On a Mathematical Analysis of a Coupled System Adapted to MRI Image Denoising

Abstract

Image denoising is a vibrant, fast-growing and emerging branch of mathematics and computer science that has enormous applications in real-world problems. This paper proposes a new anisotropic diffusion model for image denoising, especially in magnetic resonance imaging (MRI), based on a nonlinear reaction–diffusion system. This model is driven by using the decomposition strategy of the $H^{-1}$ norm, which is suitable for illustrating the small features in the textured image. Using the Schauder fixed point theorem, we have checked the well-posedness of the suggested reaction–diffusion system within a suitable framework. Finally, representative experiments and comparisons to other competitive models are performed to ensure the effectiveness of the proposed model.

Appendix

In this section, we provide some important lemmas, theorems and inequalities used in the proofs:

Theorem 2

(Hölder inequality) Let $f\in L^{p}(\Omega )$ and $g\in f\in L^{p'}(\Omega )$ where $\displaystyle {\frac{1}{p}+\frac{1}{p'}}=1$, with $1\le p \le \infty$. Then, $f.g\in L^1(\Omega )$ and

$$\begin{aligned} \int _{\Omega } |fg|\le ||f||_{L^{p}(\Omega )}||g||_{L^{p'}(\Omega )}. \end{aligned}$$

Lemma 2

(Young inequality) Let p, q be positive real numbers satisfying $\displaystyle {\frac{1}{p}+\frac{1}{q}=1}$. Then, if a,b are nonnegative real numbers,

$$\begin{aligned} ab\le \frac{a^p}{p}+\frac{b^q}{q}. \end{aligned}$$

Theorem 3

(Second Schauder Fixed-Point Theorem (1927)) Suppose that

• (i) X is a reflexive, separable B-space;

• (ii) The map $T: M \subseteq X \rightarrow M$ is weakly sequentially continuous, i.e., if $x_n \rightharpoonup x$ as $n \rightarrow \infty$, then also $T\left( x_n\right) \rightharpoonup T(x)$;

• (iii) The set M is non-empty, closed, bounded and convex.

Then, T has a fixed point.

Theorem 4

Assume that the continuous functions $u, \kappa :[0, T] \rightarrow [0, \infty )$ and $K>0$ satisfy

$$\begin{aligned} u(t) \le K+\int _0^t \kappa (s) u(s) {\text {d}} s \end{aligned}$$

for all $t \in [0, T]$. Then,

$$\begin{aligned} u(t) \le K \exp \left( \int _0^t \kappa (s) {\text {d}} s\right) . \end{aligned}$$

Theorem 5

(Aubin-Lions-Simon)(see Aubin (1963)) Let E, V and F be Banach spaces such that $E \subset$ $V \subset F$, we assume that $E \hookrightarrow V$ is compact and $V \hookrightarrow F$ is continuous. For $1 \le p \le \infty$ and $1 \le q \le \infty$, let

$$\begin{aligned} W=\left\{ u \in L^p(0, T, E): u^{\prime } \in L^q(0, T, F)\right\} . \end{aligned}$$

• (i) If $p<\infty$, the embedding $W \hookrightarrow L^p(0, T, V)$ is compact.

• (ii) If $p=\infty$ and $1<q$, then $W \hookrightarrow {\mathcal {C}}(0, T, V)$ is compact.

Theorem 6

(see Zeidler (2013)) Let E be a Banach space and F be a Hilbert space such that $E \subset F$; we assume that $E\hookrightarrow F$ is a continuous and E dense in F. We identify F with its dual space $F^{\prime }$ (so that $E \subset F = F^{\prime } \subset E^{\prime }$). Let $u \in L^{2}(0,T;E)$, and suppose that $\partial _t u \in L^{2}(0,T;E^{'})$. Then, $u \in C(0, T; F)$.