Existence and Uniqueness of Weak Solutions for Novel Anisotropic Nonlinear Diffusion Equations Related to Image Analysis

This paper establishes the existence and uniqueness of weak solutions for the initial-boundary value problem of anisotropic nonlinear diffusion partial differential equations related to image processing and analysis. An implicit iterative method combined with a variational approach has been applied to construct approximate solutions for this problem. Then, under some a priori estimates and a monotonicity condition, the existence of unique weak solutions for this problem has been proven. This work has been complemented by a consistent and stable approximation scheme showing its great significance as an image restoration technique.


Introduction
In the last three decades, nonlinear diffusion equations have inspired numerous research studies in various application ranges. Perona and Malik [1] were the first to introduce such equation in image processing and analysis in the following manner: 〈c(|∇u|)∇u, n〉 � 0, on zΩ ×(0, T], u(x; 0) � u 0 (x), in Ω, where Ω is an image domain in R 2 and c is a positive decreasing function defined on R + . When it comes to processing a digital image, Perona and Malik chose the above model to preserve meaningful features such as edges while reducing irrelevant information such as noise in the homogeneous area. Nevertheless, this model, known as an isotropic nonlinear diffusion equation, handles an image feature with the same amount of blurring in all its directions. For instance, this process cannot successfully eliminate noises at edges [2]. Accordingly, it might be wise to consider the orientation of essential features by using anisotropic diffusion. Weickert [2] introduced this property by defining an orientation descriptor using the structure tensor, which is convenient to identify features such as corners and T-junctions. Besides, digital images present some structural difficulties; that is, they are discrete in space and image intensity values. Accordingly, it would be of great interest to adapt the diffusion to digital images' structure by considering vertical, horizontal, and diagonal differential operators. Due to these reasons, we modeled and developed anisotropic nonlinear diffusion equations using a novel diffusion tensor.
Various tools can be used to examine the existence of solutions for nonlinear partial differential equations (PDEs), such as variational techniques, monotonicity method, fixedpoint theorems, iterative methods, and truncation techniques; for more detailed information, we refer to [3][4][5][6][7] and the references therein. ese PDEs have been motivated by various applications such as image restoration and reconstruction (see, for example, [3,4,[8][9][10][11]). Moreover, the image processing of the brain allows the localization of epileptogenic foci for the patient. A noninvasive method has been examined numerically as an inverse problem in [12].
Under some challenging conditions, the existence and uniqueness of weak solutions for the Perona and Malik model have been investigated in the bounded variation space BV(Ω) [3,13]. In some other functional frameworks, Wang and Zhou have thoroughly studied in [4] and proved the existence and uniqueness of weak solutions in the Orlicz space LlogL(Ω) using a new diffusion function c(s) � ((s + (s + 1)log(s + 1))/(s(s + 1))) for all s ≥ 0.
In this paper, we suppose that Ω is an open-bounded domain of R 2 with Lipschitz boundary zΩ, and T is a positive number. We denote where (e 1 , e 2 ) is the canonical basis of R 2 . We consider the following anisotropic nonlinear parabolic initial-boundary value problem: where D ∇u , the diffusion tensor, is a real symmetric positive definite matrix of R 2×2 defined as follows: and g: R + ⟶ R is a C 1 positive decreasing function. en, we can define ϕ: R + ⟶ R as a C 2 function such that satisfying 2 Journal of Mathematics To construct an adaptive diffusion tensor, the function g is approximated numerically by a cubic Hermite spline [14] that interpolates numeric data specified at 0 � k 0 < k 1 < · · · < k m with m ∈ N * : where p · and v · are the coefficients used to define the position and the velocity vector at a specific point, k i are the threshold parameters, P j,cd is the family of the basis functions composed of polynomials of degree 3 used on the interval [c, d[ such that And we may consider

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From the definition of ϕ, we can deduce where C i and C m are constants determined by the continuity of ϕ at each k i . In this case, the values of the coefficients A k i k i+1 ,j are determined experimentally provided that ϕ satisfying the above conditions on [0, k m [. Besides, we may introduce some sufficient conditions on k m and A k m,. that guarantee the properties of ϕ on [k m , ∞[: Anisotropic diffusion model (3) allows strong directional smoothing within the areas where |u x 1 |, |u x 2 |, |u x 12 |, or |u x − 12 | is small and prevents blurring boundaries, contours, or corners that separate neighboring areas, where one or a combination of these differential operators has significant value.
Moreover, the matrix D ∇u has two eigenvalues λ +/− : with θ +/− are the corresponding eigenvectors. We can then expand the first equation of (3) into Accordingly, it is clear from the expression of λ +/− that λ + ≥ λ − > 0, which means that the diffusion towards θ + is privileged over θ − . In fact, the difference 2 indicates the isotropic diffusion for zero value and anisotropic diffusion for positive values. Henceforth, we will assume that the initial value satisfies and we will introduce the following Orlicz space: Next, we define weak solutions for problem (3) on Q T ≔ Ω × (0, T] with T > 0: Definition 1. A function u: Q T ⟶ R is a weak solution for problem (3) if the following conditions are satisfied: Now, we state our main theorem.
Inspired by [4], this paper will investigate the existence and uniqueness of weak solutions for problem (3) according to the following steps: (i) First, we approximate nonlinear evolution problem (3) by nonlinear elliptic problems using an implicit iterative method (discretization in time-variable only), and then we prove the existence of a unique weak solution for each elliptic problem adopting a variational approach. ese solutions constitute approximate solutions for problem (3).
(ii) Next, we show the uniqueness of solutions for initial-boundary value problem (3) using the monotonicity of the vector field D ∇ ∇.: u ∈ R 2 ⟶ D ∇u ∇u ∈ R 2 . (iii) Finally, passing to limits in some a priori energy estimates and using the monotonicity condition (17), we demonstrate the existence of weak solutions for problem (3).

Preliminaries
In this section, we state some useful lemmas that will be used later in the proofs.
Assume Ω ⊂ R 2 is bounded, and let u i ∞ i�1 be a sequence of functions in L 1 (Ω) satisfying Suppose also en, there exist a subsequence u i j ∞ j�1 and u ∈ L 1 (Ω) such that Journal of Mathematics with u ∈ LlogL k m (Ω).
Proof. Given M > 0, we may find an l ≥ k m such that which implies that On the other hand, there exists a positive constant C such that which is true for all i and arbitrary ε > 0. It follows then that It remains to prove that u ∈ L log L k m (Ω).
We know that the function f(s) � s log(s) for s ≥ 1 is increasing and convex, and then the function f(|s|) is also convex for all s ≥ 1. erefore, we obtain Integrating the above inequality over (32) and passing to limits as j ⟶ ∞, we get en, passing to limits as N ⟶ ∞, we deduce It follows then u ∈ LlogL k m (Ω). is finishes the proof.

Approximate Solutions
In this section, we will discretize the time-variable interval [0, T] to get approximate solutions for problem (3). We denote h � (T/N) with N ∈ N * , and we designate by u n an approximate solution at time nh. We define gradually from n � 1, 2, . . . , N the following elliptic problems: To solve these equations step by step, we only need to prove the existence and uniqueness of weak solutions of the following elliptic problems: where h > 0 and u 0 ∈ L 2 (Ω).
And when φ is a constant function, we obtain In order to prove the existence and uniqueness of weak solutions for problem (36), we consider the variational problem where and when u ∈ U, the functional E is defined as It is easy to prove that (36) is the Euler-Lagrange equations of the functional E [16].

Theorem 2. Problem (36) has a unique weak solution.
Proof. Since then we can construct a minimizing sequence u q Besides, It follows then On the other hand, given ε 0 > 0, we may find l 0 � k m such that with C � (ε 0 + (1/A k m ,2 )) > 0 and i � 1, 2. It follows then that for i � 1, 2, erefore, thanks to Lemma 4 and the weak compactness of L 2 (Ω), we can find a subsequence u q j ∞ j�1 of u q ∞ q�1 and a function u 1 ∈ L 2 (Ω) ∩ W 1,1 (Ω) such that and for i � 1, 2, erefore, we have and following the reasoning in the proof of Lemma 4, it is easy to show that for any a ∈ x 1 , x 2 , x 12 , x − 12 and for a fixed ϵ > 0, there exists l ≥ k m such that Journal of Mathematics Similarly, since ϕ is increasing and convex in [0, l], then we can prove that erefore, we obtain from (52) and (53) that us, by letting ε ⟶ 0, we get for any a ∈ x 1 , x 2 , x 12 , x − 12 . It follows then that which signifies that u 1 ∈ U is a minimizer of the energy functional E(u), i.e., Furthermore, for all φ ∈ C 1 (Ω) and t ∈ R, we have Hence, we have ρ ′ (0) � 0, which means Because of (50), we get We conclude then that u 1 is a weak solution for problem (36). Now, assume that there is another weak solution u of (36). en, for every φ ∈ C 1 (Ω), we have which leads to en, if we choose φ � u − u 1 as a test function in (62), we get anks to Lemma 2, we deduce that for every h � (T/N).

Existence and Uniqueness of Weak Solutions
Proof. of eorem 1. In the beginning, we establish the uniqueness of solutions for problem (3). For this purpose, we suppose there exist two weak solutions u and v for problem (3). en, we obtain the following: By multiplying the first equation of the above problem by (u − v) and integrating over Ω and [0, t], we get for every t ∈ (0, T]. Since the second term of the above equation is nonnegative (thanks to Lemma 2), it follows then u � v a.e. in Q T . Let us now find our weak solution for problem (3). We intend to send h to zero and show that a subsequence of our solutions u h of the approximate problems (35) converges to a weak solution for problem (3). To this end, we need to find some a priori estimates.

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Besides, as in (46), for |z x 1 u h |, |z x 2 u h | ≥ k m , we may find a positive constant C such that By Lemma 4, we can find a subsequence of u h (for simplicity, we also denote it by u h ) such that [17] u h ⇀u, weakly * in L ∞ 0, T; L 2 (Ω) , So, it remains to prove that u is just a weak solution for problem (3). Let us now denote ξ h � D ∇u h ∇u h . We will show that ξ h is bounded in [L 2 (Q T )] 2 , so we may find a subsequence of ξ h that converges weakly in [L 2 (Q T )] 2 to a particular vector-valued function. en, we will prove that this vector-valued function is equal almost everywhere to D ∇u ∇u in Q T through monotonicity condition (17).
From the expression of D ∇u h , we can derive the following: Given ε 1 , ε 2 > 0, we may find l 1 � l 2 � k m such that for all s ≥ k m with M � (ε 1 + A k m ,2 ). us, we can distinguish two cases: 2 , which means that we can find a subsequence of ξ h (denote it also by ξ h ) and a function ξ ∈ [L 2 (Q T )] 2 such that Since s↦s exp(s)(s ≥ 0) is increasing and convex, then as in the proof of Lemma 4, we deduce that en, by using Lemma 1, we get It follows then ξ · ∇u ∈ L 1 (Q T ). Next, we will show that ξ � D ∇u ∇u a.e. in Q T .

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On the other hand, we let v ∈ L 1 (Q T ) with for i � 1, 2. We sum up inequalities (69): We have from Lemma 2 that en, we obtain Letting h ⟶ 0 and noting that we obtain By using φ � u in (89), we get Combining (95) with (96), we have Now, setting v � u + λw for any λ > 0, w ∈ W 1,2 (Q T ), we derive from the above inequality that By letting λ ⟶ 0 and using Lebesgue's dominated convergence theorem, we obtain for every ψ ∈ [L 2 (Ω)] 2 . It follows then ξ � D ∇u ∇u, a.e. in Q T .
erefore, we conclude from (89) that for any φ ∈ C 1 (Q T ) with φ(., T) � 0. Finally, we need to prove that u ∈ C([0, T], L 2 (Ω)). Since It follows then that u ∈ C(0, T; H − 1 (Ω)). Besides, for every h > 0, let v h (x, t) � u(x, t + h) be the weak solution for problem (3) satisfying For each t 0 ∈ [0, T], we may choose w h as a test function in the first equation for problem (104) over [0, t 0 ]: Because of Lemma 2, we deduce Now, in order to prove that u ∈ C([0, T], L 2 (Ω)), we need to prove We suppose that (107) is not true. en, there exist a positive number δ and a sequence h i with h i ⟶ 0 as i ⟶ ∞ such that From estimate (72), we have en, from (108), we get From (109), we conclude that u(x, h i ) is a bounded sequence in L 2 (Ω). en, we may find a subsequence (denote it also by u(x, h i ) ) such that there exists a function u 0 ∈ L 2 (Ω) such that u x, h i ⇀u 0 , weakly in L 2 (Ω). (111) Since u ∈ C(0, T; H − 1 (Ω)), it follows that erefore, we must have u 0 � u 0 , and since u ∈ C(0, T; H − 1 (Ω)), it follows that which is contradictory with (110). erefore, we conclude that (107) is true and u ∈ C([0, T], L 2 (Ω)). is completes the proof of eorem 1.

Consistent and Stable Symmetric Finite Difference
Approximation. In this section, we provide a consistent and stable discretization scheme using symmetric finite difference approximation: at time t n � nδ t , n ≥ 0, and the mesh points x i � iδ, y j � jδ(0 ≤ i ≤ N + 1 and 0 ≤ j ≤ M + 1), and we denote by u n i,j the finite difference approximation of u(x i , y j ; t n ). e time-space derivatives are discretized in the following manner: and n ≥ 0, with the initial condition u 0 i,j and the discrete Neumann boundary condition: A unique sequence (u n ) n∈N is produced when using filter (116) on a particular initial image u 0 [2]. Besides, due to the continuity of the function g, the sequence u n depends continuously on u 0 for every finite n. Furthermore, equation (116) satisfies the following maximum-minimum principle, which describes a stability condition for the discrete scheme. Theorem 3. Discrete extremum principle [1,2].
For an iteration step satisfying scheme (116) satisfies for all 1 ≤ i ≤ N, 1 ≤ j ≤ M, and n ∈ N.

Experimental Results.
is section will show the performance of proposed diffusion filter (116) in the image denoising process, under the boundary and initial conditions (117) while respecting the requirements concerning ϕ (Section 1), and δ t (118). We will use the Peak Signal-to-Noise Ratio (PSNR that is a positive value) [18] and the Structural SIMilarity Index (SSIM that lies in (0, 1)) [19] to evaluate the quality of the restored images. e best results for the denoising process are equivalent to the higher value of these metrics.
For comparative purposes, we will examine the proposed diffusion function with another one that has the same properties using the same filter (116). erefore, we will use the following diffusion functions: (i) e proposed diffusion function (m � 1 for instance): (ii) e Wang and Zhou diffusion function (WZ) [4]: Additionally, we will consider real test images Figure 1 and evaluate our model's performance on these images, which will be corrupted with different levels of Gaussian white noises with zero mean and variance σ 2 . Table 1 shows the quantitative results on real images, corrupted with various Gaussian noises, filtered by discrete model (116) using proposed diffusion function (120) and the one proposed by WZ (121). ese results are obtained using the optimal parameters determined experimentally, as in Table 2 for each diffusion function.
It can be seen from Table 1 and Figure 2 that the proposed model shows remarkable results against the WZ model. From a visual comparison, Figure 2 shows that the restored images using the proposed diffusion function have considerable noise removal and preserve the image essential features better than the restored images by the WZ diffusion function. Besides, compared with the WZ diffusion function,  [20]. Patient50: coronal T2 of a 50-year-old female patient [21]. Patient55: axial T2 of a 55-year-old patient [22]. (a) Patient30, (b) Patient50, and (c) Patient55.   the results from Table 1 prove that the suggested approach has higher values in SSIM, whereas the WZ model shows significant results in PSNR while σ 2 -value increases.

Conclusion
is paper principally investigates the class of anisotropic diffusion partial differential equations related to image processing and analysis. e existence and uniqueness of weak solutions for this problem have been proven under sufficient conditions satisfied by ϕ. A consistent and stable numerical approximation has been applied, and a discrete nonlinear filter has been tested and revealed its efficiency in the image restoration field.

Data Availability
No data were used to support this study.