Existence and Uniqueness of Weak Solutions for a New Class of Convex Optimization Problems Related to Image Analysis

)is paper proposes a new anisotropic diffusion model in image restoration that is understood from a variational optimization of an energy functional. Initially, a family of new diffusion functions based on cubic Hermite spline is provided for optimal image denoising. After that, the existence and uniqueness of weak solutions for the corresponding Euler–Lagrange equation are proven in an appropriate functional space, and a consistent and stable numerical model is also shown. We complement this work by illustrating some experiments on different actual brain Magnetic Resonance Imaging (MRI) scans, showing the proposed model’s impressive results.


Introduction
During the last few decades, the digital image has been significantly utilized as a noninvasive medical diagnosis or treatment tool. For instance, a different method has been developed in the electroencephalogram (EEG) (see [1]). Since a digital image contains relevant information that should be carefully extracted from it, many engineers and mathematicians have established numerous techniques and theories [2][3][4] to collect quantitative or qualitative data. At that point, by analyzing these reliable data and understanding its contents, one can make the right decisions at the diagnosis time.
Along with the meaningful details, digital images carry false information as noise and other undesirable artifacts due to various reasons. is erroneous information can occur during image formation, transmission, or recording processes. erefore, one needs good models to identify and eliminate these noises while preserving relevant information and structures.
Image denoising is a necessary preprocessing step for many applications that rely on image quality, such as image segmentation and pattern recognition. Hence, how to restore a degraded image to its original form? Many approaches have been developed to process images and showed great interest in using the variational approach [5][6][7][8][9][10][11][12][13][14][15], which attempts to establish mathematical models strictly related to the physical world by defining a diffusion equation from a particular optimization problem, which generally has the following form: where Ω ⊂ R 2 is an open bounded domain, u 0 and u are the observed and the reconstructed images defined as functions of Ω ⊂ R 2 ⟶ R that associate each pixel x ∈ Ω to the gray level u(x) or u 0 (x), φ: R + ⟶ R is a nonnegative increasing function with φ(0) � 0, and ρ ∈ R + is a weighting parameter that enables to adjust the influence of the data term in the regularizing term.
One common approach to solve the problem (1) is by seeking the steady-state solution of the following PDE, which corresponds to the Euler-Lagrange equation of the energy functional F(u): Perona and Malik (Perona-Malik) [16] were the first to propose such a model in image processing, which corresponds to nonconvex energy functional. Unfortunately, this property leads to multiple solutions with staircase effects observed in practice [6,17]. In [8], Charbonnier et al. suggested a strictly convex energy functional to circumvent the Perona-Malik model's ill-posedness. Besides, many variational approaches have also been developed during the last thirty years. e most famous one was the total variation denoising suggested by Rudin et al. in their seminal paper [5]. Several improved nonlinear diffusion models for image denoising derived from Perona-Malik or TV models have been proposed in the last twenty years; for more detailed information, we refer to [2,[10][11][12][13][14][15]18] and the references therein.
Nevertheless, because of the above model's isotropy, the diffusion equation (2) is controlled by the function s ⟶ (φ′(s)/s) and produces the same amount of blurring in all its directions.
is means that the process cannot successfully eliminate noises at the edges. us, it would be important to consider smoothing along edges by adopting anisotropic diffusion [19].
Furthermore, the type (1) model uses the gradient magnitude as a local descriptor operator for the image edge detector. However, digital images present some difficulties in their discrete structure; they are discrete in space and discrete in intensity value. Consequently, one may need to adapt to the digital image structure by considering differential operators that respond to vertical, horizontal, and diagonal edges and using a consistent approximation. erefore, motivated by the above reasoning and inspired by Weickert's anisotropic idea [19], we conceived new convex optimization problems that lead to anisotropic diffusion equations with a novel matrix diffusion tensor. erefore, this paper provides a new variational PDE model based on directional edge detectors in Section 2. e construction of a new diffusion function using cubic Hermite spline and the existence and uniqueness of weak solutions are illustrated in Section 3. Next, Section 4 implements a consistent explicit symmetric finite difference approximation that involves the 3 × 3 neighborhoods at each point and shows the experiments on different real images. We terminate this work with a conclusion in Section 5.

Novel Anisotropic Diffusion Model for Image Restoration
Let us now consider the rectangular domain Ω ≔ (0, a) × (0, b) ⊂ R 2 with Γ ≔ zΩ being its boundary. is approach's general idea is to search for a solution among the minima of some functional energy. e proposed functional energy has a regularizing term that depends on the combination of a function φ and local structures of the image expressed by the four directional derivatives: where (e 1 , e 2 ) is the canonical basis of R 2 . Hence, our problem consists of solving the following energy minimization problem: such that under the following assumptions: which leads to 2 Journal of Mathematics is is equivalent to Let g: R + ⟶ R be a positive function such that g(s) � (φ ′ (s)/s). We may reformulate equation (9) and present it as follows.

Proposition 1.
e Euler-Lagrange equation (9) is equivalent to where D ∇u is a real symmetric positive definite matrix of R 2×2 defined as follows: Proof. First, it is clear that On the other hand, we have Journal of Mathematics 12 ∇u · e 1 + e 2 e 1 + e 2 · e 1 + e 2 e 1 + e 2 Besides, the matrix D ∇u has two eigenvalues λ ± as follows: Since g > 0 and then we deduce that the matrix D ∇u is symmetric positive definite, which completes the proof. □ By applying the steepest descent method, the Euler-Lagrange equation (10) can be solved considering the steady state of the following evolution equation: In [20], the existence and uniqueness of weak solutions for problem (16) were proven if the weighting parameter ρ � 0.

Description of the Proposed Model.
Our model's main objective is to allow strong directional smoothing within the areas where |u x 1 |, |u x 2 |, |u x 12 |, or |u x − 12 | is small and prevent blurring boundaries, contours, or corners that separate neighboring areas, where one or a combination of these differential operators has substantial value. Some sufficient assumptions have to be made to ensure that our nonlinear diffusion meets acceptable behavior along these lines. erefore, we can assume, as exhibited by Perona-Malik [16], some minimal hypotheses on g: Besides, due to the decreasing positivity of the function g, it is evident that our model's behavior (16) encourages smoothing along edges in the e 1 , e 2 , (e 1 + e 2 ), or (− e 1 + e 2 ) directions and ceases across them.
Moreover, we may provide another investigation analysis of our model by examining the eigenvalues-eigenvectors of the matrix D ∇u . e matrix D ∇u has two eigenvalues λ ± with θ ± being the corresponding eigenvectors: where provided that |u x 12 | ≠ |u x − 12 |. We can then expand the first equation of (16) into Accordingly, the diffusion caused by (20) is measured by the λ ± values and oriented toward θ ± . Specifically, it is clear from the expression of λ ± that λ + ≥ λ − > 0, which means that the diffusion toward θ + is privileged over θ − . Furthermore, we can deduce the following results: (i) In flat areas, we have en, the diffusion is isotropic and linear, and the smoothing effect is the same in all directions.
For instance, we can assume that u x 1 � u x 2 and |u x 1 | ≫ 0. en, we obtain |u x − 12 | � 0 and |u x 12 | ≫ 0, which implies that Consequently, the diffusion process is anisotropic and oriented along the (− e 1 + e 2 ) direction.
In fact, the difference 2 gives insights into our diffusion model's anisotropic property. In other words, it indicates the isotropic diffusion for the zero value and the anisotropic diffusion for larger values.

New Adaptive Diffusion Function Using Hermite Spline.
For effective image denoising and better control of the diffusion via the process (16), we will approximate numerically the unknown diffusion function g that meets the requirements (H 1 + H 2 ) by using Hermite spline [21]. erefore, we can use cubic Hermite spline to interpolate Journal of Mathematics numeric data specified at 0, k, and h ≥ 1 (0 < k < h) to obtain a function that meets the requirements (H 1 + H 2 ). To this end, we propose a function g as follows: where p · and v · are the coefficients used to define the position and the velocity vector at a specific point, k and h are two threshold parameters, and P i,cd is a family of the basis functions composed of polynomials of degree 3 used on the interval [c, d[ such that And, we may consider Hence, we can reformulate the expression of g as follows: By considering the continuity of φ at k and h, and using φ(0) � 0, it follows then where Now, we establish a useful growth condition for the function φ.
where A > 0 and B ∈ R.

Existence and Uniqueness.
In this section, we investigate the existence and uniqueness of weak solutions of the Euler-Lagrange equation associated with the energy functional E(u) defined in (5). For ρ > 0 and u 0 ∈ L 2 (Ω), we consider First, we introduce a new functional space LlogL h (Ω): (32) Next, we define a weak solution for problem (31).
And when v is a constant function, we obtain Before stating our main theorem, we need first to introduce a useful lemma.
Lemma 1 (uniform integrability and weak convergence [22]). Assume that Ω is bounded and let u m ∞ m�1 be a sequence of functions in L 1 (Ω) satisfying Suppose also Then, there exist a subsequence u m j ∞ j�1 and u ∈ L 1 (Ω) such that u m j ⇀u weakly in L 1 (Ω). (37) Now, we state our main theorem.

Theorem 1. ere exists a unique weak solution for problem (31).
Proof. We consider the variational problem where and the functional E as defined in (5). It is obvious that u 0,Ω ∈ U knowing that u 0,Ω � (1/|Ω|) Ω u 0 dx. Since then we can construct a minimizing sequence u m Besides, It follows then On the other hand, from Proposition 2, we may find α > 0 and β ∈ R such that (44) en, we get Moreover, we have en, for i � 1, 2 there exist l ≥ h ≥ 1 and positive constants C and C 1 (independent of l ) such that

(47)
On the other hand, we know that en, given ϵ > 0, let η ϵ � C 1 /ϵ and choose l ≥ h ≥ 1 such that for all s ≥ l, we have φ(s) > η ϵ s. Hence, for i � 1, 2, we obtain and this is true for all m and arbitrary ϵ > 0. It follows then that, for i � 1, 2, we have From (43) (51) Additionally, knowing that the function f(s) � s log(s) for s ≥ 1 is increasing and convex, the function f(|s|) is also convex for all s ≥ 1. erefore, we obtain for i � 1, 2 Integrating the above inequality over Since f ′ (|z x i u|)χ h≤|z x i u| ≤ M ∈ L ∞ (Ω) and by letting j ⟶ ∞, we get It follows then z x i u ∈ LlogL h (Ω). Besides, we havet And by following the same reasoning as set out above, we know that the function φ(s) for s ≥ 0 is increasing and convex. en, we can easily deduce that erefore, by combining (57) with (58), we conclude which signifies that u ∈ U is a minimizer of the energy functional E(u), i.e., Furthermore, for all v ∈ C 1 (Ω) and for all t ∈ R, we have Hence, we have r ′ (0) � 0, which means Because of (52), we have We conclude then that u is a weak solution for problem (31). Now, assuming that there is another minimizer u of E and using the fact that E is strictly convex (5) and (6), we have a contradiction. us, there is only one minimizer, which completes the proof.

Consistency and Stability of Finite Difference Approximation.
For a consistent and stable discretization scheme, one can use the following accurate finite difference scheme approximation: at time t n � nδ t , n ≥ 0, and mesh points x i � iδ, y j � jδ (0 ≤ i ≤ N + 1 and 0 ≤ j ≤ M + 1), we denote by u n i,j the finite difference approximation of u(x i , y j ; t n ) and exhibit the time-space derivatives discretization as follows: By assuming δ � 1 and denoting then we may approximate the solution for problem (16) by the above scheme to obtain the following discrete diffusion filter:

Journal of Mathematics
where u 0 i,j is the degraded image, 1 ≤ i ≤ N, 1 ≤ j ≤ M, and n ≥ 0. Furthermore, we use the discrete Neumann boundary condition: en, using the filter (67) on every initial image u 0 yields a unique sequence (u n ) n∈N [19]. Besides, due to the continuity of the function g, for every finite n, u n depends continuously on u 0 .
us, under a specific condition, equation (67) satisfies the following maximum-minimum principle, which describes a stability property for the discrete scheme.

Experimental Procedures.
is section is devoted to comparing our model (16), as an image denoising algorithm, with the ones proposed by Wang and Zhou (Wang-Zhou) [10] and Maiseli [15]. All the experiments are conducted under Windows 10 and GNU Octave version 6.2.0, running on a laptop with an Intel ® Core TM i7-10510U (8 MB cache, 4 Core, up to 4.9 GHz), 16 GB memory (LPDDR3, Dualchannel, 2133 MHz), and 512 GB storage (PCIe, SSD, 3 × 4). e experiments were done on three actual MRI scans ( Figure 1) affected by different σ 2 -values of zero-mean white Gaussian noise and restored using our filter (67) with the boundary-initial conditions (68), provided that the diffusion function (23) satisfies the assumptions (H 1 + H 2 ).
We also used the discrete diffusion filter as described in [15] for Wang-Zhou and Maiseli models with the following diffusion functions: Wang-Zhou [10] diffusion function: Maiseli [15] diffusion function is a combination between the Perona-Malik diffusion function [16] and the Charbonnier diffusion function [8]: where k 1 and k 2 are positive constants. Besides, in order to evaluate the quality of the restored images from different image denoising methods, we used two image quality metrics: (i) Peak signal-to-noise ratio (PSNR) [26]: where I is the original or the uncorrupted image and u is the distorted or the restored image. PSNR is one of the oldest image quality metrics evaluating an image's signal strength relative to noise, and it is always positive. We evaluate the PSNR metric by using the Octave built-in function "psnr." However, due to its limitations and its failure in some circumstances as an adequate measure of visual quality [27], we used another metric that significantly correlates with the human visual perception. (ii) Structural similarity index (SSIM) [28]: where c 1 and c 2 are tuning parameters. μ u , σ 2 u , and σ uI stand for the mean, variance, and covariance, respectively. It is a method for measuring the similarity between a degraded image and a perfect one, and it is bounded between zero and one. For a good similarity between the original and the restored images, we need higher values of SSIM-index. In all experiments, we estimate the SSIM-index using "ssim.m" with automatic downsampling, downloaded from the website: https://ece.uwaterloo.ca/ z70wang/research/ssim/.
In all experiments and to prioritize SSIM-index over PSNR, we used a combined metric to quantify the quality of the restored images. e iterations are stopped when the IQM value reaches the maximum. e image denoising process using the proposed model (67) can be implemented as shown in Algorithm 1. First, we consider I as one of the three original brain MRI scans ( Figure 1) and generate in it a Gaussian white noise with zero-mean and σ 2 -value, using the Octave built-in function "imnoise." Next, the initial value u 0 is set to be the noisy     Table 2, when it comes to the SSIM-index, the proposed method reveals impressive results against the Wang-Zhou and Maiseli methods. Additionally, our method converges more rapidly to the solution than the others with the less iteration number.
From a visual comparison, Figure 2 shows that the denoising process using our diffusion function removes noises more efficiently and preserves essential image features. Besides, it can be seen from different images that all the restored images via Wang-Zhou and Maiseli methods are more affected by blocky artifacts than the proposed one, despite the higher values in PSNR. is can be attributed to the fact that the Wang-Zhou and Maiseli methods have lower SSIM-indices.

Conclusion
is paper presented a new anisotropic diffusion model for image restoration, eliminating the corruption caused by white Gaussian noise. e existence and uniqueness of weak solutions of functions in Orlicz-Sobolev space that minimize the energy functional E have been proven, provided that the functions φ and g satisfy some specific conditions (H 1 + H 2 ). Besides, we have established a consistent and stable numerical model for the denoising process and used the cubic Hermite spline to approximate the best possible diffusion function that revealed its efficiency regarding the optimal image denoising via (67). We have also proved that our method provides better results compared to the Wang-Zhou and Maiseli methods. e next stage of the research attempts to prove the existence and uniqueness of weak solutions for the evolution problem (16) and use different numerical methods, such as finite element and mixed finite element methods.

Data Availability
No data were used to support this study.