Abstract
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.
1. 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:where is an image domain in and is a positive decreasing function defined on .
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, fixed-point theorems, iterative methods, and truncation techniques; for more detailed information, we refer to [3–7] and the references therein. These PDEs have been motivated by various applications such as image restoration and reconstruction (see, for example, [3, 4, 8–11]). 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 [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 using a new diffusion function for all .
In this paper, we suppose that is an open-bounded domain of with Lipschitz boundary , and is a positive number. We denotewhere is the canonical basis of . We consider the following anisotropic nonlinear parabolic initial-boundary value problem:where , the diffusion tensor, is a real symmetric positive definite matrix of defined as follows:and is a positive decreasing function. Then, we can define as a function such thatsatisfying
To construct an adaptive diffusion tensor, the function is approximated numerically by a cubic Hermite spline [14] that interpolates numeric data specified at with :where and are the coefficients used to define the position and the velocity vector at a specific point, are the threshold parameters, is the family of the basis functions composed of polynomials of degree 3 used on the interval such that
And we may consider
From the definition of , we can deducewhere and are constants determined by the continuity of at each . In this case, the values of the coefficients are determined experimentally provided that satisfying the above conditions on . Besides, we may introduce some sufficient conditions on and that guarantee the properties of on :
Anisotropic diffusion model (3) allows strong directional smoothing within the areas where , , , or 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 has two eigenvalues :with are the corresponding eigenvectors. We can then expand the first equation of (3) into
Accordingly, it is clear from the expression of that , which means that the diffusion towards is privileged over . In fact, the difference indicates the isotropic diffusion for zero value and anisotropic diffusion for positive values.
Henceforth, we will assume that the initial value satisfiesand we will introduce the following Orlicz space:
Next, we define weak solutions for problem (3) on with :
Definition 1. A function is a weak solution for problem (3) if the following conditions are satisfied:(i) with for .(ii)For any with , we haveNow, we state our main theorem.
Theorem 1. Under assumption (14), there exists a unique weak solution for initial-boundary value problem (3).
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. These 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 .(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).
2. Preliminaries
In this section, we state some useful lemmas that will be used later in the proofs.
Lemma 1. For all and , we have .
Proof. If , then .
If , then , which means .
Lemma 2. Suppose is a convex function. Then, for all , we have
Proof. For each , we put . Then, we haveSince , then we obtainWe conclude thenwhich completes the proof.
Lemma 3. Uniform integrability and weak convergence [15].
Assume is bounded, and let be a sequence of functions in satisfyingSuppose alsoThen, there exist a subsequence and such that
Lemma 4. Assume is bounded, and let be a sequence of functions in such thatThen, there exist a subsequence and such thatwith .
Proof. Given , we may find an such that for all . Consequently,which implies thatOn the other hand, there exists a positive constant such thatwhich is true for all and arbitrary . It follows then thatThen, from Lemma 3, there exist a subsequence of and a function such thatIt remains to prove that .
We know that the function for is increasing and convex, and then the function is also convex for all . Therefore, we obtainIntegrating the above inequality over with , we haveSince and passing to limits as , we getThen, passing to limits as , we deduceIt follows then . This finishes the proof.
3. Approximate Solutions
In this section, we will discretize the time-variable interval to get approximate solutions for problem (3). We denote with , and we designate by an approximate solution at time . We define gradually from 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 and .
Definition 2. A function with for is called a weak solution for problem (36); if for any , we haveAnd when is a constant function, we obtainIn order to prove the existence and uniqueness of weak solutions for problem (36), we consider the variational problemwhereand when , the functional is defined asIt is easy to prove that (36) is the Euler–Lagrange equations of the functional [16].
Theorem 2. Problem (36) has a unique weak solution.
Proof. Sincethen we can construct a minimizing sequence in such that andBesides,It follows thenOn the other hand, given , we may find such thatwith and . It follows then that for ,Therefore, thanks to Lemma 4 and the weak compactness of , we can find a subsequence of and a function such thatand for ,Therefore, we haveand following the reasoning in the proof of Lemma 4, it is easy to show that for any and for a fixed , there exists such thatSimilarly, since is increasing and convex in , then we can prove thatTherefore, we obtain from (52) and (53) thatThus, by letting , we getfor any . It follows then thatwhich signifies that is a minimizer of the energy functional , i.e.,Furthermore, for all and , we have with . Then, whereHence, we have , which meansBecause of (50), we getWe conclude then that is a weak solution for problem (36).
Now, assume that there is another weak solution of (36). Then, for every , we havewhich leads toThen, if we choose as a test function in (62), we getThanks to Lemma 2, we deduce thatTherefore, a.e. in .
In conclusion, we have shown that there exists a unique weak solution satisfying (35) for every . Consequently, we define an approximate solution for problem (3) asfor every .
4. Existence and Uniqueness of Weak Solutions
Proof. of Theorem 1. In the beginning, we establish the uniqueness of solutions for problem (3). For this purpose, we suppose there exist two weak solutions and for problem (3). Then, we obtain the following:By multiplying the first equation of the above problem by and integrating over and , we getfor every . Since the second term of the above equation is nonnegative (thanks to Lemma 2), it follows then a.e. in .
Let us now find our weak solution for problem (3). We intend to send to zero and show that a subsequence of our solutions of the approximate problems (35) converges to a weak solution for problem (3). To this end, we need to find some a priori estimates.
It follows from (35) that for every ,Then, by taking as a test function in (68) and using , we getFor each , we can find such that . Then, by adding all the inequalities (69) from to , we getThen, by definition of , we obtain for thatSince is a symmetric positive definite matrix, we have alsoTherefore, after taking the supremum over , we deduce thatRecalling that for all , then we can derive the following:Besides, as in (46), for , we may find a positive constant such thatThus, we concludeBy Lemma 4, we can find a subsequence of (for simplicity, we also denote it by ) such that [17]withSo, it remains to prove that is just a weak solution for problem (3). Let us now denote . We will show that is bounded in , so we may find a subsequence of that converges weakly in to a particular vector-valued function. Then, we will prove that this vector-valued function is equal almost everywhere to in through monotonicity condition (17).
From the expression of , we can derive the following:Given , we may find such thatfor all with . Thus, we can distinguish two cases:(i)If then .(ii)If thenThen, is bounded in , which means that we can find a subsequence of (denote it also by ) and a function such thatSince is increasing and convex, then as in the proof of Lemma 4, we deduce thatThen, by using Lemma 1, we getIt follows then . Next, we will show that a.e. in .
For each with , we take as a test function in (35):with . By summing from 1 to , we obtainFrom the definition of (65), we haveTherefore,Letting tend to zero, we getOn the other hand, we let withfor . We sum up inequalities (69):We have from Lemma 2 thatThen, we obtainLetting and noting thatwe obtainBy using in (89), we getCombining (95) with (96), we haveNow, setting for any , , we derive from the above inequality thatBy letting and using Lebesgue’s dominated convergence theorem, we obtainfor every . It follows thenTherefore, we conclude from (89) thatfor any with . Finally, we need to prove that . If we choose in (89), we obtainSince , we conclude that where is the dual space of . SinceIt follows then that . Besides, for every , let be the weak solution for problem (3) satisfying . Then, satisfiesFor each , we may choose as a test function in the first equation for problem (104) over :Because of Lemma 2, we deduceNow, in order to prove that , we need to proveWe suppose that (107) is not true. Then, there exist a positive number and a sequence with as such thatFrom estimate (72), we haveThen, from (108), we getFrom (109), we conclude that is a bounded sequence in . Then, we may find a subsequence (denote it also by ) such that there exists a function such thatSince , it follows thatTherefore, we must have , and since , it follows thatwhich is contradictory with (110). Therefore, we conclude that (107) is true and . This completes the proof of Theorem 1.
5. Numerical Implementation and Experimental Results
5.1. 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 , , and the mesh points , , and we denote by the finite difference approximation of . The time-space derivatives are discretized in the following manner:
By assume and denote
Then, we may approximate problem (3) using the above scheme to obtain the following nonlinear diffusion filter:for , , and , with the initial condition and the discrete Neumann boundary condition:
A unique sequence is produced when using filter (116) on a particular initial image [2]. Besides, due to the continuity of the function , the sequence depends continuously on for every finite . 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 satisfyingscheme (116) satisfiesfor all , , and .
5.2. Experimental Results
This 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 (Section 1), and (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 ) [19] to evaluate the quality of the restored images. The 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). Therefore, we will use the following diffusion functions:(i)The proposed diffusion function ( for instance):(ii)The 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 .
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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). These 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, the results from Table 1 prove that the suggested approach has higher values in , whereas the model shows significant results in while -value increases.
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6. Conclusion
This paper principally investigates the class of anisotropic diffusion partial differential equations related to image processing and analysis. The existence and uniqueness of weak solutions for this problem have been proven under sufficient conditions satisfied by . 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.