An adaptive second-order partial differential equation based on TV equation and p-Laplacian equation for image denoising

Abstract

This paper introduces an adaptive diffusion partial differential equation (PDE) for noise removal, which combines a total variation (TV) term and a p-Laplacian (1 < p ≤ 2) term. Utilizing the edge indicator, we can adaptively control the diffusion model, which alternates between the TV and the p-Laplacian(1 < p ≤ 2) in accordance with the image feature. The main advantage of the proposed model is able to alleviate the staircase effect in smooth regions and preserve edges while removing the noise. The existence of a weak solution of the proposed model is proved. Experimental results confirm the performance of the proposed method with regard to peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) and visual quality.

1 Introduction

Image denoising is an important step in low-level image processing and computer vision. Image denoising aims to preserve important structure features including edges and corners while removing the noise in the image. The variational [4, 6, 8, 11, 12, 16, 19] and partial differential equations (PDEs) [9, 10, 13,14,15, 18, 23] based methods are very popular for removing noise during the last two decades [3]. Despite the success enjoyed by these methods, there are problems related to fine structure and edge preservation, staircase effect or over-smoothing of images.

Consider the following representative minimization problem

$$ \min\limits_{u}E(u)=\frac{1}{p}{\int}_{\Omega}|\nabla u|^{p} dx, $$

(1)

where the minimizer u is the restored image, ${\Omega }\subset \mathbb {R}^{2}$ is the image domain with 1 ≤ p ≤ 2. The formal gradient flow associated with the functional E(u) is given by,

$$ \frac{\partial u}{\partial t}=\text{div}(|\nabla u|^{p-2}\nabla u), $$

(2)

when p = 1, it is the widely used total variation (TV) based scheme proposed in 1992 [16]. In [6], the existence and uniqueness of TV minimization was proved in the space of functions of bounded variation (BV), and the corresponding gradient flow was treated in [1, 2]. The TV-based model performs better at preserving edge sharpness and location while removing noise but creates patchy or staircase effect within homogeneous, smooth or moderate gradient regions [7]. Choosing p = 2 results in isotropic diffusion, which removes the noise and solves the staircase problem but is at the expense of edge blurring. Taking a fixed value 1 < p < 2 results in an anisotropic diffusion, which is somewhere between TV model and isotropic smoothing. However, there is a trade-off between edge preservation and piecewise smooth regions reconstruction.

In order to mitigating the drawbacks of TV-based and isotropic diffusion, various remedies have been proposed [6, 8, 12, 24]. For instance, Chamboll and Lions [6] proposed minimizing the following energy functional, which combines TV-based and isotropic diffusion

$$ \min\limits_{u\in BV({\Omega})}E(u)=\frac{1}{2\beta}{\int}_{|\nabla u|\leq \beta}|\nabla u|^{2} dx+{\int}_{|\nabla u|\geq \beta}|\nabla u| -\frac{\beta}{2}dx . $$

(3)

Here β is a fixed positive number and it is a threshold of |∇u|. As |∇u|≤ β, it means isotropic diffusion. As |∇u|≥ β, the TV energy is used. This model is successful in restoring images in which homogeneous regions are separated by distinct edges. However, this model is closely linked to β and a fixed threshold can lead to bad results [8, 24]. In order to solve this problem, Zhou et al. [24] introduced a locally adaptive version of the model (3):

$$ \min\limits_{u}E(u)=\frac{1}{2\beta(x)}{\int}_{|\nabla u|\leq \beta(x)}|\nabla u|^{2} dx+{\int}_{|\nabla u|\geq \beta(x)}|\nabla u| -\frac{\beta(x)}{2}dx, $$

(4)

where the adaptive parameter β(x) is obtained by solving a separate energy minimization problem. This model can remove noise while preserve the edge. However, Zhou [24] lacks a strong theoretical foundation.

Recently, motivated by the correspondence between the variational and PDE methods for imaging problems, Surya Prasath et al. [18] proposed adaptive forward-backward diffusion equation (AFBD for short) of the following form:

$$ \frac{\partial u}{\partial t}=\text{div}\left( h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\right), $$

(5)

where p ≥ 1. h(∇G_σ ∗ u) is an edge detector and given by,

$$ h(\nabla G_{\sigma}* u)=\frac{1}{1+k| \nabla G_{\sigma}* u|^{2}}, $$

(6)

where k > 0. G_σ denotes the two-dimensional Gaussian kernel G_σ = (2πσ)^− 1 exp(−|x|²/2σ²), σ > 0 is a constant. The authors proved the existence of weak solutions for (5). As p = 1, their experimental results showed that their model overcome well-known edge smearing effects of the heat equation by using gradient dependent diffusion function. However, the resulting image in the presence of the noise also showed staircase effect. When p > 1 (for example p = 1.5, see Section 3 Figs. 3g, 4g, 5h), the model (5) alleviates the staircase effect but dose not preserve the edge.

In order to alleviate the staircase effect in smooth regions and preserve edges while removing the noise, we propose a novel adaptive PDE model based on TV equation and p-Laplacian equation. Our proposed model can have the following form

$$ \frac{\partial u}{\partial t}=\text{div}\left( h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u+\mu (1-h(\nabla G_{\sigma}* u))\frac{\nabla u}{|\nabla u|}\right) , $$

(7)

where 1 < p ≤ 2 , μ > 0 is a constant.

Now let us mention the advantage of the proposed model. Firstly, the proposed model joints TV term which is edge preserving and p-Laplacian (1 < p ≤ 2) term which helps in effective noise removal. In homogeneous regions, we make use of the p-Laplacian term to reduce the noise and avoid the staircase effect; and near edges, we utilize the TV term to preserve them. By using the edge indicator h(∇G_σ ∗ u), we roughly segment the image into two subregions, i.e., the homogeneous regions and the region’s nearby boundaries. In the homogeneous regions, the magnitude of |∇G_σ ∗ u| is small, thus the edge detector function h(∇G_σ ∗ u) is close to one, so the proposed model acts as the p-Laplacian equation (1 < p ≤ 2) to remove the noise and alleviate the staircase effect; in the region’s nearby boundaries, the magnitude of |∇G_σ ∗ u| is large, then the edge detector function h(∇G_σ ∗ u) takes smaller value (close to zero), so the proposed model acts as the TV diffusion equation to preserve the edge. Secondly, in contrast to the previous anisotropic diffusion PDEs, we prove the existence of a weak solution of the proposed equation, which is very important for the numerical computation. Thirdly, experimental results on different noisy images indicate the advantage of the proposed adaptive model. The proposed approach is proven to provide better edge preserving with effective noise removal both theoretically and illustrated experimentally using a variety of example.

The rest of the paper is organized as follows. In Section 2, we give some important lemmas and then prove the existence of a weak solution of the proposed problem. Section 3 details the numerical aspects and shows comparison denoising results on noisy images. Finally, Section 4 concludes the paper.

2 Existence of weak solutions

The problem (7) is complemented with the initial condition

$$ u(x,0)=u_{0}, $$

(8)

and boundary condition

$$ \frac{\partial u}{\partial \pmb{n}}= 0, $$

(9)

where n = (n₁,n₂) is the outward normal direction on ∂Ω, In this section, we establish the existence of a weak solution of the proposed model (7), (8) and (9).

2.1 Weak solutions and the main result

The symbol ∥⋅∥ will stand for the Euclidean norm in L²(Ω). The corresponding scalar product will be denoted by parentheses (⋅,⋅). We will also use this notation for duality between L^p(Ω) and L^p/p− 1(Ω). The symbols $C(\mathcal {J};E)$, $C_{w}(\mathcal {J};E)$, $L^{2}(\mathcal {J},E)$ etc denote the spaces of continuous, weakly continuous, quadratically integrable etc functions on an interval $\mathcal {J}\subset \mathbb {R}$ with values in a Banach space E.

Definition 1

A function u from the class

$$ u\in C_{w}\left( 0,T;L^{2}({\Omega})\right)\bigcap L^{p}(0,T;W^{1,p}({\Omega}))\bigcap W^{1,p/(p-1)}\left( 0,T;\left[ L^{2}\bigcap W^{1,p}({\Omega})\right]^{*}\right) $$

(10)

is called a weak solution to problems (7), (8) and (9) with u₀ ∈ L²(Ω) ,

if

• (i) there exists $z\in L^{\infty }({\Omega }\times (0,T);\mathbb {R}^{2})$, $\|z\|_{L^{\infty }({\Omega }\times (0,T);\mathbb {R}^{2})}\le 1$, so that for all v ∈ L² ∩ W^1,p(Ω),

  $$ \left\langle\frac{du}{dt},v\right\rangle+(h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u,\nabla v)+\mu((1-h(\nabla G_{\sigma}* u))Z,\nabla v)= 0 $$

  (11)

  a.e. on (0,T);

• (ii) for all w ∈ L^p(0,T;W^1,p(Ω)) ∩ W^1,1(0,T;L²(Ω)), one has

  $$\begin{array}{@{}rcl@{}} &&\|u(T)-w(T)\|^{2}+ 2{{\int}_{0}^{T}}{\int}_{\Omega}\frac{dw}{dt}udxdt\\ &&\quad+ 2{{\int}_{0}^{T}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p} dxdt \\&&\quad+ 2\mu {{\int}_{0}^{T}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt\\ &&\le\|u_{0}-w(0)\|^{2}+\|w(T)\|^{2}-\|w(0)\|^{2}\\&&\quad+ 2{{\int}_{0}^{T}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla w dxdt\\ &&\quad+ 2{\mu{\int}_{0}^{T}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}*u)\right)Z\cdot\nabla wdxdt; \end{array} $$

  (12)

• (iii) the initial condition (8) holds in the space L²(Ω).

Remark 1

Let us present a motivation for this definition of solution to (7), (8) and (9). Consider a pair (u,Z) of sufficiently smooth functions satisfying (11), (12) and (8). Then, testing (11) by 2(u − w) and integrating in time, we deduce

$$\begin{array}{@{}rcl@{}} &&2{{\int}_{0}^{T}}{\int}_{\Omega}\frac{d(u-w)}{dt}\cdot(u-w)dxdt+ 2{{\int}_{0}^{T}}{\int}_{\Omega}\frac{dw}{dt}\cdot(u-w)dxdt\\ &&+ 2{{\int}_{0}^{T}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla (u-w)dxdt\\ &&+ 2{\mu{\int}_{0}^{T}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)Z\cdot\nabla (u-w)dxdt= 0. \end{array} $$

we get

$$\begin{array}{@{}rcl@{}} &&\|u(T)-w(T)\|^{2}+ 2{{\int}_{0}^{T}}{\int}_{\Omega} \frac{dw}{dt}udxdt\\&&\quad+ 2{{\int}_{0}^{T}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\\ &&\quad+ 2{\mu{\int}_{0}^{T}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)|\nabla u|dxdt\\ &&=\|u_{0}-w(0)\|^{2}+\|w(T)\|^{2}-\|w(0)\|^{2}\\ &&\quad+ 2{{\int}_{0}^{T}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla wdxdt \\&&\quad+ 2{\mu{\int}_{0}^{T}}{\int}_{\Omega} (1-h(\nabla G_{\sigma}* u))Z\cdot\nabla wdxdt\\ &&\quad+ 2{\mu{\int}_{0}^{T}}{\int}_{\Omega} \left( 1-h(\nabla G_{\sigma}* u)|\nabla u|dxdt\right.\\&&\quad-2{\mu{\int}_{0}^{T}}{\int}_{\Omega} \left( 1-h(\nabla G_{\sigma}* u)\right)Z\cdot\nabla udxdt\\ \end{array} $$

Due to (12), we have

$${{\int}_{0}^{T}}{\int}_{\Omega} \left( 1-h(\nabla G_{\sigma}* u)\right)|\nabla u|dxdt {\le{\int}_{0}^{T}}{\int}_{\Omega} \left( 1-h(\nabla G_{\sigma}* u)\right)Z\cdot\nabla udxdt.$$

On the other hand,

$$(1-h(\nabla G_{\sigma}* u))|\nabla u|\ge(1-h(\nabla G_{\sigma}* u))Z\cdot\nabla u.$$

All these can be true if and only if

$$|\nabla u|=Z\cdot\nabla u$$

i.e.

$$ Z=\frac{\nabla u}{|\nabla u|} $$

(13)

Substituting (13) into (11) and integrating by parts, we infer

$$\begin{array}{@{}rcl@{}} &&{\int}_{\Omega}\left[\frac{\partial u}{\partial t}-\text{div}(h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u)-\mu\text{div}\left( (1-h(\nabla G_{\sigma}* u ))\frac{\nabla u}{|\nabla u|}\right)\right]vdx\\ &&+{\int}_{\partial\Omega}\left( h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}+\mu\frac{1-h(\nabla G_{\sigma}* u)}{|\nabla u|}\right)\frac{\partial u}{\partial\pmb{n}}vd\mathcal{H}^{1}= 0. \end{array} $$

(14)

Testing (14) by any v compactly supported in Ω, we deduce (7). By arbitrariness of v, we deduce

$$\left( h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}+\mu\frac{1-h(\nabla G_{\sigma}*u)}{|\nabla u|}\right)\frac{\partial u}{\partial\pmb{n}}= 0,\quad \mathcal{H}^{1}-a.e. \text{in} \partial\Omega.$$

Since $h(\nabla G_{\sigma }* u)|\nabla u|^{p-2}+\mu \frac {1-h(\nabla G_{\sigma }* u)}{|\nabla u|}$ is non-vanishing, the boundary condition (9) is satisfied.

Now we are ready to formulate one of our main result:

Theorem 1

Letu₀ ∈ L²(Ω).Then there exists a weak solution to (7), (8) and (9) in the class (11).

2.2 Auxiliary problem

We recall the following abstract observation [20, 25]. Assume that we have two Hilbert spaces, X ⊂ Y, with continuous embedding operator i : X → Y, and i(X) is dense in Y. The adjoint operator i^∗ : Y^∗→ X^∗ is continuous and, since i(X) is dense in Y, one-to-one. Since i is one-to-one, i^∗(Y^∗) is dense in X^∗, and one may identify Y^∗ with a dense subspace of X^∗. Due to the Riesz representation theorem, one may also identify Y with Y^∗. We arrive at the chain of inclusions

$$X\subset Y\equiv Y^{*}\subset X^{*}.$$

Both embeddings here are dense and continuous. Observe that in this situation, for f ∈ Y, u ∈ X, their scalar product in Y coincides with the value of the functional f from X^∗ on the element u ∈ X

$$(f,u)_{Y}=\langle f,u\rangle.$$

Such triples (X,Y,X^∗) are called Lions triples.

We will work with the Lions triple (H^r(Ω),L²(Ω),(H^r(Ω))^∗), where r > 2(p + 1) is a fixed number. Denote by A the Riesz bijection between the spaces H^r and (H^r)^∗ (which are not identified).

In order to prove the theorem 1 we first need to study the following auxiliary problem:

$$ \left\{\begin{array}{ll} \frac{\partial u}{\partial t}+\epsilon A(u)=\text{div}(h(\nabla G_{\sigma}*u)|\nabla u|^{p-2}\nabla u)+\mu \text{div}\left( (1-h(\nabla G_{\sigma}* u))\frac{\nabla u}{\epsilon+|\nabla u|}\right),\\ u(x,0)=u_{0}, \\ \frac{\partial u}{\partial \pmb{n}}\left|{~}_{\partial{\Omega}\times(0,T)}= 0 \right.. \end{array}\right. $$

(15)

Here 𝜖 > 0 is parameter.

The weak form of (15) is the following Cauchy problem, where the first equality is understood in the sense of the space (H^r)^∗, whereas the second and the third ones are in the sense of space L²:

$$ \frac{du}{dt}+R(u)+\mu Q_{\epsilon}(u)+\epsilon A(u)= 0,\quad\quad u\left|{~}_{t = 0}=u_{0}\right.. $$

(16)

The operators R : H^r → (H^r)^∗ and Q_𝜖 : H^r → (H^r)^∗, which respect the boundary condition, are determined by the duality

$$\begin{array}{@{}rcl@{}} \langle R(u),w\rangle&=&\left( h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u,\nabla w\right),\\ \langle Q_{\epsilon}(u),w\rangle&=&\left( (1-h(\nabla G_{\sigma}* u))\frac{\nabla u}{\epsilon+|\nabla u|},\nabla w \right),\quad\quad \forall w\in H^{r}. \end{array} $$

We do not use a notation for partial time derivative since we treat (16) as an ODE in a Banach space.

Lemma 1

Letu₀ ∈ L²(Ω).The Cauchy problem (16) admits a solution u in the class

$$ L^{2}(0,T; H^{r}({\Omega}))\cap H^{1}(0,T ;(H^{r}({\Omega}))^{*})\cap C([0,T];L^{2}({\Omega})). $$

(17)

The solution satisfies the following inequality:

$$\begin{array}{@{}rcl@{}} &&\|u(t_{*})-w(t_{*})\|^{2}+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}\frac{dw}{dt}udxdt + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega} (1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt\\ &&\le \|u_{0}-w(0)\|^{2}+\|w(t_{*})\|^{2}-\|w(0)\|^{2} + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla wdxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}*u)\right)\frac{\nabla u}{\epsilon+|\nabla u|}\cdot\nabla wdxdt\\ &&\quad+ 2\mu\epsilon{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)\frac{|\nabla u|}{\epsilon +|\nabla u|}dxdt +\frac{\epsilon}{2}{\int}_{0}^{t_{*}}\langle Aw,w\rangle dt, \end{array} $$

(18)

for every t_∗∈ [0,T] and for every sufficiently regular test function $w:\bar {\Omega }\times [0,T]\to \mathbb {R}$.

Proof

□

Note that Ω is bounded, the embeddings $H^{r}({\Omega }) \subset C^{1}(\bar {\Omega })$ is compact. The operators R : C¹ → (H^r)^∗ and Q_𝜖 : C¹ → (H^r)^∗ are continuous. Moreover,

$$\begin{array}{@{}rcl@{}} &&\|R\|_{(H^{r}({\Omega}))^{*}}\le\|R\|_{(H^{1}({\Omega}))^{*}}\le{\int}_{\Omega}\left|h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla \right.u\left|dx\right.\\&&\quad\le {\int}_{\Omega}|\nabla u|^{p-1}dx\le C\|u\|^{p}_{C^{1}(\bar{\Omega})}]\\ &&\|Q_{\epsilon}\|_{(H^{r}({\Omega}))^{*}}\le\|Q_{\epsilon}\|_{(H^{1}({\Omega}))^{*}}\\&&\quad\le{\int}_{\Omega}\left|(1-h(\nabla G_{\sigma}* u))\frac{\nabla u}{\epsilon+|\nabla u|}\right|dx\le C \end{array} $$

Here, C is independent of 𝜖. Therefore, R and Q_𝜖 : H^r → (H^r)^∗ are compact operators. These give opportunity to secure existence of solutions to (16) in (17) by an application of the Leray-Schander degree theory (a systematic approach to parabolic problems of kind (16) may be found, e.g., in [25]).

We now fix a sufficiently smooth function $w:\bar {\Omega }\times [0,T]\to \mathbb {R}$. Testing (16) with 2(u − w), 0 ≤ t_∗≤ T, in the sense of ((H^r)^∗,H^r) duality, and integrate in time to obtain

$$\begin{array}{@{}rcl@{}} &&\|u(t_{*})-w(t_{*})\|^{2}+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}\frac{dw}{dt}udxdt + 2{\int}_{0}^{t_{*}}{\int}_{\Omega} h(\nabla G_{\sigma}* u)|\nabla u|^{p} dxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)\frac{\nabla u}{\epsilon+|\nabla u|}\cdot\nabla udxdt\\ &&=\|u_{0}-w(0)\|^{2}+\|w(t_{*})\|^{2}-\|w(0)\|^{2} + 2{\int}_{0}^{t_{*}}{\int}_{\Omega} h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla wdxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega} \left( 1-h(\nabla G_{\sigma}* u)\right)\frac{\nabla u}{\epsilon+|\nabla u|}\cdot\nabla wdxdt -2\epsilon{\int}_{0}^{t_{*}}\langle Au,u-w\rangle dt. \end{array} $$

(19)

Note that the following equality holds

$$\begin{array}{@{}rcl@{}} {\int}_{0}^{t_{*}}{\int}_{\Omega}(1 - h(\nabla G_{\sigma}* u))\frac{\nabla u}{\epsilon+|\nabla u|}\cdot\nabla udxdt \!&=&{\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt\\ &&-\epsilon {\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))\frac{|\nabla u|}{\epsilon+|\nabla u|}dxdt. \end{array} $$

An application of Cauchy’s inequality yields

$$-2\langle Au,u-w\rangle\le\frac{1}{2}\langle Aw,w\rangle. $$

We derive from (19) that

$$\begin{array}{@{}rcl@{}} &&\|u(t_{*})-w(t_{*})\|^{2}+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}\frac{dw}{dt}udxdt + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega} (1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt\\ &&\le \|u_{0}-w(0)\|^{2}+\|w(t_{*})\|^{2}-\|w(0)\|^{2} + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla wdxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)\frac{\nabla u}{\epsilon+|\nabla u|}\cdot\nabla wdxdt\\ &&\quad+ 2\mu\epsilon{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)\frac{|\nabla u|}{\epsilon +|\nabla u|}dxdt +\frac{\epsilon}{2}{\int}_{0}^{t_{*}}\langle Aw,w\rangle dt. \end{array} $$

Lemma 2

Let u be a solution to problem (16), then u satisfies a priori bound

$$ \|u\|_{L^{\infty}(0,T;L^{2}({\Omega}))}+\|u\|_{L^{p}(0,T;W^{1,p}({\Omega}))}+\|\frac{du}{dt}\|_{L^{2}(0,T;(H^{r}({\Omega}))^{*})}\le C. $$

(20)

The constant C is independent of 𝜖.

Proof

□

Taking w ≡ 0 in (19), 0 ≤ t_∗≤ T, we derive

$$\begin{array}{@{}rcl@{}} &\|&u(t_{*})\|^{2} + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\\ &&+ 2\mu {\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)\frac{|\nabla u|^{2}}{\epsilon+|\nabla u|}dxdt+ 2\epsilon {\int}_{0}^{t_{*}}\left< Au,u\right>dt=\|u_{0}\|^{2}. \end{array} $$

Therefore

$$ \|u(t_{*})\|^{2}\le\|u_{0}\|^{2}, $$

(21)

$$ {\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\le\|u_{0}\|^{2}, $$

(22)

$$ \mu{\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))\frac{|\nabla u|^{2}}{\epsilon+|\nabla u|}dxdt\le\|u_{0}\|^{2}, $$

(23)

$$ 2\epsilon {\int}_{0}^{t_{*}}\left< Au,u\right>dt\le\|u_{0}\|^{2}. $$

(24)

Using (21), we deduce

$$ \|u \|_{L^{\infty}(0,T;L^{2}({\Omega}))}\le\|u_{0}\|. $$

(25)

Therefore,

$$ |\nabla G_{\sigma}*u|\le\|\nabla G_{\sigma}\|\|u\|\le C\|u\|\le C . $$

(26)

Thus,

$$ 1\ge h(\nabla G_{\sigma}* u)=\frac{1}{1+k|\nabla G_{\sigma}* u|^{2}}\ge C. $$

(27)

Using (22), we get

$$C{\int}_{0}^{t_{*}}{\int}_{\Omega}|\nabla u|^{p}dxdt\le{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p}dxdt\le\|u_{0}\|^{2} , $$

a.e.

$$ {\int}_{0}^{t_{*}}{\int}_{\Omega}|\nabla u|^{p}dxdt\le C. $$

(28)

Using (25), (28) and Lemma 1, we deduce

$$\|u\|_{L^{\infty}(0,T;L^{2}({\Omega}))}+\|u\|_{L^{p}(0,T;W^{1,p}({\Omega}))}+\|\frac{du}{dt}\|_{L^{2}(0,T;(H^{r}({\Omega}))^{*})}\le C $$

All the constants C here are 𝜖-independent.

2.3 Proof of theorem 1

Proof of Theorem 1

□

Let {u_k} be a sequence of solutions (16) with 𝜖 = 𝜖_k. Set

$$Z_{k}=\frac{\nabla u_{k}}{\epsilon_{k}+|\nabla u_{k}|}. $$

Then

$$ \|Z_{k}\|_{L^{\infty}({\Omega}\times(0,T);\mathbb{R}^{2})}\le 1. $$

(29)

Since (u_k,Z_k) are the solutions to (16) in the sense of Lemma 1, we have

$$ \left\langle\frac{du_{k}}{dt},v\right\rangle+(h(\nabla G_{\sigma}* u_{k})|\nabla u_{k}|^{p-2}\nabla u_{k},\nabla v)+\mu((1-h(\nabla G_{\sigma}* u_{k}))Z_{k},\nabla v)+\epsilon_{k}\langle Au_{k},v\rangle= 0, $$

(30)

a.e. on (0,T).

$$\begin{array}{@{}rcl@{}} &&\|u_{k}(t_{*})-w(t_{*})\|^{2}+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}\frac{dw}{dt}u_{k}dxdt + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u_{k})|\nabla u_{k}|^{p}dxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega} (1-h(\nabla G_{\sigma}* u_{k}))|\nabla u_{k}|dxdt\\ &&\le \|u_{0}-w(0)\|^{2}+\!\|w(t_{*})\|^{2} - \|w(0)\|^{2} + 2\!{\int}_{0}^{t_{*}}\!{\int}_{\Omega}h(\nabla G_{\sigma}* u_{k})|\nabla u_{k}|^{p-2}\nabla u_{k}\cdot\nabla wdxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u_{k})\right)Z_{k}\cdot\nabla wdxdt\\ &&\quad+ 2\mu\epsilon_{k}{{\int}_{0}^{t}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u_{k})\right)\frac{|\nabla u_{k}|}{\epsilon_{k} +|\nabla u_{k}|}dxdt +\frac{\epsilon_{k}}{2}{\int}_{0}^{t_{*}}\langle Aw,w\rangle dt, \end{array} $$

(31)

for all t^∗∈ [0,T], and

$$ u_{k}(0)=u_{0}, $$

(32)

for all sufficiently smooth function $v, w:\bar {\Omega }\times [0,T]\to \mathbb {R}.$ To prove the theorem, we are going to pass to limit in (30), (31) and (32) as 𝜖_k → 0.

Due to (20) and (29), without loss of generality we have

$$u_{k}\to u \quad \text {weakly}^{*} \ \text{in} \quad L^{\infty}(0,T;L^{2}({\Omega})), $$

$$\frac{du_{k}}{dt}\to \frac{du}{dt} \quad \text {weakly} \ \text{in} \quad L^{2}(0,T;(H^{r}({\Omega}))^{*}), $$

$$Z_{k}\to Z \quad \text {weakly}^{*} \ \text{in} \quad L^{\infty}({\Omega}\times(0,T);\mathbb{R}^{2}), $$

$$|\nabla u_{k}|^{p-2}\nabla u_{k}\to |\nabla u|^{p-2}\nabla u\quad \text {weakly} \ \text{in} \quad L^{\frac{p}{p-1}}({\Omega}\times[0,T];\mathbb{R}^{2}) $$

Owing to the Aubin-Lions-Simon lemma [17], without loss of generality we may assume that u_k → u in C ([0,T];[W^1,2(Ω)]^∗). See [18], we know that h(∇G_σ ∗ u_k) → h(∇G_σ ∗ u) uniformly on $C\left (\bar {\Omega }\times [0,T]\right )$.

Therefore,

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u_{k}))Z_{k}\cdot\nabla w dxdt \to{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( (1-h(\nabla G_{\sigma}* u))Z\cdot\nabla w dxdt\right.. \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u_{k})|\nabla u_{k}|^{p-2}\nabla u_{k}\cdot\nabla wdxdt \to{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p-2}\nabla u\cdot\nabla wdxdt \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt \le \lim\limits_{k\to +\infty}\inf{\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u_{k}))|\nabla u_{k}|dxdt. \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{\int}_{0}^{t_{*}}{\int}_{\Omega} h(\nabla G_{\sigma}* u)|\nabla u|^{p} dxdt \le\lim\limits_{k\to +\infty}\inf {\int}_{0}^{t_{*}}{\int}_{\Omega} h(\nabla G_{\sigma}* u_{k})|\nabla u_{k}|^{p} dxdt \end{array} $$

The first term of (31) is lower-semicontinuous (see [21]), we have

$$\|u(t_{*})-w(t_{*})\|^{2}\le \lim\limits_{k\to +\infty}\inf\|u_{k}(t_{*})-w(t_{*})\|^{2}.$$

The remaining terms of (31) are either linear, or constants, or of order O(𝜖_k), Hence, in the limit we get

$$\begin{array}{@{}rcl@{}} &&\|u(t_{*})-w(t_{*})\|^{2}+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}\frac{dw}{dt}udxdt\\ &&\quad+ 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u|^{p} dxdt + 2\mu {\int}_{0}^{t_{*}}{\int}_{\Omega}(1-h(\nabla G_{\sigma}* u))|\nabla u|dxdt\\ &&\le\|u_{0}-w(0)\|^{2}+\|w(t_{*})\|^{2}-\|w(0)\|^{2} + 2{\int}_{0}^{t_{*}}{\int}_{\Omega}h(\nabla G_{\sigma}* u)|\nabla u)|^{p-2}\nabla u\cdot\nabla w dxdt\\ &&\quad+ 2\mu{\int}_{0}^{t_{*}}{\int}_{\Omega}\left( 1-h(\nabla G_{\sigma}* u)\right)Z\cdot\nabla wdxdt; \end{array} $$

Passing to the limit in (30), we obtain (11). When t_∗ = T, (u,Z) satisfies (8), (11) and (12). Finally, by density, the test function v and w in (11) and (12) can be take from the space indicated in Definition 1.

3 Experimental results

In what follows, we provide some experimental results using our model in image restoration. The discretized version of the the PDE (7) is utilized out using standard finite difference scheme via additive operator splitting (AOS) [22].

A discrete m-dimensional image can be regarded as a vector $f\in \mathbb {R}^{N}$, whose components f_i(i = 1,⋯ ,N), display the grey values at the pixels. Pixel i represents location x_i . Let h_l denote the grid size in the l direction. We consider discrete times t_k := kτ, where $k\in \mathbb {N}$ and τ is the time step size. By ${u_{i}^{k}}$ and $\mathcal {C}_{i}^{k}$ we denote approximations to u(x_i,t_k) and $h(\nabla G_{\sigma }*u(x_{i},t_{k})|\nabla u(x_{i},t_{k})|^{p-2}+\mu (1-h(\nabla G_{\sigma }*u(x_{i},t_{k}))\frac {1}{|\nabla u(x_{i},t_{k})|}$, respectively, where the gradient is replaced by central differences.

The simplest discretization of (7) with reflecting boundary conditions is given by

$$\frac{u_{i}^{k + 1}-{u_{i}^{k}}}{\tau}=\sum\limits_{l = 1}^{m}\sum\limits_{j\in\mathcal{N}_{l}(i)}\frac{\mathcal{C}_{j}^{k}+\mathcal{C}_{i}^{k}}{2{h_{l}^{2}}}\left( u_{j}^{k + 1}-u_{i}^{k + 1}\right)$$

where N_l(i) consists of the two neighbors of pixel i along the l direction (boundary pixels may have only one neighbor). In vector matrix notation this becomes

$$\frac{u^{k + 1}-u^{k}}{\tau}=\sum\limits_{l = 1}^{m} A_{l}(u^{k})u^{k + 1}$$

where $A_{l}(u^{k})=[a^{l}_{ij}(u^{k})]$ with

$$a^{l}_{ij}(u^{k}): = \left\{\begin{array}{ll} \frac{\mathcal{C}_{i}^{k}+\mathcal{C}_{j}^{k}}{2{h_{l}^{2}}}, & j\in\mathcal{N}_{l}(i), \\ -\sum\limits_{n\in\mathcal{N}_{l}(i)}\frac{\mathcal{C}_{i}^{k}+\mathcal{C}_{n}^{k}}{2{h_{l}^{2}}}, & j=i,\\ 0,& \text{otherwise}. \end{array}\right. $$

The additive operator splitting (AOS) [22] scheme is given by

$$u^{k + 1}=\frac{1}{m}\sum\limits_{l = 1}^{m}(I-m\tau A_{l}(u^{k}))^{-1}u^{k}$$

The operators B_l(u^k) = I − mτA_l(u^k) describe one-dimensional diffusion processes along the x_l axes. Under a consecutive pixel numbering along the direction l they come down to strictly diagonally dominant tridiagonal matrices which can be efficiently inverted by the Thomas algorithm, see [22] for more details.

To compare the restoration results quantitatively, we use two error measures including peak signal-to-noise ratio (PSNR) and mean structural similarity (MSSIM), which are widely used in the image processing literature. In all these measurements, the greater number indicates better denoising performance. All the experiments are performed under Matlab R2014b with AMD FX-7500 CPU at 2.1GHz and 8 GB RAM.

For the sake of verifying the effectiveness of our proposed method for image denoising, some experiments are practiced to compare the denoising result of the proposed method with that of the other methods: PM [14] ($c(s)=\frac {1}{1+(s/K)^{2}}$), TV [16] ((2) with p = 1), Chen et al. [8], Liet al. [12], Zhou et al. [24] and AFBD [18].

In all the experiments in this paper, the time step is set as 0.2. The stopping time was chosen so that the best PSNR is obtained. In PM [14], the threshold K is calculated using Canny noise estimator [5]: K = 0.9 × mean(|∇u|). We fix p = 1.5, k = 0.02 and $\mu =\frac {1}{3}$ for our proposed model. The Gaussian convolution in our model is carried out by the discrete version with rotationally symmetric Gaussian lowpass filter of size 5 × 5 with variance 0.8.

Figure 1 shows the test images used in our experiments. The quantitative results presented in Table 1 compare the proposed method with some state-of-the-art methods using PSNR and MSSIM for 10 images illustrated in Fig. 1. The test images in this table are contaminated by AWGN with standard derivation 20 and 30. This table confirms that the proposed method has the highest PSNR and MSSIM in all cases. The comparisons of PSNR values with the comparative techniques on the image Lena are plotted in Fig. 2. We see that the proposed method always yields the highest PSNR in all noise levels.

Fig. 1

Test images used in Table 1

Table 1 Comparative quantitative results of algorithms for several images. Noise are AWGN with standard deviation 20 and 30

Fig. 2

PSNR representation as a function of noise standard deviation. Input image is Lena polluted by AWGN

Figures 3, 4, 5 make a qualitative comparison among the several denoising algorithms. Figure 3 makes a visual comparison among the denoising methods in zoomed part of Lena image. As can be seen in this figure, in PM [14] the edges are sharp while image suffers from isolated points and staircase effect. TV [16], Chen [8], Zhou [24] and AFBD [18] (p = 1) yield staircase effect. Image of AFBD [18] (p = 1.5) is blurry and contains many of noise components. The staircasing and blurring effects of Li [12] are more than the proposed method. Based on the results, the proposed method has the best visual quality.

Fig. 3

Qualitative comparative results for algorithms. Input image is Lena corrupted by AWGN with standard deviation 20. Zoomed part of Lena is shown for better visual comparison

Fig. 4

Qualitative comparative results for algorithms. Input image is Cameraman corrupted by AWGN with standard deviation 20. Zoomed part of Cameraman is shown for better visual comparison

Fig. 5

Qualitative comparative results for algorithms. Input image is Airplane corrupted by AWGN with standard deviation 25. Zoomed part of Airplane is shown for better visual comparison

Figure 4 shows a zoomed part of Cameraman image. PM [14] causes isolated points. TV [16] and Chen [8] cause staircase effect. The edges in Zhou [24] are sharper than AFBD [18] (p = 1.5), but both blur the image. The filtered images by Li [12] and AFBD [18] (p = 1) yield a little staircase effect. The proposed method removes noise while preserving the feature of image.

Figure 5 compares the methods for zoomed part of Airplane image. The edges in PM [14] are sharp, but it still suffers from isolated points. Chen [8] and Li [12] blur the images. Many noise components remain in Zhou [24] and AFBD [18] (p = 1.5). TV [16] and AFBD [18] (p = 1) still suffer from staircase effect. The proposed method outperforms other methods in term of feature preservation and improved noise reduction, see, in particular, the number region of the Airplane is much better preserved in our proposed method compared to other methods.

4 Conclusion

In this paper, we introduced an adaptive second-order PDE model which can preserve edge and alleviate the staircase effect while removing the noise. By utilizing a combination TV diffusion and p-Laplacian term an edge preserving image smoothing method is obtained. The existence of a weak solution of the proposed model was investigated. Experimental results show that the proposed method provides superior visual quality than the compared models and gets better signal-to-noise ratio and mean structural similarity with noise-free images for the final denoised images. Images expanded using our model are natural looking.