. Gven a vetor y R p, ts th omponent s denoted by y or y. We wll wrte y 0 to say that all omponents of

3 y are nonnegatve. If K I s an ordered subset of ndexes, say K = k 1,...,k #K, then y K s the restrton of y to the ndexes ontaned n K: [y K ] = y k, = 1,...,#K. 7 For defnteness, we suppose n what follows that all subsequenes are arranged n nreasng order. For any = 1,...,p, the symbol e systematally denotes the th vetor of the anonal bass of R p, namely e = 1 and e j = 0, j, for all = 1,...,p. 2 Neessary and Suffent Condtons for a Mnmum Sne F., y s ontnuous, onvex and oerve for every y R p, t always has a unque mnmum and the latter s reahed on a onvex set. See [18, 24] for detals. Frst we determne the neessary and suffent ondtons for F., y to have a mnmum at ˆx. Proposton 1 For y R p, the funton F., y gven n 2 reahes ts mnmum at ˆx R p f and only f ˆx 1 #N 1 Jˆ, 8 2α#N ˆx 1 = σ #N 2α#N for σ = sgny ˆx ˆ J, 9 where Jˆ def = Jˆx,y s defned aordng to 6. Remark 1 It s worth emphaszng that from the defnton of J ˆ,8 orresponds to ˆx = y, ˆ J. Remark 2 Equatons 8 and 9 show that the restored ˆx e.g. an mage or a sgnal nvolves a frm bound on the dfferene between eah sample ˆx and the mean of ts neghbors: ˆx 1 #N 1, I. 2α#N Aordng to the value of α, textures or non-spky noses an be preserved n the soluton ˆx. Proof Beng onvex, F., y reahes ts mnmum at ˆx f and only f see e.g. [18], vol. I, Theorem on p. 253 δf ˆx,yu 0, u R p, 10 where δf ˆx, yu s the rght-sde dervatve of F., y at ˆx n the dreton of u, see Defnton 2. Let us deompose F ˆx,y n the followng way: F ˆx,y = f y ˆx + J ˆ ˆx + F ˆx, Jˆ where f y x = x y, J ˆ x = α x x j 2, 2 Jˆ Fx = x y + α x x j 2. 2 Jˆ Clearly, f y s nondfferentable at ˆx for every Jˆ sne ˆx = y whereas J ˆ and F are dfferentable at ˆx. Usng Defnton 2, for any u R p, δf ˆx,yu = J ˆ δf y ˆx u + u x x Jˆ x=ˆx + ˆ J u Fx x. 11 x=ˆx Next we alulate the terms nvolved n these sums. For every Jˆ we have ˆx + εu y ˆx y δf y ˆx u = lm ε 0 ε = u =u sgnu. It s easy to fnd that J ˆ x = 2α#N x 1 x #N x j. Note that the onstant 2 omes from the regularzaton term n 2 whh nvolves both x x j 2 and x j x 2. For every J ˆ,wehave ˆx y and the partal dervatve below s well defned: Fx x = sgn ˆx y + 2α#N ˆx 1. x=ˆx #N Introdung these results n the expresson for δf ˆx,y n 11 yelds: δf ˆx,yu = u sgnu + 2α#N ˆx 1 #N Jˆ + ˆ J u sgn ˆx y + 2α#N ˆx 1 #N. 12

4 If Jˆ s empty, the frst sum n 12 s null and 8svod. Consder next that Jˆ s nonempty. Let us apply 10 tothe expresson of δf ˆx,y n 12 wth e and e as defned at the end of Set. 1.1 for Jˆ : δf ˆx,ye 0, Jˆ 1 + 2α#N ˆx 1 #N δf ˆx,y e 0, Jˆ 1 2α#N ˆx 1 #N 0, 0, sne e sgne = e sgn e = 1. Combnng these results leads to 1 2α#N ˆx 1 1. #N Hene 8. If Jˆ s empty, the seond sum n 12 s null and 9 s vod. Consder next that Jˆ s nonempty. Let us apply 10to the expresson of δf ˆx,y n 12 wth e and e for J ˆ: δf ˆx,ye 0, ˆ J sgn ˆx y + 2α#N ˆx 1 #N δf ˆx,y e 0, ˆ J sgn ˆx y 2α#N ˆx 1 #N Combnng the last two results shows that sgn ˆx y + 2α#N ˆx 1 #N = 0, 0, 0. whh s equvalent to ˆx 1 sgny ˆx = 0, #N 2α#N J. ˆ 13 Ths amounts to 9. Lemma 1 We assume the ondtons of Proposton 1. The onstant σ { 1, +1} n 9 satsfes σ = sgny ˆx = sgn y 1, J. ˆ #N 14 Proof We know that Jˆ we have ether y 1 < 1 < 0 #N 2α#N or y 1 > 1 > 0, #N 2α#N sne otherwse we would fnd ˆx = y and Jˆ aordng to 8. These two ases are onsdered below. Consder that y 1 #N < 0. Subtratng 13 from the latter nequalty yelds y 1 ˆx 1 sgny ˆx < 0. #N #N 2α#N Usng that u = u sgnu for any u R, ths s equvalent to sgny ˆx ˆx y + 1 < 0. 2α#N Sne the expresson between the bg parentheses s postve, we fnd that σ = sgny ˆx = sgn y 1 = 1. #N Consder that y 1 #N > 0. Subtratng 13 from the latter nequalty yelds y 1 ˆx 1 #N #N Ths s equvalent to sgny ˆx ˆx y + 1 2α#N sgny ˆx 2α#N > 0. It follows that σ = sgny ˆx = sgn y 1 = 1. #N The proof s omplete. > 0. Now we an state a more handy formulaton of the mnmalty ondton gven n Proposton 1. Theorem 1 For y R p, the funton F., y gven n 2 reahes ts mnmum at ˆx R p f and only f y 1 #N 1 Jˆ, 15 2α#N

5 Fg. 1 Illustraton of Example 1 ˆx 1 #N Jˆ for σ = sgn = 1 #N y j + Jˆ y 1 #N σ 2α#N where Jˆ def = Jˆx,y s defned aordng to 6. ˆ J, 16 Proof Usng Remark 1, we obtan 15 dretly from 8 and splt the frst equaton n 9 to obtan the frst equaton n 16. Then we use the expresson for σ derved n Lemma 1. Remark 3 Usng 8 n Proposton 1, t s easy to see that [ J ˆ = ] [ Jˆ = I ] [ ˆx = y, I ] [ ] y W I, where { def W I = y R p : y 1 #N y j 1, 2α#N } I. 17 Obvously, W I s a polyhedron enlosed between p = #I pars of affne hyperplanes n R p. Its Lebesgue measure n R p s learly postve. Obtanng a soluton ˆx = y s useless, so α should be suh that y/ W I, hene we need α>α mn, where def α mn = mn 2#N y 1 I #N y j 1. Example 1 Consder the ost-funton F : R R R, Fx, y = x y + α 2 x2, 18 llustrated n Fg. 1a and b. For any y R, the energy F., y above s mnmzed by y f y 1 ˆx = α, sgny f y > 1 19 α α. The soluton ˆx as a funton of y s plotted n Fg. 1. observe that ˆx fts the data y whenever y 1/α; otherwse, for y α 1, there s a threshold effet, so the value of ˆx s ndependent of the exat value of y, t depends only on ts sgn. The funton y ˆx s lnear on eah one of the subsets, 1 α, [ 1 α, 1 α ] and 1 α, +. Example 2 s qute pathologal sne under some ondtons, the mnmzer s not unque. Example 2 Consder the ost-funton F : R 2 R 2 + R of the form 2, Fx, y = 2 x y + α 2 =1 2 =1 x x j 2 = x 1 y 1 + x 2 y 2 +αx 1 x 2 2, 20 where the smplfaton omes from the fats that N 1 ={2} and N 2 ={1}, and hene #N = 1for = 1, 2. Several ases arse aordng to the values of y 1 and y 2. If y 1 y 2 2α 1, Proposton 1 shows that ˆx 1 = y 1 and ˆx 2 = y 2,soJ ˆ =. Suh a ase s llustrated on Fg. 2a. For defnteness, let y 2 y 1 > 2α 1. Usng Proposton 1, we fnd that the soluton s gven by the segment [y 1,y α, y 2 2α 1,y 2] see Fg. 2b. Its extreme ponts orrespond to Jˆ ={1} and Jˆ ={2}. Ths pathologal behavor s due to the fat that n the nteror of the segment, ˆx 1 and ˆx 2 are alulated usng 9 whh s smplfed to a unque equaton, 1 + 2αˆx 1 ˆx 2 = 0. Note also that the onneted omponent ζ see Defnton 1 reads ζ = J ˆ = I ={1, 2}. More generally, f ˆ J = I, then ˆx s alulated usng 9 only and the mnmzer may not be unque. As seen n Set. 7

6 Fg. 2 Both fgures present the level lnes of F n 20for α = 0.5 near the data and the mnmum. In both fgures, the data pont y s plotted wth a. On the left, the mnmzer ˆx math the data y. On the rght, the mnmzer s any element of the segment [0, 1, 1, 2] plotted wth a thk lne Experments, the ase when Jˆ s empty.e. J ˆ = I sexeptonal; otherwse mnmzers are well defned, and hopefully, nterestng for pratal applatons. Remark 4 We know from Proposton 1 n [22] that f Jˆ s nonempty and f the ondton n 8 nvolves at least one strt nequalty, then the mnmzer ˆx of F s unque. We an onjeture that the ases when Jˆ s empty or 8 nvolves only equaltes are qute exeptonal n prate. If for some y R p the set Jˆ s empty, then the mnmalty ondtons are gven by 9 for all I. Sne F., y admts a mnmzer for every y R p, ths system admts at least one soluton; the latter t s not neessarly unque. In any ase, suh a soluton has no pratal nterest sne t s quas-ndependent of the data; t also ndates that the parameter α s too large. 3 Random Matres from Restrted Neghborhoods Lemma 2 Let ζ ={k 1,...,k q } I, where q def = #ζ <p, be any onneted omponent of I w.r.t N I see 1 and Defnton 1. Then the q q matrx L gven below: 1 f = j, L, j = 1 f k j N k ζ, #N k 0 otherwse, 21 s nvertble and ts nverse has non-negatve entres,.e. 1, L j 0,, j. 22 Remark 5 Observe that L, = 1 for all whle for j we have L,j < 0 L j, < 0. Indeed, f k ζ and j N k ζ, we fnd L, j = #N k 1. By 1, we also have k N kj ζ, hene Lj, = #N kj 1. Note that we an have L,j L j, sne dfferent neghborhoods an have dfferent szes. A matrx of the form 21 s presented n the example below. Example 3 Consder an mage where for every pxel whh s not at the boundary of the mage, N s omposed of ts 8 adjaent neghbors. However, the pxels at the boundares have less neghbors. Consder a onneted omponent ζ = {k 1,k 2,k 3,k 4,k 5 } as presented below n a:... k 2... k 1 k 3 k 4 k 5 a Conneted omponent ζ at the lower rght end of the mage. 1 1/8 1/8 1/8 0 1/8 1 1/8 0 0 L = 1/8 1/8 1 1/8 1/8 1/5 0 1/5 1 1/ /3 1/3 1 b The matrx L aordng to 21. Observe that k 5 s on the last olumn and on the last row of the mage, and that k 4 s on ts last row. Then #N k4 = 5 and #N k5 = 3, whle #N k = 8, = 1, 2, 3.

7 We have: N k1 ζ ={k 2,k 3,k 4 }, N k2 ζ ={k 1,k 3 }, N k3 ζ ={k 1,k 2,k 4,k 5 }, N k4 ζ ={k 1,k 3,k 5 } and N k5 ζ = {k 3,k 4 }. The matrx L orrespondng to 21 s presented n b. Proof The proof of the lemma s based on the followng equvalene result 1 [11]: a matrx L R q q s nvertble and satsfes 22 f and only f the mplaton below holds true: [Lv 0] [v 0], 23 where the expresson v 0 means that v 0, for every = 1,...,q. In what follows, we wll prove that the matrx L defned by 21 satsfes 23. Note that [Lv] = v 1 v j. 24 #N k j N k ζ Below we show several prelmnary mplatons that wll help us to prove 23. Let b<0 and u R r for r N, we have the mplaton [ b ] r u 0 =1 [ j {1,...,r}:u j 1r ] b<0. 25 Indeed, f the statement on the rght s false,.e. u > b r for all, then r u > =1 r =1 b r = b. Then the nequalty on the left hand sde annot be satsfed. Let us prove the followng mplaton: a def = v < 0 and N k ζ [ ] AND Lv 0 v j = a<0, j N k {k } OR. 26 j N k suh that v j <v = a<0 1 For ompleteness, we remnd the proof gven n [11]. If L s nvertble and 22 s true, and Lv 0, then learly v = L 1 Lv 0. Conversely, let 23 holds. If Lv = 0thenL v = 0, hene v 0 and v 0,.e. v = 0 whh shows that L s nvertble. Sne the th olumn of L 1 def s v = L 1 e,wehavelv = e 0 and hene v 0. The same holds true for all olumns v,1 q, hene L 1, j 0,, j. By 25, j 1 N k suh that 1 #N k v j1 1 #N k v. If v j1 <v, then 26 holds. Else v j1 = v = a<0. We an wrte down that 1 1 v 1 v j 0. #N k #N k j N k \{j 1 } Usng 25 yet agan, j 2 N k \{j 1 } suh that 1 1 v j2 1 1 v. #N k #N k 1 #N k Hene v j2 v k = v j1 = a<0. If v j2 <v, then 26 holds. Otherwse, v j2 = v = v j1 = a<0. Iteratng ths reasonng #N k tmes shows the result. We have a reproal of n 26: v j = a<0, j N k {k } AND [ Lv ] 0 [ ] N k ζ. 27 Suppose that the rght sde of 27 sfalse,.e.# N k ζ #N k 1 then [Lv ] = a #N k ζ #N k a = a #N k #N k ζ #N k < 0 where the last nequalty omes from the fat that the fraton s strtly postve. We have the followng mplaton: [ def ] a = v < 0 and N k ζ N k [ j N k : v j <v = a<0 ]. 28 Note that #N k ζ<#n k. Then usng 25, j N k suh that 1 1 v j #N k #N k ζ v whh shows 28. Wth the help of these mplatons, we wll prove 23 by ontradton. So, suppose that Lv 0 but that j 1 {1,...,q} suh that v j1 def = a<0.

8 Aordng to N j1 ether N j1 ζ or N j1 ζ N j1 we apply 26or28 and fnd that j 2 N j1 suh that v j2 v j1 def = a<0. Sne ζ s onneted, we an n the same way vst all N j for {1,...,q} and thus fnd a dereasng sequene, say of length n q, denoted v j n =1 whose elements are a<0. Sne ζ I, there exsts k n and j k ζ suh that N jk ζ N jk n whh ase 27 mples that v jk <v jk 1. We have thus found that b def = v jn v jk <v jk 1 j 1 v j1 = a<0. Sne we have vsted all N j for {1,...,q}, we an wrte that b = mn 1 q v. 29 Consder now N jn.by29, v j v jn, j N jn. 30 If we had v jn = v j = b, j N jn, then 27 shows that N jn ζ n whh ase we an agan terate 26 and 28, hene we have not vsted all N j for {1,...,q}. If follows that j N kjn suh that v j >b. 31 Usng 30 and 31, we an wrte down the followng: [ Lv = v ]j jn 1 v n j #N kjn j N kjn ζ = b 1 #N kjn j N kjn ζ <b #N k jn ζ #N kjn b v j = b #N k jn #N kjn ζ #N kjn 0. Thus [ Lv ] j n < 0 whh ontradts the assumpton that Lv 0. It follows that 23 s true. The proof s omplete. 4 Sem-explt Expresson for Mnmzers Gven ˆx a mnmzer of F., y let ζ ={} Jˆ be a onneted omponent wth respet to N, n the sense of Defnton 1, whh n partular mples that N Jˆ. Then Theorem 1 tells us that ˆx = 1 y j + σ, 32 #N 2α#N where σ { 1, +1} s gven by Lemma 1 and now reads σ = sgny 1 #N y j. A more general result, onsderng arbtrary onneted omponents ζ s presented next. Sne Lemma 2 holds for onneted omponents ζ suh that #ζ <p, we wll exlude the ase when Jˆ s empty.e. J ˆ = I and the unque onneted omponent s ζ = I, hene #ζ = p. Proposton 2 For y R p, let F., y reah ts mnmum at ˆx wth J ˆ = Jˆx,y, Jˆ I, where J s defned n 6. Let ζ ={k 1,...,k q } ˆ J 33 be any onneted omponent w.r.t. N I see Defnton 1 and ts neghborhood read N ζ ={n 1,...,n #Nζ }. Let us defne L ζ R q q, Q ζ R q #N ζ and d ζ R q as t follows: 1 f = j L ζ, j = 1 f k j N k ζ, #N k 0 otherwse, 1, j q; 34 1 Q ζ f n j N k N ζ,, j = #N k 0 otherwse, 1 q, 1 j #N ζ ; 35 d ζ = σ k, 2α#N k σ k = sgny k ˆx k, 1 q. 36 Then ˆx ζ reads ˆx ζ = A ζ y Nζ + b ζ, where a the matrx A ζ R q #N ζ satsfes A ζ, j 0,, j and reads A ζ = L ζ 1 Q ζ ; b the vetor b ζ R q reads b ζ = L ζ 1 d ζ. It s easy to see that A ζ depends only on {N : ζ N ζ }, and that b ζ depends only on {N : ζ N ζ } and {σ : ζ }. Proof Sne ζ ˆ J,9 n Proposton 1 shows that the entres of ˆx ζ satsfy ˆx k 1 = σ k, = 1,...,q. 37 #N k 2α#N j N k k

9 For every k ζ, we an deompose N k as N k = N k ζ N k N ζ. Sne N ζ ˆ J see Defnton 1, we have = y j for every j N ζ. Introdung ths nto 37 yelds ˆx k 1 #N k k j N k ζ ˆx kj = 1 y j + σ k, #N k 2α#N n j N k N k ζ = 1,...,q. 38 Ths s a system of q affne equatons wth q unknowns, whh an be expressed n matrx form: L ζ ˆx ζ = Q ζ y Nζ + d ζ, 39 where L ζ, Q ζ and d ζ are gven n 34, 35 and 36, respetvely. The matrx L ζ s learly of the form 21 and q<p sne by assumpton, Jˆ s strtly nluded n I. Aordng to Lemma 2, L ζ s nvertble and the omponents of L ζ 1 are 0. Hene A ζ n statement a and b ζ n statement b of the proposton are well defned. Moreover, t s lear that Q ζ, j 0 for all, j whh entals that A ζ, j 0, for all, j. The proof s omplete. Usng Proposton 2, we an formulate a sem-explt expresson for ˆx, the mnmzer of F., y over R p. The theorem below furnshes another formulaton of the mnmalty ondtons stated n Theorem 1. Theorem 2 For y R p, let F., y reah ts mnmum at ˆx wth J ˆ = Jˆx,y, Jˆ I, for J as defned n 6. Let m be the number of all onneted omponents of JN ˆ I see Defnton 1, say ζ l for l = 1,...,m: ˆ J = m ζ l. 40 l=1 Then ˆx reads ˆx ζl = A ζ l y Nζl + b ζ l, l= 1,...,m, 41 ˆx = y, ˆ J, 42 where A ζ l and b ζ l, for every l = 1,...,m, are as exhbted n Proposton 2. Furthermore, for any Jˆ there exst a R # Jˆ wth a 0 and β R suh that the system n 41 s equvalent to ˆx = a,y ˆ J +β, ˆ J. 43 Note that an be derved dretly from Theorem 1 and Remark 1; however, ths s not enough to say that all entres of a are nonnegatve, for all J ˆ. Proof If Jˆ s empty, then 41 s vod and 42 holds for every I = Jˆ. Consder next that Jˆ s nonempty. Equatons 41 and 42 are a straghtforward onsequene of Theorem 1 and Proposton 2. The statement n 43 explots the observaton that N ζl Jˆ, l {1,...,m}. Va a reorderng of the omponents of eah ζ l and eah N ζl, the mnmzer ˆx an be put nto the form where every a ontans a row of the matrx A ζ l suh that ζ l, the remanng terms, beng null, and every β s an element of b ζ l suh that ζ l. Sne the omponents of every A ζ l are 0, t follows that all omponents of a are 0, for every J ˆ. The proof s omplete. Observe that for every J ˆ, the lnear operator a depends only on N I and that the onstant β depend only on N I and {σ : J ˆ}. Remark 6 The pxels belongng to a onneted omponent ζ l Jˆ are alulated only based on the data ponts y whh are neghbors of ζ l, namely y for N ζl. All data samples y I \{ζ l N ζl } have no ontrbuton. In ths sense, the restoraton of eah ˆx ζl s loal. 5 Mnmzer ˆx Is a Loally Affne Funton of the Data y In ths seton we exhbt subsets of data n R p leadng ether to the same mnmzer pont ˆx, or that satsfy the same system of equatons, as exhbted n Proposton 3 Gven y R p, let F., y reah ts mnmum at ˆx wth Jˆ def = Jˆx,y, for J defned aordng to 6 and put σ = sgny ˆx, Consder the subset gven below: { VJ ˆ = y R p : y = y, Jˆ, [ y > 1 2α#N + 1 #N y < 1 2α#N + 1 #N ˆ J. 44 } f J,σ ˆ =+1 f J,σ ˆ. = 1 Then for every y V ˆ J, the funton F., y reahes ts mnmum at ˆx. Proof Let us onsder ˆx that mnmzes F., y. We wll show that ˆx satsfes the ondtons for a mnmum of F., y, for any y V ˆ J.

10 Consder an arbtrary y V Jˆ for J ˆ. From the defnton of VJ ˆ, t s seen that for every Jˆ we have y 1 > 1 > 0 f σ =+1; #N 2α#N y 1 < 1 < 0 #N 2α#N Hene y V ˆ J, fσ = 1. sgn y 1 = σ, J, ˆ 45 #N where σ s gven n 44. Takng nto aount that for every ˆ J we have y = y =ˆx,16 n Theorem 1 reads ˆx 1 #N ˆ J, ˆ J = 1 #N sgny ˆx +, 2α#N Jˆ where σ satsfes 45. By Theorem 1, F., y reahes ts mnmum at ˆx. If Jˆ s empty, the system above amounts to say that 9 n Proposton 1 holds for all I. So the onluson s the same. Remark 7 Let us emphasze that VJ ˆ s a subset whose Lebesgue measure n R # J ˆ s nfnte and that t an be seen as a one whose orgn s translated. Remark 8 If for y R p, F., y reahes ts mnmum at ˆx suh that J ˆ = I.e. Jˆ s empty, Proposton 3 tells us that for every y W I,σ, where W I,σ WJ ˆ,σ = y R p : I, y > 1 2α#N + 1 #N f σ =+1 y < 1 2α#N + 1 #N f σ = 1 and σ = sgny ˆx, I,F., y reahes ts mnmum at ˆx, that s ˆx =ˆx, y W I,σ. Next we determne a set WJ,σ ˆ Rp as large as possble, suh that for every y WJ,σ ˆ, the energy F., y reahes ts mnmum at an ˆx alulated by usng the same matres A ζ l and vetors b ζ l. For arbtrary sets Jˆ I and σ { 1, +1} # J ˆ, Jˆ I t s possble that there s no ˆx satsfyng ˆx = y and y ˆx for all Jˆ wth sgny ˆx = σ for all ˆ J. For ths reason, we start wth the mnmzer ˆx relevant to agveny and then we determne a set of data y suh that the relevant soluton ˆx s alulated usng exatly the same affne equaton appled to every y n ths set. Theorem 3 Gven y R p, let F., y reah ts mnmum at ˆx wth Jˆ def = Jˆx,y, Jˆ I, where J s defned by 6. Put σ = sgny ˆx, Let a R # ˆ J and β R be suh that ˆx = a,y ˆ J +β, ˆ J. 46 ˆ J, 47 aordng to 43 n Theorem 2. For all I, defne the onstants β and the affne applatons h : span { e, Jˆ } R by β = 1 #N ˆ J h,y ˆ J = 1 β j, 48 #N + 1 #N ˆ J y j Jˆ Consder the subset gven below: a j,y ˆ J, y R p. 49 W ˆ J,σ = y R p : b a y h,y ˆ J β 1 2α#N { y > 1 } 2α#N + h,y ˆ J +β f σ =+1 y < 1 2α#N + h,y ˆ J +β f σ = 1 Jˆ ˆ J. 50

11 Then for every y W ˆ J,σ, the funton F., y reahes ts mnmum at ˆx gven by ˆx = y, ˆx = a,y ˆ J +β, ˆ J, 51 ˆ J, 52 where a and β, ˆ J, are the same as those gven n 47. Observe that the frst sum n 49 nvolves only y j for j Jˆ,soh s defned on span { e, Jˆ }, ndeed. Remnd that from the defnton of J ˆ,wehave ˆx = y for every Jˆ. Proof For an arbtrary y W ˆ J,σ,let ˆx read as n We wll show that ˆx satsfes the mnmalty ondtons for F., y stated n Theorem 1. From the onstruton of β and h,see48 and 49, respetvely, we derve h,y ˆ J +β = 1 #N y j + Jˆ 1 #N ˆ J a j,y ˆ J +β j = 1, I, 53 #N where the last expresson omes from Combnng 53 wth a n the defnton of WJ,σ ˆ n 50 shows that y h,y ˆ J +β = y 1 ˆx j #N 1, 2α#N ˆ J. 54 If Jˆ s empty, then ˆx = y. Usng51, the nequalty n 54 reads y 1 #N y j 1 2α#N, Jˆ = I. By Proposton 1, F., y reahes ts mnmum at ˆx =ŷ. Note that n ths ase 47, 50b and 52arevod. Consder next that Jˆ s nonempty. Combnng 53 wth b n 50 shows that for every Jˆ we have y h,y ˆ J +β = y 1 > 1 > 0 fσ =+1, 2α#N #N < 1 < 0 fσ = 1. 2α#N It follows that for all Jˆ we have y 1 #N ˆx j > 1 2α#N and sgn y 1 = sgny ˆx = σ. 55 #N By Theorem 2 we know that 47 s equvalent to 16 n Theorem 1. Hene ˆx, J ˆ, as defned n 52, s equvalent to ˆx 1 #N ˆ J ˆx j = 1 #N y j + Jˆ σ 2α#N, ˆ J, 56 for σ as gven n 46, for all J ˆ. Combnng 54, 55 and 56 shows that ˆx satsfes all ondtons for a mnmzer of F., y, aordng to Theorem 1. Remark 9 Observe that WJ,σ ˆ s a nonempty polyhedron n R p whh s n addton unbounded hene ts Lebesgue measure s nfnte. By Remark 8, ths holds true for J ˆ = I as well. For every y R p we an exhbt a WJ,σ ˆ suh that the relevant F., y are mnmzed by solutons satsfyng the same system of affne equatons and n partular, Jˆ remans the same. The set of all feasble sets J,σ ˆ s fnte. The unon of all possble WJ,σ ˆ orresponds to a partton of R p the set of all possble data. 6 A Fast Exat Numeral Method Sne F s non-smooth, the alulaton of ˆx needs a spef optmzaton sheme. Smlar optmzaton problems are enountered along wth total-varaton methods where Fx, y = Ax y 2 + α G x, where for every = 1,...,p, G s a dsrete approxmaton of the gradent operator at, orfx, y = Ax y 1 + αx where s an edge-preservng regularzaton term. Some authors use ontnuaton methods [9, 20, 25]. In these ases, the nonsmooth. s approxmated by a famly of smooth funtons ϕ ν, ν>0, e.g. ϕ ν t = t 2 + ν. For every ν>0, the mnmzer ˆx ν of the relevant F ν x, y an be alulated usng lassal optmzaton tools. It an be shown that ˆx ν onverges to the sought-after ˆx as ν dereases to zero. However, the onvergene s qute slow, espeally when ν approahes zero. Most of the authors just fx ν>0 and mnmze a smooth approxmaton of the orgnal energy, let us te among many others [3 5, 8, 27]. Note that n suh a ase, the soluton annot exhbt the propertes relevant to the nonsmooth term ˆx = y n our ase as proven n [21, 23]. Subgradent methods see e.g. [12, 26], are slow and fnd the features of ˆx where F., y s nondfferentable after a huge amount of teratons. A method for a ase slghtly smlar to the our was dervedn[19] based on atve sets and Lagrange multplers. An adaptaton to our ontext mght be possble; but our optmzaton problem s muh smpler so we prefer to take a

12 beneft from ths. The methods n [13, 14] suppose that the mage s quantzed whh s not our ase. The mnmzaton method n [16] s exat, but ts mplementaton usng an nteror pont method ntrodues an approxmaton towards the ponts where the energy s nondfferentable. We fous on our method presented n [22] an extenson of a method exposed n a textbook [17] publshed n 1976 sne t fnds n prorty the ponts of ˆx where F., y s non-dfferentable. Its spealzaton to the energy n 2 has an appreable smplty sne all steps are fully explt. It s of the type one oordnate at a tme. Eah teraton onssts of p steps where all omponents are updated suessvely. Let the urrent soluton read x. Atstep, we mnmze the salar funton t Fx + t x e,y wth respet to t. The updatng equatons ome from Proposton 1. The numeral sheme reads: Start teratons wth ntal guess x 0 = y. For every teraton k = 1, 2,..., onsder = k modulo p and ompute φ k = y 1 x k 1 j. 57 #N Then update x k aordng to the followng rule: f k φ 1 then x k = y, 58 2α#N else x k = y φ k Stop when x k 1,...,p. + sgnφk 2α#N. 59 x k 1 s small enough for all = Theorem 4 The sequene x k defned by 57 and satsfes lm k x k =ˆx where ˆx s suh that F ˆx,y Fx, y, for every x R p. Proof The proof s based on Theorem 2 n [22]. The assumptons H1, H2 and H3 requred there are now trvally satsfed. The last one, H4, amounts to requre that the regularzaton term s loally strongly onvex wth respet to eah omponent x ndependently, for all I ;thssalso true n the present ase. The rest of the proof s to alulate the ntermedate steps. These ome dretly from Proposton 1. Next we gve some omment on the mnmzaton algorthm. The onvergene of the algorthm presented above s guaranteed for any ntal x 0. However, after the developments n Set. 2, we an expet that ˆx satsfes ˆx = y for numerous ndexes. So ntalzng wth x 0 = y should speed up the onvergene. Note that the algorthm an easly be extended to onstrants of the form d x d +, where d <d +, 1 p. Let us emphasze the extreme smplty of the numeral sheme t nvolves only summatons and omparsons to a fxed threshold, and there s no lne searh. Ths explans the speed of the method. For all data ponts whh math the soluton the relevant pxels are updated aordng to 58, whh s just a omparson to a threshold. Hene the mnmzaton wth respet to the pxels suh that ˆx = y s exat. Ths s rual sne n prate numerous samples of the soluton ft exatly the relevant data samples, as evoked n 3. Remark 10 The algorthm presented above s straghtforward to adapt to energes form Fx, y = x 1 + β Ax y 2, where A s a wavelet bass, or a frame, or a lnear operator. Suh problems are often enountered n approxmaton, ompresson, odng and ompressve sensng. Runnng Tme The omputaton tme depends learly on the sze of the mage and the neghborhoods, the stoppng rteron and the value of α; t s hgher for larger values of α. We worked on a PC Dell Lattude, D620, Genune IntelR CPU T2500, 2.00 GHz and 1.00 RAM runnng on Wndows XP Professonal and used Matlab 7.2. We dd some omparsons on a mage Fg. 4g, onsdered the four adjaent neghbors for N I, α = 0.01 and the stoppng rule was based on the value of x k x k 1. A preson of 0.1 was reahed after 24 teratons and needed 3 seonds CPU tme. For a preson of 0.001, we had 76 teratons and 9 seonds. Let us note that the algorthm an be mplemented n a parallel way whh an speed up ts onvergene usng Matlab. 7 Experments The results on nonsmooth data-fttng along wth a smooth regularzaton shown n ths paper, as well n [21, 22] learly ndate that proessng data by mnmzng suh an energy annot be suessful unless there are some nearly nose-free data samples. The man reason omes from the property skethed n 3. Satsfyng results wth suh energes were obtaned for denosng and deblurrng under mpulse nose

13 Fg. 3 Peewse polynomal under random-valued mpulse nose [3 5, 19, 22], as well as for the denosng of frame oeffents [15], among others. For ths reason, n our experments we fous on sgnals and mages ontamnated wth spky noses. New applatons wll ertanly appear, espeally sne strong results on the propertes of the solutons are already avalable. 7.1 A Toy Illustraton The orgnal peewse polynomal sgnal s plotted n Fg. 3a wth a sold lne. Data y, plotted n Fg. 3a wth a dashed lne, orresponds to 5% random-valued mpulse nose. The mnmzer ˆx for α = 0.02, shown n Fg. 3b, stll ontans some outlers; however the resdual y ˆx n Fg. 3 shows that the set J ˆ ={ :ˆx y } mathes the loatons of the outlers ontaned n y. In the next Fg. 3d we apply a smple medan flter loally, only n the neghborhood of the samples J ˆ. The resultant restoraton fts the orgnal sgnal. Fgure 3e dsplays the mnmzer ˆx for α = 0.1: the outlers are suffently smoothed but the edges n the orgnal sgnal are oversmoothed ths s not surprsng sne the regularzaton term s quadrat, hene t s not edge-preservng. The resduals n Fg. 3f show that Jˆ n ths ase s larger than the set of the outlers n data y. Ths effet s not surprsng at all sne the regularzaton s not edge-preservng. Ths example suggest that an undersmoothed ˆx an be a good ndator to loate the outlers n the data. At a seond step, the samples belongng to Jˆ need a pertnent postproessng. 7.2 Image Denosng under Random-Valued Impulse Nose The mage n Fg. 4b s ontamnated wth 40% randomvalued mpulse nose. All results presented here orrespond to the best hoe of the parameters for eah method. The soluton n s obtaned usng 5 teraton enter-weghted medan 2 CWM flter wth a 5 5 wndow and multplty parameter 8. The restoraton n d orresponds to permutaton-weght 3 PWMflteron7 7 wndow wth rank parameter 22. A detaled desrpton of the CWM and the PWM flters an be found e.g. n the textbook [6]. The mage n e s obtaned usng the two-phase method desrbed n [7]. The restoraton n f s the mnmzer F., y for α = 0.08 outlers are removed but the edges are slghtly oversmoothed. The mage n g s the mnmzer of F., y for α = As expeted from the prevous example, the outlers are not leaned but Jˆ approxmates well the loatons of the outlers n the data. The fnal restoraton n h s obtaned usng a loal medan smoother near eah omponent n J ˆ. The zoom presented below shows that the latter result has better preserved edges and a more natural appearane, ompared wth the other methods. 7.3 Cleanng Nosy Data from Outlers In dfferent applatons, data y result from outler-free degradatons e.g., dstorton, blurrng, quantzaton errors, eletron nose, plus mpulsve nose. It s well known that lassal reonstruton methods fal n the presene of outlers. A prelmnary proessng, amed at removng the outlers, s usually needed before to apply standard reonstruton methods. Preproessng s often realzed usng medanbased flters. It s rtal that preproessng keeps ntat all the nformaton ontaned n the outler-free data. In our experment, the sought mage x, shown n Fg. 5b, s related to x o n a, by x = x o + n, where n s whte Gaussan nose wth 20 db SNR. The hstogram of 2 CWM flter. Fx a number of teratons T, an nteger multplty parameter μ 1 and a wndow N, / N of neghbors for eah I.At μ tmes teraton t,pxelx s replaed by Medan{ {x,...,x } {x j,j N }}. 3 PWM flter. Choose a number of teratons T, a wndow N, / N of neghbors for eah I and a rank parameter ρ suh that 1 ρ 0.5#N. At teraton t,allpxels{x j,j N {}} are sorted n nreasng order. If the rankng r of x satsfes ether r<ρor r>#n ρ, then x s replaed by Medan{x j,j N {}}; otherwse x remans unhanged.

14 Fg. 4 Random-valued mpulse nose leanng usng dfferent methods Fg. 5 Data wth two-stage degradaton n s plotted n d, top. Our goal s to restore x the mage ontamnated wth Gaussan nose only based on the data y, shown n, whh ontan 10 % salt-and-pepper nose. Restorng x s a hallenge sne the whte Gaussan nose must be preserved. The relevane of an estmate ˆx s evaluated by both, the error ˆx x and the loseness of the estmated nose ˆn =ˆx x o to the ntal nose n. To ths end, we use the Mean-square error MSE defned by MSEn, ˆn = 1 p n ˆn 2. All mages n ths subseton are dsplayed usng the same gray value salng. All results n Fg. 6 orrespond to parameters leadng to the best removal of the outlers. The mage n a orresponds to one teraton of medan flter over a 3 3 wndow. Almost all data samples are altered and the estmated nose, ˆn =ˆx x o, s qute onentrated near zero. The mage n b s alulated usng a enter weghted medan CWM flter wth a 5 5 wndow and multplty parameter for the entral pxel equal to 14. Even though the error MSE ˆx,x s redued, the hstogram of the nose estmate ˆn, plotted on the rght, top, devates onsderably from the ntal dstrbuton, shown n Fg. 5. The mage n orresponds to one teraton of permutaton weghted medan PWM flter, for a5 5 wndow and rank parameter 4. In all these estmates, the dstrbuton of the nose estmate ˆn s qute dfferent from the dstrbuton of n. Fgure 6d dsplays the ssue of the mnmzaton of F as gven n 2 forα = 1.3. It aheves a good preservaton of the statsts of the nose n x,as seen from the hstogram of the estmated nose ˆn Fg. 6d, rght, top. Moreover, the error ˆx x s essentally onentrated around zero, as seen n Fg. 6d, rght, down. Note that ths experment takes a beneft from our Remark 2. 8 Conlusons and Perspetves In ths paper we show that the mnmzers ˆx of energes F., y of the form 2 are loally affne funtons of the

15 Fg. 6 Varous restoratons data, as exhbted n Theorem 3. Energes nvolvng a nonsmooth data-fttng are known to produe solutons that partally ft the data, as evoked n 3. An mportant property that we found s that the pxels ˆx ζ of any onneted subset ζ that do not ft 3 s restored usng a smple funton of the form ˆx ζ = A ζ y Nζ + β ζ where all entres of A ζ are 0 and β ζ s a fxed vetor that depends only on the sgn of y ˆx for ζ. If we have some knowledge that the data y follow a smple dstrbuton on a bounded doman of R p, t should be possble to evaluate the probablty to obtan a soluton gven by the same set of affne equatons. Even though we propose a fast mnmzaton sheme, we hope that the obtaned theoretal results an help to oneve faster mnmzaton shemes. Sem-explt solutons of ths knd are hard to exhbt for general regularzaton terms. The results of ths paper suggest how are restored the pxels satsfyng ˆx y under more general regularzaton terms. We onsder ths study as a startng pont for the analyss of more elaborated energes nvolvng l 1 data-fttng. The approah adopted n ths paper an be used to analyze, as well as to derve new mnmzaton shemes, for energes of the form Fx, y = x 1 + β Ax y 2,or mnmzaton problems suh as: mnmze x 1 subjet to Ax y 2 τ for τ>0, or mnmze x 1 subjet to Ax = y, where A s a wavelet bass or a frame, or any lnear operator. Suh problems arse ustomarly n approxmaton, n odng and ompresson, and n ompressve sensng. In these ases, # Jˆ /p orresponds to the level of sparsty of the obtaned soluton. Aknowledgement The author would lke to thank the Revewers for ther pertnent remarks enouragement. The author would lke to thank Janfeng CAI Natonal Unversty of Sngapore for provdng the experment wth the two-phase method n Fg. 4. Referenes 1. Allney, S.: Dgtal flters as absolute norm regularzers. IEEE Trans. Sgnal Proess. SP-40, Allney, S.: A property of the mnmum vetors of a regularzng funtonal defned by means of absolute norm. IEEE Trans. Sgnal Proess. 45, Bar, L., Brook, A., Sohen, N., Kryat, N.: Deblurrng of olor mages orrupted by mpulsve nose. IEEE Trans. Image Proess. 16, Bar, L., Sohen, N., Kryat, N.: Image deblurrng n the presene of salt-and-pepper nose. In: Proeedng of 5th Internatonal Conferene on Sale Spae and PDE methods n Computer Vson. LNCS, vol. 3439, pp Sprnger, Berln Bar, L., Sohen, N., Kryat, N.: Sem-blnd mage restoraton va Mumford-Shah regularzaton. IEEE Trans. Image Proess. 15, Bovk, A.C.: Handbook of Image and Vdeo Proessng. Aadem, New York Chan, R., Hu, C., Nkolova, M.: An teratve proedure for removng random-valued mpulse nose. IEEE Sgnal Proess. Lett. 11, Chan, T., Chen, K.: An optmzaton-based multlevel algorthm for total varaton mage denosng. SIAM J. Multsale Model. Smul. 5, Chan, T., Esedoglu, S.: Aspets of total varaton regularzed l 1 funton approxmaton. SIAM J. Appl. Math. 65, Chan, T., Esedoglu, S., Nkolova, M.: Algorthms for fndng global mnmzers of mage segmentaton and denosng models. SIAM J. Appl. Math. 66, Carlet, P.G.: Introduton à l analyse numérque matrelle et à l optmsaton, 5th edn. Colleton mathématques applquées pour la maîtrse. Dunod, Pars Combettes, P., Luo, J.: An adaptve level set method for nondfferentable onstraned mage reovery. IEEE Trans. Image Proess. 11, Darbon, J., Sgelle, M.: Image restoraton wth dsrete onstraned total varaton part I: Fast and exat optmzaton. J. Math. Imagng Vs. 26, Darbon, J., Sgelle, M.: Image restoraton wth dsrete onstraned total varaton part II: Levelable funtons, onvex and non-onvex ases. J. Math. Imagng Vs. 26, Durand, S., Nkolova, M.: Denosng of frame oeffents usng l1 data-fdelty term and edge-preservng regularzaton. SIAM J. Multsale Model. Smul. 6,

16 16. Fu, H., Ng, M., Mhael, K., Nkolova, M., Barlow, J.L.: Effent mnmzaton methods of mxed l 1 l l and l 2 l 1 norms for mage restoraton. SIAM J. S. Comput. 27, Glownsk, R., Lons, J., Trémolères, R.: Analyse numérque des néquatons varatonnelles, vol. 1, 1st edn. Dunod, Pars Hrart-Urruty, J.-B., Lemaréhal, C.: Convex Analyss and Mnmzaton Algorthms, vol. I and II. Sprnger, Berln Kärkkänen, T., Kunsh, K., Majava, K.: Denosng of smooth mages usng l1-fttng. Computng 74, Nkolova, M.: Markovan reonstruton usng a GNC approah. IEEE Trans. Image Proess. 8, Nkolova, M.: Mnmzers of ost-funtons nvolvng nonsmooth data-fdelty terms. Applaton to the proessng of outlers. SIAM J. Numer. Anal. 40, Nkolova, M.: A varatonal approah to remove outlers and mpulse nose, J. Math. Imagng Vs Nkolova, M.: Weakly onstraned mnmzaton. Applaton to the estmaton of mages and sgnals nvolvng onstant regons. J. Math. Imagng Vs. 21, Rokafellar, R.T.: Convex Analyss. Prneton Unversty Press, Prneton Rudn, L., Osher, S., Fatem, C.: Nonlnear total varaton based nose removal algorthm. Physa D 60, Shor, N.Z.: Mnmzaton Methods for Non-Dfferentable Funtons, vol. 3. Sprnger, New York Vogel, C.R., Oman, M.E.: Fast, robust total varaton-based reonstruton of nosy, blurred mages. IEEE Trans. Image Proess. 7, Mla Nkolova reeved the Ph.D. degree from the Unversté de Pars- Sud, Frane, n She got a Habltaton to dret researh n Currently, she s Senor Researh Fellow, 1 st lass, wth the Natonal Center for Sentf Researh CNRS, Frane, and performs her researh at the Centre de Mathématques et de Leurs Applatons CMLA, ENS de Cahan, Frane. Her researh nterests are n mathematal Image and sgnal reonstruton, Inverse problems, Regularzaton and varatonal methods and the propertes of ther solutons, Sentf omputng.