Numerische Mathematik

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1 Nmer. Math. 1 Digital Object Identifier DOI 1.17/s11164 Nmerische Mathemati Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations in image rocessing Karol Mila 1, Narisoa Ramarosy 1 Deartment of Mathematics, Slova University of Technology, Radlinseho 11, Bratislava, Slovaia; mila@vox.svf.stba.s Laboratoire d Analyse Nmériqe et E.D.P, Université Paris Sd, Orsay, France Received Janary 7, 1999 / Revised version received May 31, / Pblished online March, 1 c Sringer-Verlag 1 Smmary. We roose and rove a convergence of the semi-imlicit finite volme aroximation scheme for the nmerical soltion of the modified in the sense of Catté, Lions, Morel and Coll Perona Mali nonlinear image selective smoothing eqation called anisotroic diffsion in the image rocessing. The roof is based on L a-riori estimates and Kolmogorov s comactness theorem. The imlementation asects and comtational reslts are discssed. Mathematics Sbject Classification 1991: 35K55, 65P5 1. Introdction In this aer we stdy the convergence of the semi-imlicit finite volme scheme for the following nonlinear initial-bondary vale roblem t.g G σ =f in Q T I, ν = on I,, = in, where R d is a rectanglar domain, I =[,T] is a scaling interval, and g is a decreasing fnction, g = 1, < gs 1.4 for s,g t is smooth, Corresondence to: K. Mila

2 K. Mila, N. Ramarosy G σ C R d is a smoothing ernel with G σ xdx =1 R d and G σ x δ x for σ, δ x the Dirac measre at oint x, f is a Lischitz continos, nondecreasing fnction, f=, L. We assme that 1.8 s G σ x B σ B σ is a ball centered at withradis σ and by the term G σ in 1.1 we mean R d G σ x ξũξ,tdξ, where ũ is an extension of given by eriodic reflexion throgh the bondary of in the region σ = x B σx and by in R d σ. In the image rocessing, arises in the nonlinear data filtration, edge detection and image enhancement and restoration [14], [5]. The initial condition x reresents the greylevel intensity fnction of the image which we want to rocess. The soltion t, x of reresents the family of scaled filtered, smoothed versions of x; t is nderstood as an abstract arameter called scale. In general, the rocessing of by evoltionary PDE lie 1.1 is called image mltiscale analysis [1,, 11, 15] and, in a sense, it reresents an embedding of the initial image to the so called nonlinear scale sace. In or case, reresent a slight modification of the well-nown Perona Mali eqation called also anisotroic diffsion in comter vision commnity. It selectively diffses an image in the regions where the signal is of constant mean in site of those regions where the signal changes its tendency. This diffsion rocess is governed by the shae of the fnction g and its deendence on which is in a sense an edge indicator [14]. We note that in original Perona Mali formlation stands in the lace of G σ in 1.1. However, if the rodct gss is decreasing, the Perona Mali eqation can behave locally lie the bacward heat eqation, which is an ill osed roblem. So, for g s sed in ractice gs =1/1+s,gs =e s bothexistence and niqeness of a soltion cannot be obtained. One way how to reveal that mathematical disadvantage has been roosed by Catté, Lions, Morel and Coll in [5]. They have introdced the convoltion with the Gassian ernel G σ into the decision rocess for the vale of the diffsion coefficient. This slight modification for σ small, the models are close and in a sense G σ for σ allowed them to rove the existence and niqeness of the wea soltion for the modified model and to ee all ractical advantages of original formlation. Moreover, the sage of the Gassian gradient maes the rocess more stable in the resence of noise. It has made exlicit a resmoothing inclded imlicitly in nmerical realizations of Perona Mali eqation, too. De to homogeneos Nemann bondary conditions the soltion tends to

3 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations a constant fnction withtime evoltion, rovided f. By means of f on the right hand side of 1.1, the soltion is forced to be close to, which can weaen the inflence of the stoing time T. In [1], fs s is roosed. Several aroaches for the nmerical soltion of have been sggested and stdied regarding stability of the schemes, convergence to a wea soltion of continos roblem and efficiency of imlementations. There are sed finite difference aroximations see e.g. [16] as well as methods based on finite element techniqe allowing adativity by coarsening of the discrete comtational grid [8], [3]. The convergence of the semiimlicit scheme combined with finite elements in sace to the niqe wea soltion of has been roven in [8]. The finite difference schemes for , exlicit, semi-imlicit or imlicit, have been stdied only de to the stability roerties on given satial discrete grids [16]. In this aer we derive aroximation scheme, corresonding to , sing the so called finite volme method [13], [6]. In finite volme method, the discrete aroximations are iecewise constant er control volmes corresonding to ixel/voxel strctre of the discrete image, so sch aroach is the most natral in image rocessing. The nonlinearity of the eqation is treated from the revios discrete scale ste, ths or scheme is semi-imlicit and leads to a soltion of sarse linear systems in eachdiscrete scale ste of the algorithm. That can be done in efficient way sing reconditioned iterative solvers. Moreover, the scheme allows to derive L a-riori estimates for flly discrete soltions which force s to se Kolmogorov s comactness theorem in order to rove the convergence of the aroximations to the niqe wea soltion of The organization of the aer is as follows. In Sect. we resent the flly discrete semi-imlicit finite volme scheme and in Sect. 3 we rove its convergence to the niqe wea soltion of the roblem. Finally, in Sect. 4 we discss some comtational reslts.. The finite volme scheme Let τ h be a meshof. The elements of τ h are the so called control volmes. For every air, q τh with q, we denote their common interface by e q, i.e. e q = q which is sosed to be inclded in a hyerlane of R d not intersecting either or q. Let m e q denote the measre of e q, and n q the nit vector normal to e q oriented from to q. We denote by E the set of airs of adjacent control volmes, defined by E = {, q τh, q, m e q }. We also se the notation N ={q,, q E}. We assme that there exists family of oints x τh, x for every τ h, schthat for every, q E,

4 K. Mila, N. Ramarosy x q x x q x = n q. Let δ denote the diameter of the control volme, m the measre in R d, of the control volme, its bondary and let h = max τ h δ. We denote by d q = x q x the Eclidean distance between x and x q and by x q a oint of e q intersecting the segment x x q. Finally we define T q = meq d q. Now, we are ready to write the semi-imlicit finite volme scheme for solving reglarized Perona-Mali roblem : Let =t t 1... t Nmax = T denote the time discretization with t n = t n 1 +, where is the time ste. For n =,..., N max 1 we loo for, τ h satisfying.1 starting with. where.3 n m = q N gq σ,n h, T q q +f n m = 1 x dx, τ h, m g σ,n q h, =g G σ ũ h, x q,t n and ũ h, is an extension of the iecewise constant fnction h, defined as follows N max.4 h, x, t = n χ {x } χ {tn 1 <t t n} n= τ h { 1 if A is tre with the fnction χ {A} = The extension is realized by eriodic reflexion throgh the bondary of in the region σ and by in elsewhere. R d σ. Remar.1. The fnction h,, constrcted sing discrete vales given by the scheme.1, is considered as the aroximation of the soltion of and its convergence to a wea soltion of , as h,, will be stdied in Sect. 3.

5 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Remar.. The scheme.1 is dedced from initial-bondary vale roblem after integration of 1.1 over τ h and over [t n,t ]: t t n t dxdt = t t n g G σ. n dxdt+ t t n f dxdt.5 and taing into accont the homogeneos Nemann bondary conditions 1.. Remar.3. The gradient of the convoltion term in.1, i.e. G σ ũ h, x q,t n = G σ ũ h, x q,t n, x i i=1,...,d where x i are sace variables, is comted sing the convoltion derivative roerty x i G σ ũ h, x q,t n = G σ x i ũ h, x q,t n. Then we have G σ G σ ũ h, x q,t n = x q sũ h, s, t n ds x i R d x i = n G σ.6 r x q sds r r x i and ths.7 G σ ũ h, x q,t n = r n r r G σ x q s ds where the sm is evalated on control volmes r τ h and on control volmes contained in the reflexion of τ h throgh bondary of which are arrond x q. Hereby, the sm is restricted to control volmes intersecting B σ x q, the ball centered at x q withradis σ. Theorem.1. Existence and niqeness of the discrete soltion There exists niqe soltion h, given by the scheme.1. Proof. We can rove the existence and niqeness of h, once we rove it for each n, τ h, n N max. We se an indction argment for that

6 K. Mila, N. Ramarosy rose. First is given for each τ h. Next, we sose that n, τ h is nown. By.1 we have, for each τ h, that m +.8 q N q h, T q g σ,n = m n + f n m. q N g σ,n q h, T q q = let s define P = card τ h and fnction α : τ h N [1, P] whichnmbers eachvolme, i.e. α. Then we can constrct R P P -matrix A =A ij P P coming from.8 for which A αα = m + q N g σ,n q h, T q, and otherwise A ααq = g σ,n q h, T q, A ij =. q N Let R P -vector U corresond to the discrete soltion, i.e. U = α and set R P -vector F to F = m α n + f n m. Using these definitions we can rewrite.8 into linear system in the matrix form.9 AU = F. Since the matrix A is symmetric and strictly diagonally dominant, there exists niqe R P vector U satisfying.9 which in trn imlies the existence and niqeness of, τ h.

7 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Definition.. A wea soltion of the reglarized Perona-Mali roblem is a fnction L,T; H 1 satisfying the identity.1 T T ϕ t x, t dxdt + g G σ ϕdxdt + x ϕ x, dx T f ϕ x, t dxdt = for all ϕ Φ, where the fnction sace { Φ = ϕ L,T; H 1, ϕ } t L,T; H 1,ϕ., T =.11 and H 1 denotes the dal sace of H 1. Remar.4. In [5] Catté, Lions, Morel and Coll roved that there exists niqe soltion of with f in the distribtional sense which is also the classical soltion of the roblem at the same time. Their reslt can be simly adoted for Lischitz continos right hand side f. Toget existence they sed Schader s fixed oint theorem with iterations in entire arabolic eqation. In the next section, we will find sch soltion in a comtationally natral and efficient way sing semi-imlicit finite volme scheme. 3. Convergence of the scheme to the wea soltion 3.1. L - a riori estimates Lemma 3.1. The scheme.1 leads to the following estimates: There exists a ositive constant C sch that i ii max n N max N max n=,q E n m C τ h n n q m e q C d q iii n m C n= τ h hold for every sfficiently small with a constant C which does not deend on h,.

8 K. Mila, N. Ramarosy Proof. Let s consider n schthat n<n max. We mltily the scheme.1 by to obtain n m = [ g σ,n q h, T q q N +f n m. q ] 3.1 Using the roerty a b a = 1 a 1 b + 1 a b on the left hand side of 3.1 and after smming over τ h,wehavethat 1 1 m n m τ h τ h + 1 n m 3. τ h = τ h q N + τ h f n [ g σ,n q h, T q m. q ] Since g σ,n q h, T q = g σ,n q h, T q we can rearrange the smmation of the first term of the right hand side of 3. to obtain 3.3 τ h q N = 1 [ g σ,n q h, T q,q E q [ g σ,n q h, T q q ] = ]. Alying 3.3 in 3. and after smming over n =,..., m 1 <N max, we have = 1 1 m 1 m 1 m+ n m τ h τ h + 1 m 1 n=,q E 1 m+ τ h n= [ gq σ,n h, T q ] q m 1 n= f n m. τ h

9 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Then we se Yong s ineqality and the Lischitz continity of f by K f > we denote the Lischitz constant of f in the right hand side to obtain 1 m 1 m+ τ h + K f 1 m 1 n= n= K f m+ τ h τ h f n m 1 n= m 1 n 1 m + τ h n= m τ h m m. τ h Since L, there exists C 1 > schthat m C1, τ h and one can show that there exists a ositive constant C schthat m 1 m 1 m+ n m τ h τ h + 1 m 1 n=,q E C 1 + C n= [ gq σ,n h, T q ] q m n= τ h n m. Now, we can aly the discrete Gronwall lemma to state the reslt i of the lemma, i.e. there exists a ositive constant C 3 schthat m 3.6 m C3 for all m N max τ h hold for every sfficiently small where the constant C 3 does not deend on h,. We get also x i G σ ũ h, x q,t n 3.7 x i G σ x q ξũξ,t n dξ R d 1 x i G σ x q ξ dξ + 1 ũξ,t n dξ C σ R d R d +C 4 ũξ,t n dξ C σ + C 4 n m C5 σ τ h

10 K. Mila, N. Ramarosy It comes from 3.7 that G σ ũ h, x q,t n <, which in trn imlies that there exists a ositive constant α schthat 3.8 gq σ,n h, >α>. Using 3.8 and 3.6 in 3.5, one can dedce assertions ii and iii of the lemma. 3.. Sace and time translate estimates In order to show relative comactness in L Q T of h, h, verifying.1 -.4, we need to establishthe estimates of differences in sace and in time for the set of discrete soltions. Lemma 3.. Sace translate estimate For all vector ξ R d, there exists a ositive constant C sch that 3.9 h, x + ξ,t h, x, t dxdt C ξ ξ +h ξ,t where ξ = {x,[x, x + ξ] }. Proof. Let ξ R d be a given vector. For all, q E, we denote by ξ q the following vale ξ q = ξ ξ.n q. For all x ξ, we denote by E x,, q the fnction defined as follows 1 if the segment [x, x + ξ] intersects E x,, q = e q, and q; and ξ q > otherwise. For any t,t there exists n N schthat n 1 <t n. Then for almost all x ξ we can see that h, x + ξ,t h, x, t = n x+ξ n x = E x,, q n q n,q E 3.1 where x is the volme τ h where x. We introdce the term ξq d q in 3.1 by mltilying and dividing by it the right hand side. Using the Cachy-Schwartz ineqality we obtain 3.11,q E h, x + ξ,t h, x, t E x,, q ξ q d q n q n E x,, q. ξ q d q,q E

11 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Using the fact that ξ q d q = ξ ξ n qd q = ξ ξ x q x, we have that 3.1,q E E x,, q ξ q d q = ξ xx+ξ x ξ x. In order to bond the difference between the two volme center oints in the right hand side of 3.1, we add and sbtract x + ξ to obtain x x+ξ x x = x x+ξ x + ξ x x x + ξ x x+ξ x + ξ + x x x + ξ h + ξ since x + ξ x + ξ and x x. This reslt imlies that 3.13 E x,, q ξ q d q h + ξ.,q E Now, we integrate the relation 3.11 on ξ,t and se 3.13 to obtain h, x + ξ,t h, x, t dxdt 3.14 ξ,t N max h + ξ n=,q E n q n ξ q d q ξ E x,, q dx since h, is iecewise constant for eachinterval n, n +1. By the geometrical argment given in [7], we have that E x,, q dx m e q ξ ξ q ξ and alying this reslt in 3.14 we obtain h, x + ξ,t h, x, t dxdt 3.15 ξ,t N max h + ξ ξ n=,q E n q n d q m e q. Finally, sing the discrete a riori estimate ii of Lemma 3.1 we end the roof.

12 K. Mila, N. Ramarosy Lemma 3.3. Time translate estimate There exists a ositive constant C sch that h, x, t + s h, x, t dxdt Cs, for all s,t. Proof. Let s,t be a given nmber. Let s define the following fnctions of time t At = h, x, t + s h, x, t dxdt, t t + s n t = and n t+s =, where. means the er integer art of ositive real nmber. Since h, is iecewise constant fnction we have that m which can be written as At = 3.16 n t+s τ h At = τ h n t+s nt nt n m. t n<t+s We se the aroximation scheme.1 in 3.16 to have At = t n<t+s τ h n t+s t n<t+s nt [ g σ,n q h, T q q ] + q N + n t+s nt f n m τ h which after rearranging of the sm concerning the control volme variable leads to the relation At = n t+s nt n t+s q + nt q t n<t+s,q E 3.17 gq σ,n h, T q q + + n t+s nt f n m τ h t n<t+s

13 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Alying Yong s ineqality in terms of the revios exresion and sing the relation 1 a + b a + b yield 3.18 At 1 A t+ 1 A 1t+ 1 4 A t+a 3 t where A t = A 3 t = A t = A 1 t = t n<t+s t n<t+s t n<t+s t n<t+s,q E,q E,q E τ h n t+s T q n t q T q n t+s q nt, n t+s, g σ,n q h, Tq q, nt f n m. Now, we integrate 3.18 according to the time variable to obtain Atdt 1 A tdt + 1 A 1 tdt A tdt + A 3 tdt. Next ste is to give a bond for each term on the right hand side of 3.3. We begin with A term. A tdt = = It is clear that 3.4 T s,q E t n<t+s T q n t q,q E T q n t q nt dt = nt χ {t n<t+s} dt. n N χ {t n<t+s} = χ {n s<t n}.

14 K. Mila, N. Ramarosy Then we slit the integration over,t s into a sm of time ste intervals to have that A tdt n t= n t+1 n t,q E T q n t q χ {n s<t n} dt n N nt since n t deends on t and the integrated fnction is ositive. Bt we have that 3.5 J = n t+1 n t n N χ {n s<t n} dt = n N n t+1 and if we change the variable to w = t n + s, we have that n t χ {n s<t n} dt 3.6 J = n N n t+1 n+s n t n+s χ {<w s} dw = s Then it yields that 3.7 A tdt s N max n t=,q E T q n t q Alying the estimate ii of Lemma 3.1 in 3.7 gives nt. 3.8 A tdt Cs. Similarly, only changing n t into n t+s, one can show that 3.9 A 1 tdt Cs. Using the definition 3.1 we have that A tdt = T s n N,q E g σ,n q h, T q q χ{t n<t+s} dt.

15 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Since t varies over,t s, we can restrict the smmation only to n =,..., N max 1 and alying 3.4 we have 3.3 n=,q E A tdt g σ,n q h, Tq q However, one can show that for all n N max 1 holds 3.31 χ {n s<t n} dt. χ {n s<t n} dt = min T s, n max,n s s. Alying the estimate ii of Lemma 3.1 and 3.31 in 3.3 yield 3.3 A tdt Cs. For the term A 3 t we se the assmtions for f stated in 1.6. Using Yong s ineqality and the relation 1 a b a + b in the Definition 3. we have 3.33 A 3 t t n<t+s n t+s + τ h + nt + f n m. Since f is Lischitz continos and f =we can dedce 3.34 A 3 t t n<t+s τ h n t+s + K f + Kf n m. + n t + Now we integrate it over,t s in order to obtain 3.35 A 3 tdt B 1 +B +K f B 3 +K f B 4

16 K. Mila, N. Ramarosy where B i,i=1,..., 4, corresond to 3.36 B 1 = T s t n<t+s n t+s m dt, τ h B = B 3 = B 4 = T s T s T s t n<t+s t n<t+s t n<t+s n t m dt, τ h τ h m dt, τ h n m dt. We se the same argment as in the estimate of A t to state that B 1 n t+s = τ h n t+s n t+s +1 m n t+s χ {n s<t n} dt. n N The identities 3.5, 3.6 and the estimate i of Lemma 3.1 imly that 3.4 B 1 CTs and, similarly, 3.41 B CTs. In order to give an er bond of B 3, one can se the estimates 3.4 and 3.4 to have 3.4 B 3 C 1 n= which together with 3.31 imlies that χ {n s<t n} dt 3.43 B 3 C 1 Ts. In order to estimate the last term B 4, we se that 3.44 B 4 n= T s n m χ {n s<t n} dt τ h

17 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations which together with the estimate i of Lemma 3.1 and 3.31 leads to 3.45 B 4 CTs. Ths, sing 3.4, 3.41, 3.43 and 3.45, we can dedce that 3.46 A 3 tdt T C + K f C 1 + C s. Finally, alying 3.8, 3.9, 3.3 and 3.46 in 3.3 we have roved the lemma. Let s define the set E ext = {κ, sch that there exists τ h,κ }, and let κ := where τ h,κ. The following lemma reresents the so called trace ineqality given in [6]: Lemma 3.4. Let be an oen bonded olygonal connected sbset of R d. Let γ h, be defined by γ h, = κ a.e. for the d 1 Lebesge measre on κ E ext. Then there exists ositive C, deending only on, sch that γ h, L C h, 1,τh + h, L where h, 1,τh =,q E n q n m e q d q 1. Lemma 3.5. Convergence of h, There exists L Q T sch that for some sbseqence of h, as h,. h, in L Q T Proof. From the estimate i of Lemma 3.1. we have that h, L Q T C and we have roved the time and sace translate estimates given in Lemmas 3. and 3.3. In order to se Kolmogorov s comactness criterion [4], Theorem IV.5, it will be sfficient to rove that 3.47 K = h, x + ξ,t h, x, t dxdt C ξ.,t

18 K. Mila, N. Ramarosy In fact, we can write and ths =,q E,q E h, x + ξ,t h, x, t = E x,, q n q n + χ [x, x + ξ] κ n κ κ E ext h, x + ξ,t h, x, t E x,, q ξ q d q n q n E x,, q ξ q d q,q E + χ [x, x + ξ] κ n κ. κ E ext Using the same techniqe as in the roof of Lemma 3. one obtains that N max K h + ξ ξ C + from where n= κ E ext χ [x, x + ξ] κ n κ dxdt N max K h + ξ ξ C + ξ n κ m κ n= κ E ext which in trn gives N max K h + ξ ξ C + ξ γ h, L. Now, sing Lemma 3.4 we have n= n= N max K h + ξ ξ C +C ξ h, 1,τh + h, L Then, the a-riori estimates of Lemma 3.1 give s that there exists C> sch that 3.47 holds tre. Then h, is relatively comact in L Q T. This imlies that there is a sbseqence of h, converging to a limit in L Q T.

19 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations 3.3. Convergence of the discrete soltion to the wea soltion In this section we consider the sbseqence hm, m of h, that converges to when h m, m see Lemma 3.5. Next ste is to rove that is the wea soltion of For the sae of simlicity, we still call this sbseqence h,. First let s define the set of fnctions Ψ = { ϕ C,1 [,T], ϕ. n =on,t,ϕ., T = } which is dense in the set Φ defined in.11. Let ϕ Ψ be given. In order to have a discrete analogy of the wea soltion identity.1, we mltily the scheme.1 by ϕ x,t n. Then we sm the reslting identity over all τ h and n =,..., N max 1. It yields 3.48 = Nmax 1 n= n= τ h ϕ x,t n τ h + Nmax 1 n= n q N ϕ x,t n m τ h ϕ x,t n f n [ g σ,n q h, T q q ] m. Next we mae a discrete integration by art of eachterm of the relation 3.48 in order to aroach the wea soltion form. Then we transfom the term on the left hand side by rearranging the smmation over n, i.e., by tting the time difference in ϕ instead of in. We also tae into accont the fact that ϕ x,t=for all τ h. We obtain n= τ h n ϕ x,t n m 3.49 N max = n τ h n=1 ϕ x,t n ϕ x,t n 1 m τ h ϕ x, m. For the first term of the right hand side of 3.48, we gather the sm over τ h and over q N to have n= τ h ϕ x,t n q N [ g σ,n q h, T q q ]

20 = 1 n=,q E K. Mila, N. Ramarosy gq σ,n h, T q q 3.5 ϕ x q,t n ϕ x,t n since gq σ,n h, T q = gq σ,n h, T q. Then we can write the scheme in its discrete wea form analogos to the identity.1, i.e. N max n= τ h n n= ϕ x,t n ϕ x,t n 1 m + ϕ x, m τ h,q E + n= gq σ,n h, T q q ϕ x q,t n ϕ x,t n τ h ϕ x,t n f n m =. In the seqel, we rove the convergence of each term of 3.51 to its continos analogy in.1 for all test fnctions ϕ Ψ. Lemma 3.6. We have that N max n=1 τ h n as h, for all ϕ Ψ. ϕ x,t n ϕ x,t n 1 T m ϕ t x, t dxdt Proof. Let ϕ Ψ be given. Then we define the difference of discrete and continos terms of lemma by [ N max T 1 = n ϕ x,t n ϕ x,t n 1 m n=1 τ h t n t n 1 We add and sbstract n N max 3.5 T 1 n=1 ϕ t t n t n 1 τ h n t n t n 1 ] x, t dxdt. ϕ t x, t dxdt in the smmation to obtain ϕ t x,t ϕ x, t dxdt + t

21 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations + T h, ϕ t x, t dxdt. To estimate the first term of the right hand side of 3.5 we aly the Cachy-Schwartz ineqality in order to searate n and to have its discrete L Q T norm. Then sing the fact that ϕ t x,t ϕ x, t hm t since ϕ C,1 [,T], we can dedce that this term tends to as h,. From Lemma 3.5, which gives a strong convergence, h, to in L Q T, we have the same assertion also for the second term on the right hand side of 3.5. Ths we roved that T 1 when h,. Lemma 3.7. For a given and for as defined in. we have that ϕ x, m x ϕ x, dx τ h as h, for all ϕ Ψ. Proof. Using. we have that ϕ x, m τ h = τ h x ϕ x, dx = ϕ x, ϕ x, x dx Since ϕ C,1 [,T], there exists a ositive constant M schthat ϕ x, ϕ x, hm and one can dedce the assertion of the lemma. Using the definition of Ψ, we aly the Green formla to obtain 3.53 g G σ ϕdx = div g G σ ϕ dx, which will be sed in the seqel for the convergence roof.

22 K. Mila, N. Ramarosy Lemma 3.8. We have that N 1 max 1 n=,q E ϕ x q,t n ϕ x,t n as h, for all ϕ Ψ. Proof. We consider that N 1 max 1 n=,q E T gq σ,n h, T q q T div g G σ ϕ dxdt gq σ,n h, T q q ϕ xq,t n ϕ x,t n div g G σ ϕ dxdt = where R i,i=1,..., 5 comes from the slitting of the left hand side difference into several arts adding and sbstracting some extra terms and which will be defined and estimated in the seqel. First, we define 5 i=1 R i 3.54 R 1 = 1 n=,q E q g σ,n q h, Rqm n e q, where R n q reresents the difference between discrete and continos normal derivative evalated on x q,t n, i.e ϕ Rq n xq,t n ϕ x,t n = ϕ x q,t n n q. d q Since ϕ C,1 [,T], one can show that there exists a constant M 3 > schthat R n 3.56 q hm3. Then, sing 1.4 we have 3.57 R 1 h M 3 n= q m e q.,q E

23 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations We mltily and divide the right hand side of 3.57 by d q and aly Cachy-Schwartz ineqality to obtain 3.58 R 1 h M 3 n= n=,q E,q E T q q d q m e q It comes from geometrical argments that there exists a ositive constant M 4 schthat 3.59 d q m e q M 4.,q E The estimate ii of Lemma 3.1 combined with 3.59 imlies that R 1 h M 3 M 4 TC 1 and one can conclde that R 1 as h,. Next, we set 3.61 R = 1 n=,q E q g σ,n q h, t t n e q n Rqdxdt where 3.6 R n q = ϕ x q,t n ϕ x, t n q. Thans to the reglarity of ϕ, one can show that for any x e q holds h + M5 R n q witha ositive constant M 5 deending only on ϕ. We aly this reslt to relace R n q in 3.61 and by the same argment as in estimating of the term R 1 we derive that 3.63 R as h,. Now, we denote 3.64 G n q = g G σ ũ h, x q,t n g G σ ũ h, x, t

24 K. Mila, N. Ramarosy and define the third term 3.65 R 3 = 1 n=,q E q t t n e q G n q ϕ n q dxdt. To rove the convergence of R 3 to, first we bond G n q. For that rose, we se the fact that g is Lischitz continos. Let L g be the Lischitz constant of g, i.e., for any ositive real nmbers ζ 1 and ζ hold 3.66 g ζ 1 g ζ L g ζ 1 ζ. Then we have L g G σ ũ h, x q,t n G σ ũ h, x, t G n q and one can se the trianglar ineqality for the Eclidean norm to obtain L g G σ ũ h, x q,t n G σ ũ h, x, t. G n q Using the form of the convoltion as given in.7 and as t t n,t, one can show that for any x e q holds G n q Lg n G σ x q s G σ x s ds r r where the sm is evalated only on control volmes r τ h and r in reflexion of τ h throgh the bondary of intersecting B σ x q, the ball centered at x q withradis σ. Thans to the hyotheses on G σ, which is in C R d, the Cachy- Schwarz ineqality and the estimate i of Lemma 3.1 we obtain G n q hm6 witha ositive constant M 6. Since ϕ is a continos fnction, S =s ϕ <,wehavethat Q T R 3 h M 6S n= q m eq,q E which together with Cachy-Schwarz ineqality and 3.59 leads to the desired reslt 3.67 R 3 as h,. Let the forth term be given as 3.68 R 4 1 n=,q E q

25 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations t t n e q g G σ ũ h, g G σ ϕn q dxdt. The Lischitz continity of g and the trianglar ineqality for Eclidean norm give that 3.69 g G σ ũ h, g G σ L g G σ ũ h, Using Cachy-Schwartz ineqality in the convoltion term in 3.69 and the definition of extension lead to the following reslt g G σ ũ h, g G σ 3.7 C G σ x s ds R d 1 h, L Q T. Since G σ is C, and G σ has a comact sort, one can show that there exists a constant M 7 > schthat 1 G σ x s ds M7. R d Then, we se the same techniqe as for the estimate of R 1 to see that there exists a ositive constant M 8 schthat 3.71 R 4 M 8 h, L Q T and since h, converges to strongly in L Q T, we dedce that 3.7 R 4 as h,. The last term is defined by 3.73 R 5 = T div g G σ ϕ h, dx. The form of the first term of 3.73 is slightly different from its eqivalent in 3.68 bt it can be jstified by the fact that div g G σ ϕ h, x, t dx 3.74 = τ h div g G σ ϕ dx

26 K. Mila, N. Ramarosy by the definition of h, and by the vale of t which belongs to the interval [t n,t ]. Alying the Green formla in 3.74 imlies that div g G σ ϕ dx τ h = τ h q N e q g G σ ϕ n q dx. Since g G σ ϕn q = g G σ ϕn q, we finally obtain div g G σ ϕ h, x, t dx 3.75 = 1,q E q e q g G σ ϕn q dx. By the hyotheses, g. C R +, and convoltion roerty state that G σ C R d since G σ C R. Then div g G σ ϕ L Q T and it comes from strong convergence of h, to that 3.76 R 5 as h,. Finally, according to 3.6, 3.63, 3.67, 3.7 and 3.76,we can conclde that N 1 max 1 n=,q E ϕ x q,t n ϕ x,t n gq σ,n h, T q q T in L Q T. This ends the roof of Lemma. div g G σ ϕ dxdt Lemma 3.9. We have that n= ϕ x,t n f T n m τ h as h, for all ϕ Ψ. f ϕ x, t dxdt

27 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Proof. The difference between the seqence and its desired limit can be written as where N 1 = N = n= n= T τ h ϕ x,t n f n m τ h ϕ x,t n n= τ h f ϕ x, t dxdt = N 1 + N t t n t t n [ f n f ] dxdt, f [ϕ x,t n ϕ x, t] dxdt. Or rose is to rove that these two qantities tend to as h,. De to the Lischitz continity of f we have t N 1 K f ϕ x,t n n dxdt n= τ h t n 3.79 and alying Cachy-Schwartz ineqality and the relation 1 a b a + b we obtain 3.8 Nmax 1 N 1 K f T from where we have 3.81 n= τ h ϕ x,t n m 1 h, x, + h, dxdt N 1 as h,. Thans to the reglarity of ϕ, one can show the existence of a ositive constant M 9 schthat ϕ x,t n ϕ x, t h + M 9. 1

28 K. Mila, N. Ramarosy Then it comes that N h + M 9 K f dxdt Q T which in trn imlies that 3.8 N as h,. Ths the lemma is roved. Theorem 3.1. Convergence to the wea soltion The seqence h, converges strongly in L Q T to the niqe wea soltion defined in.1 as h,. Proof. Using Lemma 3.5 and Lemma 3., we now from [6] that the limit of the seqence h, is in sace L,T; H 1. Then we can se Green s theorem in the reslt of Lemma 3.8 and together with Lemmas we can dedce that satisfies the wea identity for all test fnctions ϕ Ψ. Bt Ψ is dense in Φ which imlies the convergence reslt. The niqeness of the wea soltion is given in [5] see Remar.4 and so not only sbseqence bt the seqence h, itself converges to. 4. Nmerical exeriments In this section we resent nmerical exeriments obtained by the scheme.1. We have choosen 1 gs = 1+Ks witha constant K>and the convoltion is realized with ernel G σ x = 1 Z e x x σ, where the constant Z is choosen so that G σ has nit mass. In order to comte the diffsion coefficientg q σ,n h, in.1 we se concet described in Remar.3. The terms G σ x q s ds in.7 are comted sing r comter algebra system e.g. Mathematica. For any given σ they can be recomted in advance. The sarse linear systems corresonding to.1 can be solved by any efficient linear solver. In Fig. 1 we resent embedding of the initial image noisy corrted for-etal shae, 56 x 56 ixels into the so called nonlinear scale sace

29 Semi-imlicit finite volme scheme for solving nonlinear diffsion eqations Fig. 1. Fig.. given by We resent the initial image and its rocessing in the scales t =1,, 3, 4, 5. We see the simlification denoising of the image together with reserving of imortant edges in the seqence of discrete scale stes. In Fig., the scanned image coat-of-arms from a boo withnot aer nor colors of good qality is rocessed. We resent scanned original left and rocessed image right after 4 discrete scale stes. In bothexeriments = h, σ = 1, K =and h is a ixel size. References 1. Alvarez, L., Gichard, F., Lions, P.L., Morel, J.M. 1993: Axioms and Fndamental Eqations of Image Processing. Archive for Rat. Mech. Anal. 13, 57. Alvarez, L., Morel, J.M. 1994: Formalization and comtational asects of image analysis. Acta Nmerica, 1 59

30 K. Mila, N. Ramarosy 3. Bänsch, E., Mila, K. 1997: A coarsening finite element strategy in image selective smoothing. Comting and Visalization in Science 1, Brezis, H. 199: Analyse fonctionelle, Theorie et alications. Masson 5. Catté, F., Lions, P.L., Morel, J.M., Coll, T. 199: Image selective smoothing and edge detection by nonlinear diffsion. SIAM J. Nmer. Anal. 19, Eymard, R., Galloet, T., Herbin, R.: Finite Volme Methods. to aear 7. Eymard, R., Gtnic, M., Hilhorst, D. 1998: The finite volme scheme for the Richards eqation. Prerint University Paris-Sd 8. Kačr, J., Mila, K. 1995: Soltion of nonlinear diffsion aearing in image smoothing and edge detection. Al. Nmerical Math. 17, Kačr, J., Mila, K. 1: Slow and fast diffsion effects in image rocessing. Comting and Visalization in Science 3, Kačr, J., Mila, K. 1997: Slowed anisotroic diffsion. In: Romeny, B.t.H, Florac, L., Koenderin, J., Viergever, M. eds Proceedings of Scale-Sace Theory in Comter Vision, Lectre Notes in Comter Science 15, Sringer, Berlin 11. Lions, P.L. 1994: Axiomatic derivation of image rocessing models. M 3 AS 4, Nordström, K.N. 1989: Biased anisotroic diffsion-a nified aroachto edge detection. Prerint, Deartment of Electrical Engeneering and Comter Sciences, University of California, Bereley 13. Patanar, S. 198: Nmerical heat transfer and flid flow. Hemishere Pbl. Com. 14. Perona, P., Mali, J.1987: Scale sace and edge detection sing anisotroic diffsion. In Proc. IEEE Comter Society Worsho on Comter Vision 15. Romeny, B.M.t.H. ed. 1994: Geometry driven diffsion in comter vision. Klwer, Dordrecht 16. Weicert, J., Romeny, B.M.t.H., Viergever, M.A. 1998: Efficient and reliable schemes for nonlinear diffsion filtering. IEEE Trans. Image Processing 7,