International Journal of Analysis and Applications Volume 19, Number 2 (2021), 228-238 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-228 IMAGE RESTORATION USING A NOVEL MODEL COMBINING THE PERONA-MALIK EQUATION AND THE HEAT EQUATION SAMIRA LECHEHEB∗, MESSAOUD MAOUNI, HAKIM LAKHAL Laboratory of Applied Mathematics and History and Didactics of Maths ”LAMAHIS”, Department of Mathematics, University of 20 August 1955, Skikda, Algeria ∗Corresponding author: lecheheb.samira24@gmail.com; s.lecheheb@univ-skikda.dz Abstract. This article is devoted to the mathematical study of a new proposed model based on a Perona- Malik equation combined with a heat equation. This study shows how system of partial differential equations can be used to restore a digital image. By using compactness method and the monotonicity arguments, with suitable assumptions on the nonlinearities, we prove the existence of the weak solution for the proposed model which its consistency is given in our work. 1. Introduction In recent years, the application of partial differential equations in image processing have attracted at- tention of many authors in computer vision. Various techniques have been developed in image processing during the last decades. Now these techniques are used for all kinds of tasks in all kinds of domains: indus- trial inspection, medical visualization, human computer interfaces, artistic effects, etc. Texture extraction and image restoration are the two fundamental problems that have made a significant contribution to this discipline, as can be ascertained from recent survey papers [1,4,5,8,10,11,16,17,19,21]. The question here is: how to preserve the contours of an image while the elimination of the noise. In 1990, Perona and Malik [20] answered this question in their model which is one of the first attempts to derive a model that incorporates local information from an image within a PDE framework. The numerical results of this model represent Received January 6th, 2021; accepted February 1st, 2021; published February 22nd, 2021. 2010 Mathematics Subject Classification. 35Q30, 65N12, 76M25. Key words and phrases. topological degree; nonlinear system; homotopy. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 228 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-228 Int. J. Anal. Appl. 19 (2) (2021) 229 an efficient and effective tool for image denoising. Nevertheless, two important phenomena have been ob- served. The first one proved by Kichenassamy [14] that the Perona-Malik model is an ill-posed problem in the sense of Hadamard, this paradoxical result is sometimes referred to as the Perona-Malik Paradox [14]. The second one is the phenomenon known as the staircase effect. To make this problem well posed, Catté et all [9] suggested introducing the regularization in space and time directly into the continuous equation. There is an extensive literature for the purpose of denoising in image processing. Let us mention the work of Aboulaich et al [2], concerning new diffusion models for the image processing. The proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. In their valuable monograph [7], Atlas et al. have presented a new model for image restoration. The proposed model is an interpolation of two classical models, Perona-Malik and p-Laplacian. By using the monotonicity arguments, they proved the existence and the uniqueness of solutions for p large enough. They also studied the asymptotic behavior of the solution as p →∞ and they proved that the limit problem coincides with the Perona-Malik model in the some subregion. In 2016, Afraites et al [3] proposed a model removing noise while preserving the edges and reducing staircase effect where they combined a nonlinear regularization of total variation (TV) operator’s with a decomposition approach of H−1 norm suggested by Guo et al. ( [12, 13]). The generalization of this work was made by Atlas et al [6]. Until the work of Lecheheb et al. [15] combining the Perona-Malik equation with the heat equation, the authors were able to demonstrate the existence and consistency of the their proposed model. We build up on their works by providing a generalization to the case of systems. In this article, we present a novel model for image denoising, which combines the Perona- Malik equation and the heat equation. Our model is well-posed. By using the compactness method and the monotonicity arguments, we prove the existence of solutions for the following system (1.1)   −div ( g1(|∇v|)∇u ) − 1 λ21 ∆u = f(x,u,v) in Ω, −div ( g2(|∇u|)∇v ) − 1 λ22 ∆v = h(x,u,v) in Ω, ( g1(|∇v|) + 1 λ21 ) ∇u ·~n = ( g2(|∇u|) + 1 λ22 ) ∇v ·~n = 0 on ∂Ω, where Ω ⊆ RN is a bounded domain with smooth boundary ∂Ω, with Neumann boundary conditions. ~n is the unit outward normal at the image boundaries ∂Ω and 0 < λ ≤ 1 suth that λ = (λ1,λ2). The function g(·) = (g1,g2) is defined by one of the following expressions: g(k) = 1 1 + ( k λ )2 or g(k) = exp ( − k2 2λ2 ) , Int. J. Anal. Appl. 19 (2) (2021) 230 It is clear that the function g(k) is a decreasing non-negative function satisfying the following conditions (1.2)   lim g(k) k→0 = 1, lim g(k) k→+∞ = 0. We remark that, if gi = 1 for i = 1, 2 we recover the linear diffusion. For the rest of this article, we assume that f,h : Ω × R × R → R are carathéodory functions satisfying the growth condition (1.3)   |f(x,s,z)| ≤ d1(x) + 1 2λ21 |s| + 1 2λ22 |z|, |h(x,s,z)| ≤ d2(x) + 1 2λ21 |s| + 1 2λ22 |z|, where d = (d1,d2) is in (L 2(Ω))2 and 0 < λ ≤ 1, (λ = (λ1,λ2)). The content of this paper is arranged as follows. In the next section we will present the main results. In the section 3, we will study the existence of the solutions of the problem (1.1) under some different conditions on the nonlinear terms. This existence is obtained by using the compactness method [18] and the monotonicity arguments. 2. Main results In this section, we are ready to present the definition of a weak solution for problem (1.1) and the main result. At beginning, let V = H10 (Ω) ×H 1 0 (Ω), which is a Banach space endowed with the norm ‖(u,v)‖2V = ‖u‖ 2 H10 (Ω) + ‖v‖2H10 (Ω), and let W = L2(Ω) ×L2(Ω). In the sequel, ‖ · ‖H10 (Ω) and ‖ · ‖L2(Ω) will denote the usual norms of H 1 0 (Ω) and L2(Ω), respectively. We give now the definition of a weak solution for problem (1.1) Definition 2.1. We say that (u,v) ∈ V is a weak solution for the system (1.1) if for any (φ,ψ) ∈ V we have (2.1) ∫ Ω (g1(|∇v|) + 1 λ21 )∇u∇φ dx + ∫ Ω (g2(|∇u|) + 1 λ22 )∇v∇ψ dx = ∫ Ω f(x,u,v)φ dx + ∫ Ω h(x,u,v)ψ dx. The main result of this article read as follows. Int. J. Anal. Appl. 19 (2) (2021) 231 Theorem 2.1. Assume conditions (1.2) and (1.3) are fulfilled. Then problem (1.1) has at least one solution. 3. Proof of main result In the present section, we will use the compactness method to obtain existence results from finite- dimensional approximations. Proof. Let X be a finite-dimensional subspace of V endowed with the V-norm, and X∗ its dual. Define de mappings L : X × [0, 1] −→ X∗ by (3.1) 〈L(u,v,t), (φ,ψ)〉V = ∫ Ω ( g1(t|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ Ω ( g2(t|∇u|) + 1 λ22 ) ∇v∇ψ dx − ∫ Ω f(x,tu,tv)φ dx− ∫ Ω h(x,tu,tv)ψ dx, for all (φ,ψ) ∈ X, L is well defined. Next, we divide the proof into four steps: Step 1:: In this step, we show{ (u,v) ∈ X : L(u,v,t) = 0 for some t ∈ [0, 1] } ⊂ B̄ ( 2 min(k1,k2) ‖(d1,d2)‖W ) is established. Indeed, if L(u,v,t) = 0 for some (u,v,t) ∈ X × [0, 1], then 0 = 〈L(u,v,t),u,v〉≥ min(k1,k2)‖(u,v)‖V − 2‖(d1,d2)‖W, which implies that ‖(u,v)‖V ≤ 2 min(k1,k2) ‖(d1,d2)‖W. Consequently, for any R > 2 min(k1,k2) ‖(d1,d2)‖W, we have (3.2) L(u,v,t) 6= 0 if (u,v,t) ∈ ∂BX(R) × [0, 1]. Step 2: In this step, we show L ( B̄X(R) × [0, 1] ) ⊂ B̄X ∗ ( MR + 2‖(d1,d2)‖W ) is established. If (u,v,t) ∈ B̄X(R) × [0, 1], we have |〈L(u,v,t), (φ,ψ)〉| ≤ ( max (2λ21 + 3 2λ21 , 2λ22 + 3 2λ22 ) ‖(u,v)‖V + 2‖(d1,d2)‖W ) ‖(ϕ,ψ)‖V ≤ ( max (2λ21 + 3 2λ21 , 2λ22 + 3 2λ22︸ ︷︷ ︸ M ) R + 2‖(d1,d2)‖W ) ‖(ϕ,ψ)‖V ≤ ( MR + 2‖(d1,d2)‖W ) ‖(ϕ,ψ)‖V , for all (ϕ,ψ) ∈ V, and hence (3.3) L ( B̄X(R) × [0, 1] ) ⊂ B̄X ∗ ( MR + 2‖(d1,d2)‖W ) . Int. J. Anal. Appl. 19 (2) (2021) 232 Step 3: In this step, we prove that L is a continuous mapping on B̄X(R) × [0, 1]. Let (un,vn, tn) ∈ B̄X(R)×[0, 1] converge to (u,v,t) in X×[0, 1], i.e in V×[0, 1]. Since (L(un,vn, tn)) is bounded because of (3.3), to prove that L(un,vn, tn) → L(u,v,t), it is sufficient to show that L(u,v,t) is the unique cluster point of (L(un,vn, tn)). Let l ∈ X∗ be such a cluster point, still we denote by (tn), (un) and (vn) a subsequence of (tn), (un) and (vn) respectively such that L(un,vn, tn) → l in X∗. Since (un,vn) → (u,v) in V, it follows that (un,vn) → (u,v) in W, and hence, going if necessary to a subsequence, we may assume that (un,vn) → (u,v) a.e. in Ω . On the other hand, (∂iun,∂ivn) → (∂iu,∂iv) in W, therefore (∇un,∇vn) → (∇u,∇v) a.e in Ω. This implies that g1(tn|∇vn|) → g1(t|∇v|) a.e. in Ω, g2(tn|∇un|) → g2(t|∇u|) a.e. in Ω, and hence, for any (φ,ψ) ∈ X, g1(tn|∇vn|)∇φ → g1(t|∇v|)∇φ in L2(Ω), g2(tn|∇un|)∇ψ → g2(t|∇u|)∇ψ in L2(Ω). For the last term, f(x,tnun, tnvn) → f(x,tu,tv) a.e.. Using Lebesgue dominated convergence theorem and (1.3), we arrive at f(x,tnun, tnvn) → f(x,tu,tv) in L2(Ω). Consequently ∫ Ω f(x,tnun, tnvn)φ dx → ∫ Ω f(x,tu,tv)φ dx. Similarly we have ∫ Ω h(x,tnun, tnvn)φ dx → ∫ Ω h(x,tu,tv)φ dx. We conclude that 〈L(un,vn, tn), (φ,ψ)〉V = ∫ Ω ( g1(tn|∇vn|) + 1 λ21 ) ∇un∇φ dx + ∫ Ω ( g2(tn|∇un|) + 1 λ22 ) ∇vn∇ψ dx − ∫ Ω f(x,tnun, tnvn)φ dx− ∫ Ω h(x,tnun, tnvn)ψ dx → ∫ Ω ( g1(t|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ Ω ( g2(t|∇u|) + 1 λ22 ) ∇v∇ψ dx − ∫ Ω f(x,tu,tv)φ dx− ∫ Ω h(x,tu,tv)ψ dx = 〈L(u,v,t), (ϕ,ψ)〉V . Int. J. Anal. Appl. 19 (2) (2021) 233 Thus l = L(u,v,t). All those properties allow us to apply the homotopy invariance property to (3.4) degB ( L(·, ·, 1),B(R), 0 ) = degB ( L(·, ·, 0),B(R), 0 ) . But L(u,v, 0) = 0 is equivalant to the problem (1 + 1 λ21 ) ∫ Ω ∇u∇φ dx + (1 + 1 λ22 ) ∫ Ω ∇v∇ψ dx = ∫ Ω f(x)φ dx + ∫ Ω h(x)ψ dx, for all (φ,ψ) ∈ X, whose solution is unique because of the boundedness of the set of its possible solutions. Consequently, degB ( L(·, ·, 0),B(R), 0 ) = ±1, and from (3.4) and the existence property of degree, there exists (u,v) ∈ BX(R) which satisfies (3.5) ∫ Ω ( g1(|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ Ω ( g2(|∇u|) + 1 λ22 ) ∇v∇ψ dx = ∫ Ω f(x,u,v)φ dx + ∫ Ω h(x,u,v)ψ dx, |(u,v)‖V ≤ 2 min(k1,k2) ‖(d1,d2)‖W, for all (φ,ψ) ∈ X. Step 4: We now show the passage to the limit. Consider the function bi : RN → RN defined by bi(ζi) = ( gi(ζi) + 1 λ2i ) ζi for any ζi ∈ RN and i = 1, 2. To prove the passage to the limit, we need the following lemma: Lemma 3.1. [16] Let 0 < λi ≤ 1, for any ζi,ζ′i ∈ R N such that ζi 6= ζ′i we have (bi(ζi) − bi(ζ′i))(ζi − ζ ′ i) > 0 for i = 1, 2. The proof of the above lemma can be found in [16]. Lemma 3.2. If b ∈ C(RN,RN ), b(ζ) ≤ (1 + 1 λ2 )ζ for all ζ ∈ RN and if un → u in H10 (Ω) then b(∇un) → b(∇u) in L2(Ω). Lemma (3.2) is proved by the dominated convergence theorem of Lebesgue. Now, it is well known that one can write V = ⋃ n≥1 Xn where Xn ⊂ Xn+1(n ≥ 1) and Xn has dimension n. Consequently, given any (φ,ψ) ∈ V , there exists a sequence (φn,ψn) with (φn,ψn) ∈ Xn Int. J. Anal. Appl. 19 (2) (2021) 234 which converges to (φ,ψ). On the other hand, by (3.5) applied to X = Xn, there exists, for each n ≥ 1, some (un,vn) ∈ Xn such that∫ Ω b1(∇un)∇ϕ1 dx + ∫ Ω b2(∇vn)∇ϕ2 dx = ∫ Ω f(x,un,vn)ϕ1 dx + ∫ Ω h(x,un,vn)ϕ2 dx, ‖(un,vn)‖V ≤ 2 min(k1,k2) ‖(d1,d2)‖W, for all (ϕ1,ϕ2) ∈ Xn. In particular, taking (ϕ1,ϕ2) = (φn,ψn) introduced above, (3.6) ∫ Ω b1(∇un)∇φn dx + ∫ Ω b2(∇vn)∇ψn dx = ∫ Ω f(x,un,vn)φn dx + ∫ Ω h(x,un,vn)ψn dx, ‖(un,vn)‖V ≤ 2 min(k1,k2) ‖(d1,d2)‖W, for all n ≥ 1. The estimate in (3.6) implies that, going if necessary to subsequences, we can assume that there exists (u,v) ∈ V such that (un,vn) → (u,v) weakly in V, (un,vn) → (u,v) strongly in W and (un,vn) → (u,v) a.e. in Ω. As ( b1(∇un) ) n∈N is bounded in L 2(Ω), then there exists ξ1 ∈ L2(Ω) such that b1(∇un) → ξ1 weakly in L2(Ω). Similarly, we obtain b2(∇vn) → ξ2 weakly in L2(Ω), and (∇φn,∇ψn) → (∇φ,∇ψ) strongly in W. On the other hand, as f(x,un,vn) → f(x,u,v) in L2(Ω) and h(x,un,vn) → h(x,u,v) in L2(Ω) , one can let n →∞ in (3.6) to obtain (3.7) ∫ Ω ξ1∇φ dx + ∫ Ω ξ2∇ψ dx = ∫ Ω f(x,u,v)φ dx + ∫ Ω h(x,u,v)ψ dx. It remains to show that (3.8) ∫ Ω ξ1∇φ dx = ∫ Ω b1(∇u)∇φ dx, and (3.9) ∫ Ω ξ2∇ψ dx = ∫ Ω b2(∇v)∇ψ dx. To prove the two equalities, we use the trick of Minty [16]; we begin by studying the limit of∫ Ω b1(∇un)∇un dx, and ∫ Ω b2(∇vn)∇vn dx. Int. J. Anal. Appl. 19 (2) (2021) 235 Indeed ∫ Ω b1(∇un)∇un dx = ∫ Ω f(x,un,vn)un dx → ∫ Ω f(x,u,v)u dx,∫ Ω b2(∇vn)∇vn dx = ∫ Ω h(x,un,vn)vn dx → ∫ Ω h(x,u,v)v dx, because (un,vn) → (u,v) weakly in V. But we know that (u,v) satisfies (3.7), and hence∫ Ω f(x,u,v)u dx = ∫ Ω ξ1∇u dx, and ∫ Ω h(x,u,v)v dx = ∫ Ω ξ2∇v dx. Therefore (3.10) lim n→+∞ ∫ Ω b1(∇un)∇un dx = ∫ Ω f(x,u,v)u dx = ∫ Ω ξ1∇u dx, and (3.11) lim n→+∞ ∫ Ω b2(∇vn)∇vn dx = ∫ Ω h(x,u,v)v dx = ∫ Ω ξ2∇v dx. Let (φ,ψ) ∈ V, it exists (φn,ψn)n∈N such that (φn,ψn) ∈ Xn for all n ∈ N and (φn,ψn) → (ϕ,ψ) in V when n → +∞. Thanks to Lemma 3.1, we will pass to the limit in the two terms∫ Ω b1(∇un)∇φn dx, and ∫ Ω b2(∇vn)∇ψn dx. Indeed, for the first equation 0 ≤ ∫ Ω (b1(∇un) − b1(∇φn))(∇un −∇φn) dx =∫ Ω b1(∇un)∇un dx− ∫ Ω b1(∇un)∇φn dx− ∫ Ω b1(∇φn)∇un dx + ∫ Ω b1(∇φn)∇φn dx = T1,n − T2,n − T3,n + T4,n, we saw in (3.10) that T1,n → ∫ Ω ξ1∇u dx when n →∞. We have lim n→+∞ T2,n = ∫ Ω ξ1∇φ dx. Similarly lim n→+∞ T3,n = ∫ Ω b1(∇φ)∇u dx. Int. J. Anal. Appl. 19 (2) (2021) 236 Finally, we also have lim n→+∞ T4,n = ∫ Ω b1(∇φ)∇φ dx, when n → +∞. The passage to the limit therefore gives:∫ Ω (ξ1 − b1(∇φ))(∇u−∇φ) dx ≥ 0 for all φ ∈ H10 (Ω). Similarly, we obtain∫ Ω (ξ2 − b2(∇ψ))(∇u−∇ψ) dx ≥ 0 for all ψ ∈ H10 (Ω). We now choose judicious test functions φ and ψ. We take φ = u + 1 n w, with w ∈ H10 (Ω) and n ∈ N ∗, and ψ = v + 1 n w̃, with w̃ ∈ H10 (Ω) and n ∈ N ∗. We thus obtain: − 1 n ∫ Ω ( ξ1 − b1(∇u + 1 n ∇w) ) ∇w dx ≥ 0, and − 1 n ∫ Ω ( ξ2 − b2(∇v + 1 n ∇w̃) ) ∇w̃ dx ≥ 0, then ∫ Ω ( ξ1 − b1(∇u + 1 n ∇w) ) ∇w dx ≤ 0, and ∫ Ω ( ξ2 − b2(∇v + 1 n ∇w̃) ) ∇w̃ dx ≤ 0. But u + 1 n w → u in H10 (Ω), v + 1 n w̃ → v in H10 (Ω), from Lemma 3.2, we get b1 ( ∇u + 1 n ∇w ) → b1(∇u) in L2(Ω), and b2 ( ∇v + 1 n ∇w̃ ) → b2(∇v) in L2(Ω). Passing to the limit when n → +∞, we then obtain∫ Ω (ξ1 − b1(∇u))∇w dx ≤ 0, ∀w ∈ H10 (Ω), Int. J. Anal. Appl. 19 (2) (2021) 237 and ∫ Ω (ξ2 − b2(∇v))∇w̃ dx ≤ 0, ∀w̃ ∈ H10 (Ω). By linearity (can change w into −w and w̃ into −w̃), we have∫ Ω (ξ1 − b1(∇u))∇w dx = 0, ∀w ∈ H10 (Ω), and ∫ Ω (ξ2 − b2(∇v))∇w̃ dx = 0, ∀w̃ ∈ H10 (Ω). We deduce that ∫ Ω ξ1∇w dx = ∫ Ω b1(∇u)∇w dx, ∀w ∈ H10 (Ω),∫ Ω ξ2∇w̃ dx = ∫ Ω b2(∇v)∇w̃ dx, ∀w̃ ∈ H10 (Ω). 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