Mathematical and numerical analysis of a nonlinear diffusion model for image restoration

Transcription

1 Global Journal of Pure Applied Mathematics. ISSN Volume, Number 3 06, pp Research India Publications Mathematical numerical analysis of a nonlinear diffusion model for image restoration R. Aboulaich LIRIMA-LERMA Laboratories, Mohammadia School of Engineering, Mohammed V University in Rabat, Ibn Sina Str., POB 765 Agdal, Rabat, Morocco. S. Boujena MACS, Mathematics computing Department, Ain Chock Sciences Faculty, Km 8 Route El Jadida POB 5366 Maârif, Casablanca, Morocco. E. EL Guarmah Royal Air School, Informatics Mathematics Department, LIRIMA-LERMA Laboratories, DFST, BEFRA, POB 4000, Marrakech, Morocco. M. Ziani Numerical Analysis Group, Applied Mathematical Laboratory, Dept. of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, 4, Ibn Battouta Avenue, PO Box 04, PC 0090, Rabat, Morocco. Corresponding author

2 Abstract The purpose of this paper is to study, analytically numerically, a nonlinear diffusion model of partial differential equation in image restoration based on that of Perona Malik. The originality of this work is to prove the existence uniqueness, in an appropriate Hilbert space, of a solution to the PDE problem proposed with a nonhomogeneous Dirichlet boundary condition. A first numerical approach based on an explicit scheme is given. The stability of this scheme is established by means of a CFL condition. In order to overcome this shortcoming a semi-implicit scheme based on so-called additive operator splitting AOS method is used. The results obtained for large ranges of the parameters, show the efficiency of this AOS method. AMS subject classification: E46, 53C35, 57S0. Keywords: Nonlinear diffusion, non homogeneous Dirichlet boundary condition, image restoration, Hilbert space, AOS scheme.. Introduction In last decades, image restoration has been the subject of several researches. The contours preservation imposes the introduction of nonlinear models, see [], [4], [5], [7], [8], [9], [0], [], [], [4] [5]. We present in this work a modified Perona-Malik model using a nonlinear PDE. The idea is to maintain, for this model, the efficiency of Perona Malik s one to preserve the contours while having existence uniqueness of a solution in a suitable Hilbert space. We recall that several models using regularization by convolution have been studied before in order to prove the existence uniqueness of regularized problems for image restoration see for more details [9], [4] [5]. In [3], authors prove existence uniqueness for Perona-Malik model in Orlicz space. In [] [4], existence uniqueness of solution for the modified Perona Malik model with homogeneous Dirichlet boundary condition is proved in a Hilbert space. In this work, the same model is considered with non homogeneous Dirichlet boundary conditions. For the numerical resolution, we purpose at first, an explicit scheme we prove, for its stability, a CFL condition. In order to improve the numerical resolution to avoid the stability condition, we remplace the above explicit sheme by a semi-implicit approximation based on the AOS algorithm proposed in [6]. We end this work by some numerical simulations a comparison between the proposed explicit semi-implicit schemes.

3 Mathematical numerical analysis of a nonlinear diffusion model 46. Nonlinear diffusion model We are interested to restore a noisy image u 0 on using the following nonlinear PDE model with non homogeneous Dirichlet condition on. Find φ L 0,T,H such that: φ div[µ φ φ] =0 in Q, t φx,t = u 0 x, 0 x, t [0,T], φx,0 = u 0 x, 0 x. is an open bounded subset of IR n, n = orn = 3, with boundary Q = x[0,t] for some given T>0. We set ux, t = φx,t u 0 x, 0 for all x, t Q, then the problem is equivalent to the following problem: find u L 0,T,H such that: u t div[µ u + u 0 u + u 0 ]=0 in Q, ux, t = 0 x, t [0,T], ux, 0 = 0 x. A weak formulation of the problem is to Find u L 0,T,Vsuch that: ut vdx + µ[ ut + u 0 ] ut + u 0. vdx = 0 v V, 3 t u0 = 0. V = H0 will be provided with the inner product u, v = u. vdx.. is its associeted norm. We indicate by H the adherence of V in L. The space H is provided with the scalar product of L defined by: u, v = uv dx. The associated norm is noted by.. The existence uniqueness of a solution of the considered PDE problem with homogeneous boundary conditions are establish see [] []. In the following we will establish the existence uniqueness of the weak solution of in L 0,T,H under the same hypothesis on µ:. µ : IR + IR +,. µ is a continuously differentiable function, 3. lim s + [µs] =µ 0 with µ 0 > 0, 4. s µ s µs s IR +. The hypothesis involve that µ is bounded.

4 46 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani 3. Existence Uniqueness theorems We set sup µs = a inf µs = b, with a 0 b 0. s IR + s IR + We denote by A the operator defined by Av, w = µ v + u 0 v + u 0. wdx for v, w V. 4 Where u 0 H is given. According to the assumptions on µ,wehaveav H for all v V. To prove the existence of a solution for the problem 3, we need the next lemma, see [] [6]. Lemma 3.. A is an operator monotone hemicontinuous, satisfying for all v, w V Proof. Let be v, w H0 Av Aw, v w = Av Aw, v w b v w. 5 µ v + u 0 [ v + u 0 ]. v wdx µ w + u 0 [ w + u 0 ]. v wdx. For α [0, ], we take α = µs[w + u 0 + αv w] [w + u 0 + αv w]. v wdx, with su = u. For each fixed x, we consider the function f defined by: fx,α= µs[w + u 0 + αv wx] [w + u 0 + αv wx]. v wx on [0, ]. We need to prove that α fx,αis differentiable function on ]0, [ for each fixed x. Let x fixed in α fixed in ]0, [. The following situations are then discussed: a 0 < sw + u 0 + αv w x < +. b sw + u 0 + αv w x = 0. In the situation a f α x,α = µ{s[w + u 0 + αv w]} v w + µ {s[w + u 0 + αv w]} sw + u 0 + αv w α [w + u 0 + αv w]. v w.

5 Mathematical numerical analysis of a nonlinear diffusion model 463 We have: sw + u 0 + αv w = [w + u 0 + αv w] { [w } + u0 + αv w] { } [w + u0 + αv w] = +. x x Then with s[w + u 0 + αv w] α = C C. C α, C = { } [w + u 0 + αv w] + x { } [w + u0 + αv w], x f α x,α = µ{sw + u 0 + αv w} v w + µ {sw + u 0 + αv w} sw + u 0 + αv w { w + u 0 + αv w. v w}. 6 In the situation b sw + u 0 + αv w = 0 w + u 0 + α v w = 0 w + u 0 = x = w + u 0 x = v w = 0 x x v w b or x w + u 0 = α x v w w + u 0 = α v w x x b If we re in the situation b then using the definition of f, we conclude that f x,δ = 0, δ ]0, [ f x,δ = 0, δ ]0, [. δ If we re in the situation b then [ ]/[ ] w + u 0 v w = α, x x

6 464 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani lim k + t k [ ]/[ ] w + u 0 v w = α, x x for all t ]0, [ such that t = α we have sw + u 0 + tv w x > 0, then the function t f x,t is differentiable. Show that t f x,t is also diffrentiable in t = α. Let t k k IN a sequence of elements of the ]0, [ so that t k = α for all k IN lim t k = α. k + For all k IN the function t f x,t is derived in t k its derivative is given by 6, then [ ] f x,tk = lim [µ{sw + u 0 + t k v w} v w k + + µ {sw + u 0 + t k v w} { w + u 0 + t k v w. v w} ]. sw + u 0 + t k v w Notice that, since s µ are continuous then lim k + lim [µ{sw + u 0 + t k v w} v w ]=µ0 v w, k + [ ] f x,tk = µ0[ v w] t [ k µ {sw + u 0 + t k v w} + lim k + sw + u 0 + t k v w We have µ {sw + u 0 + t k v w} sw + u 0 + t k v w { w + u 0 + t k v w. v w} ]. { w + u 0 + t k v w. v w} µ {sw + u 0 + t k v w} sw + u 0 + t k v w v w, µ is continuous, we deduce that µ {sw + u 0 + t k v w} µ 0, lim k + f x,t k t = µ0 v w. k So t f x,t is differentiable in α we have f x,α α = µ0 v w.

7 Mathematical numerical analysis of a nonlinear diffusion model 465 It s now easy to show that f x,t t K v w, with K = µ0 in the case b K = a in the case a. Then is differential on ]0, [ f x, α α = dx, α ]0, [. α Applying the mean value theorem to on [0,], we deduce that there exists λ ]0, [ such that Av Aw, v w = 0 = f x, λ λ = dx. λ We take ={x /0 <s[w + u 0 + λv w] < + }, ={x /s[w + u 0 + λv w] =0}. Then = =. Furthermore, µ{sw + u 0 + λv w} v w f x, λ λ + µ {sw + u 0 + λv w} sw + u 0 + λv w = { w + u 0 + λv w. v w} if x µ0 v w if x f x, λ Av Aw, v w = dx λ f x, λ + dx λ µ[sw + u 0 + αv w] v w dx µ [sw + u 0 + αv w] sw + u 0 + αv w sw + u 0 + αv w v w dx + µ0 v w dx b v w 0.

8 466 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani Theorem 3.. Existence theorem Let µ satisfying. 4. u 0 a function defined on Q such that u 0., 0 H. Then there exists at least one weak solution φ of problem such that φ L 0,T,H L 0,T,H. Proof. To show the existence, we use Faedo-Galerkin method. We consider the spectral problem w, v = λw, v v V. 7 Since the injection of V in H is compact, the problem 7 admits a sequence of eigenvalues λ j associated to eigenvectors w j such that w j,v= λ j w j,v v V, 8 {w j } j IN is orthonormal in H orthogonal in V. We denote by u N t the approximate solution of 3 defined by u N x, t = u N tx [w,...,w N ] N u N x, t = Cj N tw jx j= u N., 0 = 0 [w,...,w N ]. 9 We have then u N t, w j + µ u N t + u 0 u N t + u 0, w j = 0, j N, t [0,T], u N., 0 = u 0N [w,...,w N ]. u 0N u 0 in H. 0 Each C N j t verifies dc N j t dt = G j t, C N t,...,cn N t. Knowing that G j is a continuous function, then by using the Cauchy Péano theorem we deduce that there exists a local solution u N t of 0 on [0,T N ] [0,T]. By multiplying 0 by Cj N t by adding, we deduct that: u N t u N tdx + µ[ u N t + u 0 ] u N t + u 0. u N tdx = 0. t Then d dt u Nt + µ[ u N t + u 0 ] u N t + u 0. u N tdx = 0,

9 Mathematical numerical analysis of a nonlinear diffusion model 467 Case : b>0. As d dt u Nt + µ[ u N t + u 0 ] u N t dx. + µ[ u N t + u 0 ] u 0. u N tdx = 0. from we have d dt u Nt +b Then µ[ u N t + u 0 ] u N t dx b u N t dx, 3 u N t dx µ[ u N t + u 0 ] u 0. u N tdx. 4 d dt u Nt +b u N t a b u 0 + b u Nt, 5 u N t a b T u 0 + u 0, 6 from 4 we deduce that t u N t +b u N t dt a 0 b T u 0 + u 0, 7 t 0 for all t [0,T N ]. There exists thus a constant u N t dt a T b u 0 + b u0, 8 a C = b T u 0 + u 0 > 0 a constant a C = b T u 0 + b u 0 > 0 depending of a, b, T u 0 such that Case : b = 0. u N t C t 0 u N τ dτ C t [0,T N ]. 9

10 468 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani In, we will rewrite the second term in the form µ{ u N t + u 0 } u N t dx = {µ[ u N t + u 0 ] µ 0 } u N t dx + µ 0 u N t dx. Where µ 0 is given by the assumption 3. on µ. Then becomes d dt u Nt +µ 0 u N t = {µ[ u N t + u 0 ] u 0. u N tdx. µ[ u N t + u 0 ] µ 0 u N t dx, 0 d dt u Nt +µ 0 u N t a u 0 u N t + µ u N t + u 0 µ 0 u N t dx. Let us set µ s = µs µ 0, then lim µ s = 0. Take ε>0, therefore there exists s + B>0so that for all x, t [0,T N ] µ { u N x, t + u 0 } < ε if u N x, t + u 0 >B. For t [0,T N ] fixed, we consider the following sets t ={x / u Nx, t + u 0 B}, t ={x / u Nx, t + u 0 >B}, t t = t t =. We have then: µ { u N x, t + u 0 } u N x, t dx = µ { u N x, t + u 0 } u N x, t dx t + µ { u N x, t + u 0 } u N x, t dx t B + u 0 µ u N t + u 0 dx + ɛ t t u N t dx.

11 Mathematical numerical analysis of a nonlinear diffusion model 469 From we have d dt u Nt +µ 0 u N t a u 0 u N t + 4B + u 0 a + µ 0 mes + ɛ u N t. We take ε = µ 0 we set 4 a + 4µ 0 + a mes u 0 +4B mesa + µ 0 = ξ,µ 0,a,u 0, µ 0 then d dt u Nt + µ 0 u Nt ξ,µ 0,a,u 0, hence, by the same reasoning as in the first case, we deduce that there exists a constant C = ξ,µ 0,a,u 0 + u 0 > 0 a constant C = µ 0 ξ, µ 0,a,u 0 T + u 0 >0 depending of, µ 0,a,u 0 T such that u N t C t 0 u N t dt C t [0,T N ]. Thus in both cases there exists two constants C > 0 C > 0 independent of N such that u N t C t [0,T N ], t u N τ dτ C. 0 Moreover the problem 0 admits a global unique solution u N on [0,T] according to the monotony the hemicontinuity of the operator A we deduce that the approximate solutions u N of the problem 0 converge towards a weak solution u of the problem 3, see [4], [6], [3] φ = u + u 0 is a weak solution of. Theorem 3.3. Uniqueness theorem Under Hypothesis of the existence theorem 3., the weak solution φ of the problem is unique φ = dφ L 0,T,H. dt Proof. Let φ φ two weak solutions of the problem. Then u = φ u 0 u = φ u 0 are two solutions of the problem. We have for all v V u t t u t t, v + Au t Au t, v = 0. 3

12 470 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani Taking u u = w v = wt in 3, we have then from : Then d dt wt = Au t Au t, wt 0 wt w0 = 0 Thus wt = u t u t = 0. In conclusion, uniqueness of weak solution for the problem is obtained. Besides, let u = φ u 0 then u t = Aut in V, from 4, for all v V,wehave It follows that Aut, v a ut v. Aut V a ut for all t [0,T]. Thus Au L 0,T,V u L 0,T,V. Which allows to conclude that φ L 0,T,V. 4. Numerical approach In order to restore a noisy image u 0, we use a model, with µx = + α, + x which verifies the asymptions.-4.: u t div + α u = 0 in Q = x]0,t[, + u ux, 0 = u 0 x, 0 x, ux, t = u 0 x, 0 x t [0,T], 4 where u represents the restored image. 4.. An explicit scheme 4.. Algorithm We denote respectively by h τ the spatial time steps sizes. In the following, we take h =. The discretization of the time derivative in 4 is given by: u t i, j = un+ i, j u n i, j, τ the superscript n n + denote the time levels t n t n+, respectively.

13 Mathematical numerical analysis of a nonlinear diffusion model 47 A first-order explicit scheme employed for the spatial derivative approximation: u n i, j = u x u y ni, j = un i +,j u n i, j, h ni, j = un i, j + u n i, j. h Then u u n n i +,j u n i, j u n i, j + u n i, j i, j = +. h h We define for every field p = p,p IR, the discrete divergence approximation: p n i, j pn i,j= 0 if <i<n divp i,j = p n i, j if i = p n i,j if i = N p n i, j pn i, j = 0 if <j<n + p n i, j if j = p n i, j if j = N where N is an integer greater than p n i, j = p n + u i, j = n i, j + α p n i, j = + u n i, j + α A classical explicit finite difference scheme for 4 is then given by n u i, j, x n u i, j, y u n+ i, j = u n i, j + τ divp n i, j, n M, 5 6 where p n i, j = p n i, j, pn i, j u n i, j = ux i,y j,t n, x i = ih, y j = jh, t n = nτ τ = T M is the time step size.

14 47 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani This scheme can be explicitly solved for the unknown u n+ : u n+ i, j = u n i, j + τ + u n i, j + α u n i, j + + u n i +,j u n i, j + τ + u n i,j + α u n i,j u n i, j + τ + u n i, j + α u n i, j u n i, j. 7 We suppose that h =. 4.. Stability of the numerical scheme In the following, we will study the stability of the proposed explicit scheme using the infinity norm. After reorganizing the terms, we obtain from equation 7 [ u n+ i, j = u n i, j τ + u n i, j + α τ + u n i,j + α ] τ + u n i, j + α + τ + u n i, j + α u n i, j + + u n i +,j + τ + u n i,j + α u n i,j + τ + u n i, j + α u n i, j. We have Then α + α + α i, j. + u n i, j + α + u n i, j α α.

15 Mathematical numerical analysis of a nonlinear diffusion model 473 Finally, we have 4τ + α τ + u n i, j + α τ + u n i,j + α τ + u n i, j + α Notice that if τ 4 + α, then u n+ i, j 4τ + α u n i, j +τ + α u n i, j + + u n i +,j Then Which implies that Therefore + τ + α u n i,j + u n i, j. u n+ i, j 4τ + α u n + 4τ + α u n. u n+ u n. u n+ u 0. And we can conclude that the CFL-like condition of the stability is given by: τ 4 + α. 8 In order to avoid this condition, we use in the next section a semi-implicit scheme proposed in [6]. 4.. Semi-implicit scheme In the following, we consider the discretization defined by: u k+ u k τ = A l u k u k+. 9 l= Where A u k = x A u k = y + u k + α + u k + α u k, x u k. y

16 474 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani Figure : Restored image with α = τ = 0.. This scheme requires to solve a linear system at each time step. For this reason it is called a linear semi-implicit scheme. The solution u k+ is given by u k+ = I τ A l u k u k. 30 l= This semi-implicit scheme still has a major drawback at each iteration one needs to solve a large linear system whose matrix is not tridiagonal. Let us now consider a modification of Equation 30, namely the additive operator splitting AOS scheme, see [6]. u k+ = I τa l u k u k. 3 l= The operators B l = I τa l u k describe one-dimensional diffusion processes along the spatial axes. The above splitting scheme is efficient because at each time step a single tridiagonal matrix inversion is performed for each spatial dimension. Under a consecutive number of pixels along the direction they are reduced to tridiagonal matrices strictly diagonally dominant that can be effectively reversed by the Thomas algorithm.

17 Mathematical numerical analysis of a nonlinear diffusion model Figure : Restored image with α = 0.5 τ = 0.. Figure 3: Restored image by explicit scheme with α =. 475

18 476 R. Aboulaich, S. Boujena, E. EL Guarmah, M. Ziani Figure 4: Restored image with α = Numerical results We recall that the SNR is the Signal-to-noise ratio used to estimate the quality of an image I with respect to a reference image I, defined by the expression: SNRI /I = 0 log0 σ I, σ I I where σ is the variance. Figures are used as a test for processing images corrupted by a Gaussian Noise.

19 Mathematical numerical analysis of a nonlinear diffusion model 477 We remark in figure that the Additive Oprator Splitting scheme provides better restoration results than the explicit scheme for α = τ = 0.. On the other h, we observe that both the explicit the AOS schemes give good results for α = 0.5 τ = 0. becuase in this case, the stability condition is satisfied. In Figure 3 we present the value of the coefficient SNR, for the explicit scheme, according to the variation of the additional Gaussian noise. We notice that the explicit scheme is only stable provides a better results for very small time steps. This leads to poor efficiency. Next we consider Figure 4 which depicts the filtering by a Gaussain noise. The situation is similar as in Figure : the Additive Oprator Splitting AOS scheme provides a better restoration results than an explicit scheme for α = with τ = 0.3 τ = 0. when the stability condition is not satisfied. 5. Conclusion We proposed in this work a nonlinear diffusion model with non homogeneous boundary conditions in image restoration. After establishing the existence uniqueness of a solution for PDE problem, we tested the model numerically by both an explicit a semi-implicit schemes using an additive operator splitting AOS method. The obtained results show that an instability occurs for particular values of the parameters τ α for the explicit scheme if the stability condition is not verified. The semi-implicit scheme with an additive operator splitting AOS method permits to avoid this difficulty. Acknowledgment This work was supported by Euromediterranean Scomu project, the LIRIMA Laboratory by the Franco-Moroccan Henri Curien project governed by EGIDEMA//46. References [] R. Aboulaich, S. Boujena, E. El Guarmah, A nonlinear diffusion model with non homogeneous boundary conditions in image restoration, ECS0 Milan Italie, June [] R. Aboulaich, S. Boujena, E. El Guarmah, A nonlinear parabolic model in processing of medical image, Mathematical Modelling of Natural Phenomena, 3, [3] R. Aboulaich,D. Meskine, A. Souissi, New diffusion models in image processing, Journal of Computers Mathematics with Applications, 564, [4] R. Aboulaich, S. Boujena, E. El Guarmah, Sur un modèle non-linéaire pour le débtruitage de l image, C.R. Acad. Sci. Paris. Ser., 345,

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