Coupling local and nonlocal diffusion equations for image denoising

Abstract

Due to the strong ability of restoring textures and details in images, nonlocal equations have attracted an extensive interest for image denoising. However, the lack of regularity causes residual noise in the restored images. In this paper, we propose and study an evolution equation consisting of the weighted local and the weighted nonlocal $p$-Laplacian equations. The existence and uniqueness of solutions for the proposed equation are proven under the assumption that the weights vanish in sets of positive measure. We also show that solutions of the proposed equation converge to the solution of the usual $p$-Laplacian equation if the kernel is rescaled appropriately. Comparisons with local and nonlocal diffusion equations for removing Gaussian noise in images are presented.

Introduction

In this paper, we investigate the properties of the evolution equation

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}=\text{div}\left(\left(1-g\left(x\right)\right){|\nabla u|}^{p-2}\nabla u\right)+\lambda {\int }_{\Omega }\frac{g\left(x\right)+g\left(y\right)}{2}\times \hfill \\ J(x-y){|u(y,t)-u(x,t)|}^{p-2}\left(u(y,t)-u(x,t)\right)dy,(x,t)\in {\Omega }_T,\hfill \\ (1-g\left(x\right)){|\nabla u|}^{p-2}\nabla u\cdot \overrightarrow{n}=0,(x,t)\in \partial \Omega \times (0,T),\hfill \\ u(x,0)=f\left(x\right),x\in \Omega ,\hfill \end{array}\right.$$

and discuss its applications in image restoration under Gaussian noise. Here $\Omega \subset {\mathbb{R}}^N$ ($N\ge 2$) is a bounded domain with smooth boundary $\partial \Omega$, $\overrightarrow{n}$ is the unit outward normal on $\partial \Omega$, ${\Omega }_T=\Omega \times (0,T)$, $T>0$, $\lambda >0$ is a constant and $1<p<+\infty$. The kernel $J:{\mathbb{R}}^N\to \mathbb{R}$ is a nonnegative continuous radial function with compact support, $J\left(0\right)>0$, and ${\int }_{{\mathbb{R}}^N}J\left(z\right)dz=1$. Given a noisy image $f\left(x\right)$, the solution $u(x,t)$ is the restored image with scale variable $t$.

Eq. (1) is coupled with a weighted $p$-Laplacian equation and a weighted nonlocal $p$-Laplacian equation. The $p$-Laplacian equation

$$\frac{\partial u}{\partial t}=\text{div}\left({|\nabla u|}^{p-2}\nabla u\right),1\le p<\infty ,$$

has been widely investigated for image processing. If $p=2$, this is the heat equation, whose solution is given as the convolution of the initial value $f$ with a Gaussian Kernel [1]. That is to say, image denoising by the heat equation is equivalent to the Gaussian filter and always causes over-smoothness in the restored images. For edge-preserving, it is straightforward to consider nonlinear equations. The total variation (TV) flow ($p=1$ in Eq. (2)) [2], which comes from the gradient descent scheme of the famous ROF model [3], has been thoroughly studied in the past two decades. The main advantage of the TV flow for image denoising is the powerful ability to restore edges and homogeneous regions. However, it utilizes only the image’s local information, which means that it fails to preserve textured structures and repeated patterns in images.

The nonlocal diffusion equation

$$\frac{\partial u}{\partial t}={\int }_{\Omega }J(x-y){|u(y,t)-u(x,t)|}^{p-2}\left(u(y,t)-u(x,t)\right)dy,1\le p<\infty ,$$

was proposed as a remedy [4], [5], [6]. It is often referred to as the nonlocal $p$-Laplacian equation, since the solutions of Eq. (3) converge strongly in $L^p\left({\Omega }_T\right)$ to the solution of Eq. (2) with homogeneous Neumann boundary condition if the kernel $J$ is rescaled appropriately [7]. In image denoising problems, the kernel $J$ represents the nonlocal weight that measures the similarity between two patches [8]. Consequently, Eq. (3) avoids the drawbacks of Eq. (2) and restores textures and details in images effectively. It is noticed that a main difference between the usual $p$-Laplacian equation (2) and the nonlocal $p$-Laplacian equation (3) is the lack of regularizing effect [9] of the latter. As a result, Eq. (3) causes residual noise in homogeneous regions of the restored images.

Inspired by the local method and the nonlocal method mentioned above, we propose Eq. (1) to reduce artifacts of each method for image denoising. The local $p$-Laplacian equation and the nonlocal $p$-Laplacian equation in (1) are coupled through a space varying function $g\left(x\right)$. In regions with rich textures and repeated structures, we let $g\left(x\right)=1$, which leads to nonlocal diffusion

$$\lambda {\int }_{\Omega }\frac{1+g\left(y\right)}{2}J(x-y){|u(y,t)-u(x,t)|}^{p-2}\left(u(y,t)-u(x,t)\right)dy.$$

In regions with cartoon structures, we let $g\left(x\right)=0$, which leads to

$$\text{div}\left({|\nabla u|}^{p-2}\nabla u\right)+\lambda {\int }_{\Omega }\frac{g\left(y\right)}{2}J(x-y){|u(y,t)-u(x,t)|}^{p-2}\left(u(y,t)-u(x,t)\right)dy.$$

If $g\left(y\right)J(x-y)=0$ for any $y\in \Omega$ (recall that $J$ has a compact support), then the nonlocal term in the above disappears and it becomes local diffusion. At last, $g\left(x\right)\in (0,1)$ shall be used for complex regions.

In the nonlocal part of Eq. (1), the diffusion coefficient is constructed as $\frac{g\left(x\right)+g\left(y\right)}{2}$. The authors of [10] suggested using $g\left(\frac{x+y}{2}\right)$ as the weight for nonlocal equations. As we shall see in Theorem 9, both of them are well-defined in the sense that the nonlocal problem converges to the local problem if the kernel $J$ is rescaled appropriately. Note that our construction has the nice symmetric property

$${\int }_{\Omega }{\int }_{\Omega }\frac{g\left(x\right)+g\left(y\right)}{2}J(x-y){|u\left(y\right)-u\left(x\right)|}^{p}dydx={\int }_{\Omega }{\int }_{\Omega }g\left(y\right)J(x-y){|u\left(y\right)-u\left(x\right)|}^{p}dydx={\int }_{\Omega }{\int }_{\Omega }g\left(x\right)J(x-y){|u\left(y\right)-u\left(x\right)|}^{p}dydx.$$

Then we could define weak solutions for Eq. (1) straightforwardly. Besides, our construction is a more appropriate approach for digital images, since $\frac{x+y}{2}$ is not necessary on the grid.

The weighted $p$-Laplacian equation in Eq. (1) has a smoothing effect. It can be replaced by other weighted nonlinear diffusion equations for image denoising. In this paper, we also consider the evolution equation

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}=\text{div}\left(\left(1-g\left(x\right)\right)\frac{\nabla u}{1+{|\nabla u_{\sigma }|}^2∕K^2}\right)+\lambda {\int }_{\Omega }\frac{g\left(x\right)+g\left(y\right)}{2}J(x-y)\left(u(y,t)-u(x,t)\right)dy,(x,t)\in {\Omega }_T,\hfill \\ (1-g\left(x\right))\frac{\partial u}{\partial \overrightarrow{n}}=0,(x,t)\in \partial \Omega \times (0,T),\hfill \\ u(x,0)=f\left(x\right),x\in \Omega ,\hfill \end{array}\right.$$

which is coupled with the weighted regularized Perona–Malik equation [11] and the weighted linear nonlocal equation. Notice that the regularized Perona–Malik equation has a better smoothing effect than the $p$-Laplacian equation and the nonlocal $p$-Laplacian equation (3) performs the best for image denoising when $p=2$. Eq. (5) outperforms Eq. (1) for image denoising.

To finish the introduction, we briefly review some related works. Coupling local and nonlocal methods for image denoising have been proposed in [12], [13]. The variational model

$$\lambda {\int }_{\Omega }|\nabla u|dx+{\int }_{\Omega }{\int }_{\Omega }J(x,y){|u\left(y\right)-f\left(x\right)|}^{p}dydx$$

combines the total variation regularization term and a nonlocal $L^p$ data fidelity term to reduce the undesirable artifacts caused by each method. The model can be viewed as the adaptive regularization of the nonlocal means filter [8]. Recently, qualitative properties for the gradient flow of

$$\frac{1}{2}{\int }_{{\Omega }_l}{|\nabla u|}^{2}dx+\frac{\lambda }{4}{\int }_{{\Omega }_{nl}}{\int }_{\Omega }J(x-y){(u\left(y\right)-u\left(x\right))}^{2}dydx$$

has been studied in [14], where ${\Omega }_l\cap {\Omega }_{nl}=\Omega$. Notice that Eq. (1) is the gradient flow of the energy functional

$$\frac{1}{p}{\int }_{\Omega }(1-g\left(x\right)){|\nabla u|}^{p}dx+\frac{\lambda }{2p}{\int }_{\Omega }{\int }_{\Omega }\frac{g\left(x\right)+g\left(y\right)}{2}J(x-y){|u\left(y\right)-u\left(x\right)|}^{p}dydx.$$

If $g\left(x\right)={\chi }_{{\Omega }_{nl}}$ and $p=2$, then it follows from (4) that functional (8) coincides with (7).

The rest of this paper is organized as follows. In Section 2 we define solution spaces and state main results. In Section 3 we prove the existence and uniqueness of weak solutions for Eq. (1). We also prove that solutions of Eq. (1) converge to the solution of the usual $p$-Laplacian equation if the kernel $J$ is rescaled appropriately. Section 4 is devoted to the applications of Eqs. (1), (5) for image denoising. We conclude the paper in Section 5.