Well-posedness study of a non-linear hyperbolic-parabolic coupled system applied to image speckle reduction

TL;DR

This paper analyzes a non-linear hyperbolic-parabolic coupled system for image despeckling, establishing its well-posedness and demonstrating its effectiveness through numerical experiments on noisy images.

Contribution

It introduces a new coupled system based on telegraph diffusion for despeckling and proves its well-posedness using Schauder's fixed point theorem.

Findings

The model effectively reduces speckle noise in images.

Numerical results outperform recent models.

The system's well-posedness is rigorously established.

Abstract

In this article, we consider a non-linear hyperbolic-parabolic coupled system based on telegraph diffusion framework applied to image despeckling. A separate equation is used to calculate the edge variable, which improves the quality of the despeckled images. A well-posedness result of the proposed coupled system is settled via Schauder's fixed point theorem. Numerical experiments are reported to illustrate the effectiveness of the proposed model, with recently developed models, over a set of gray level test images contaminated by speckle noise.

Tables

Table 1. Table 1: Left table: Comparison of MSSIM and PSNR values of despeckled images. Right table: Parameter values for the numerical experiments.

Image $L$ Shan Model[26] TDM [22] Proposed Model MSSIM PSNR MSSIM PSNR MSSIM PSNR Circle 1 0.9581 34.29 0.9643 34.69 0.9651 35.43 3 0.9734 38.09 0.9771 39.52 0.9782 40.00 5 0.9764 39.35 0.9805 40.72 0.9810 41.23 Texture 1 0.8125 27.61 0.8355 27.71 0.8360 27.83 3 0.8782 30.83 0.8925 31.05 0.8967 31.31 5 0.8979 31.87 0.9109 32.15 0.9164 32.47 Peppers 1 0.5827 17.56 0.5895 17.64 0.5905 17.83 3 0.7018 22.46 0.7019 22.55 0.7061 22.85 5 0.7155 23.73 0.7334 24.22 0.7395 24.67

Image $L$ Shan[26] TDM [22] Proposed $\alpha$ $\beta$ $\gamma$ $\nu$ $K$ $\gamma$ $\alpha$ $\beta$ $\iota$ $\nu$ Circle 1 1.5 2 10 1 1 1 1.5 1.8 2.5 0.1 3 1.5 2 10 1 1 2 1.7 2 2.5 0.1 5 2 2.25 5 1 1 2 1.7 2.2 2 0.1 Texture 1 1.5 1.8 2 1.5 2 1 2 1 3 0.1 3 1.8 2 2 1.5 2 2 2 1 3 0.1 5 1.8 2 2 1.5 2 5 2.5 1 3 0.1 Peppers 1 1.5 2 1 1.5 2 1 2 1 3 0.1 3 1.5 2 1 1.5 2 2 2 1 3 0.1 5 2 2.4 2 1.5 2 2 2.5 1 3 0.1

Equations

$$J=I\eta ,$$

$$I_t=\text{div}(g(t,x)\nabla I)+\lambda f(J,I)\text{in}{\Omega }_T:=(0,T)\times \Omega ,$$

$$I_{tt}+\gamma I_t-\text{div}\left(\frac{\nabla I}{1+|\frac{\nabla I}{K}{|}^2}\right)=0,\text{in}{\Omega }_T,$$

$$I_{tt}+\gamma I_t=\text{div}\left(\theta (I)\frac{\nabla I}{|\nabla I|}\right)-\lambda \left(1-\frac{J}{I}\right),\text{in}{\Omega }_T,$$

$${\partial }_{n}I=0,\text{in}\partial {\Omega }_T:=(0,T)\times \partial \Omega ,$$

$$I(0,x)=J(x),I_t(0,x)=0,\text{in}\Omega ,$$

$$I_{tt}+\gamma I_t-\text{div}\left(g\left(I_{\xi },|\nabla I_{\xi }|\right)\nabla I\right)=0,\text{in}{\Omega }_T,$$

$$\displaystyle g\left(I_{\xi},|\nabla I_{\xi}|\right)=\dfrac{2|I_{\xi}|^{\nu}}{\big{(}M^{I}_{\xi}\big{)}^{\nu}+|I_{\xi}|^{\nu}}\cdot\dfrac{1}{1+\left(\frac{|\nabla I_{\xi}|}{K}\right)^{2}}\,.$$

$$\displaystyle I_{tt}+\gamma I_{t}-{\rm div}\Big{(}\frac{s^{\alpha}}{1+s^{\alpha}}\frac{1}{1+\iota|u_{\xi}|^{\beta}}\nabla I\Big{)}=0,\hskip 71.13188pt{\rm in}~{}~{}~{}\Omega_{T}\,,$$

$$u_t=h(|\nabla I_{\xi }|)-u+\frac{{\nu }^2}{2}\Delta u,\text{in}{\Omega }_T,$$

$$\{\begin{array}{l}I(0,x)=I_0(x),I_t(0,x)=0,u(0,x)=G_{\xi }\ast |\nabla I_0{|}^2\mathrm{in}\Omega ,\\ {\partial }_{n}I=0={\partial }_{n}u\mathrm{on}\partial {\Omega }_T,\end{array}$$

$$W_1(0,T)$$

$$W_2(0,T)$$

$$\displaystyle\big{\langle}I_{tt},\phi\big{\rangle}+\int_{\Omega}I_{t}\phi\,dx+\int_{\Omega}\frac{s^{\alpha}}{1+s^{\alpha}}\frac{1}{1+|u_{\xi}|^{\beta}}\nabla I\cdot\nabla\phi\,dx=0\,,$$

$${\int }_{\Omega }{u}_t\varphi dx+\frac{1}{2}{\int }_{\Omega }\nabla u\cdot \nabla \varphi dx+{\int }_{\Omega }u\varphi dx={\int }_{\Omega }h(|\nabla I_{\xi }|)\varphi dx.$$

$$0\le h(\tilde{u})\le 1\forall \tilde{u}\in {\mathbb{R}}^{+}.$$

$${\mathcal{B}}_{M_1,M_2}=\{\begin{array}{l}\bar{I}\in W_1(0,T):\Vert \bar{I}{\Vert }_{L^{\infty }(0,T;H^1)}+\Vert {\bar{I}}_t{\Vert }_{L^{\infty }(0,T;L^2)}\le M_1\Vert I_0{\Vert }_{H^1},\\ 0<\rho \le \bar{I}(t,x)\mathrm{for}a.e.(t,x)\in {\Omega }_T,\\ \bar{u}\in W_2(0,T):\Vert \bar{u}{\Vert }_{L^{\infty }(0,T;L^2)}+\Vert {\bar{u}}_t{\Vert }_{L^{\infty }(0,T;L^2)}\le M_2\Vert I_0{\Vert }_{L^2}.\end{array}$$

$$\displaystyle I_{tt}+I_{t}-{\rm div}\big{(}\bar{g}(t,x)\nabla I\big{)}=0\hskip 128.0374pt{\rm in}~{}~{}~{}\Omega_{T}\,,$$

$$u_t=h(|\nabla {\bar{I}}_{\xi }|)-u+\frac{1}{2}\Delta u\text{in}{\Omega }_T,$$

$$\displaystyle\bar{g}(t,x)\equiv g_{\bar{I},\bar{u}}(t,x):=\frac{|\bar{I}_{\xi}|^{\alpha}}{\big{(}M_{\xi}^{\bar{I}}\big{)}^{\alpha}+|\bar{I}_{\xi}|^{\alpha}}\cdot\frac{1}{1+|\bar{u}_{\xi}|^{\beta}}\,.$$

$$\mathrm{i})0<\kappa \le \bar{g}\le 1,$$

$$\mathrm{ii})|{\bar{g}}_t|\le C,$$

$$\displaystyle\|u\|^{2}_{H^{1}}+\int_{0}^{t}\|u_{t}\|^{2}_{L^{2}}\,ds\leq C\big{(}1+t\,|\Omega|\big{)}\,.$$

$$\displaystyle\|u\|^{2}_{H^{1}}+\|u_{t}\|^{2}_{L^{2}}\leq C\,e^{t}\big{(}1+t\,|\Omega|\big{)}\,.$$

$$\displaystyle W_{0}=\Bigg{\{}$$

$$\Vert v{\Vert }_{L^{\infty }(0,T;H^1)}+\Vert v_t{\Vert }_{L^{\infty }(0,T;L^2)}\le C\Vert I_0{\Vert }_{H^1};$$

$$\displaystyle\hskip 56.9055pt~{}~{}0<\rho\leq w(t,x)~{}{\rm for~{}a.e.}~{}(t,x)\in\Omega_{T}\,,~{}~{}\text{and}~{}~{}(w,v)~{}{\rm satisfies}~{}\eqref{mainc}\Bigg{\}}\,.$$

$$\mathcal{P}:$$

$$(w,v)\mapsto (I_w,u_v).$$

$$w_k\to w,v_k\to v\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$G_{\xi }\ast w_k\to G_{\xi }\ast w\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$|G_{\xi }\ast w_k{|}^{\alpha }\to |G_{\xi }\ast w{|}^{\alpha }\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$\displaystyle\dfrac{|G_{\xi}\ast w_{k}|^{\alpha}}{\big{(}M_{\xi}^{w_{k}}\big{)}^{\alpha}+|G_{\xi}\ast w_{k}|^{\alpha}}\rightarrow\dfrac{|G_{\xi}\ast w|^{\alpha}}{\big{(}M_{\xi}^{w}\big{)}^{\alpha}+|G_{\xi}\ast w|^{\alpha}}\hskip 5.69046pt{\rm in}~{}~{}L^{2}(0,T;L^{2})\hskip 5.69046pt{\rm and~{}a.e.~{}on}~{}~{}\Omega_{T}\,,$$

$${\partial }_{x_i}{G}_{\xi }\ast w_k\to {\partial }_{x_i}{G}_{\xi }\ast w(i=1,2)\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$h(|\nabla G_{\xi }\ast w_k|)\to h(|\nabla G_{\xi }\ast w|)\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$|G_{\xi }\ast v_k|\to |G_{\xi }\ast v|\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$\frac{1}{1+|G_{\xi }\ast v_k{|}^{\beta }}\to \frac{1}{1+|G_{\xi }\ast v{|}^{\beta }}\mathrm{in}{L}^2(0,T;L^2)\mathrm{and}a.e.\mathrm{on}{\Omega }_T,$$

$$I_k\to I,u_k\to u\text{weakly}\ast \text{in}{L}^{\infty }(0,T;H^1),$$

$$I_k\to I,u_k\to u\text{in}{L}^2(0,T;L^2),$$

$${\partial }_t{I}_k\to {\partial }_{t}I,{\partial }_t{u}_k\to {\partial }_{t}u\text{weakly}\ast \text{in}{L}^{\infty }(0,T;L^2),$$

$${\partial }_{tt}{I}_k\to {\partial }_{tt}I\text{weakly}\ast \text{in}{L}^2(0,T;(H^1{)}^{\prime }).$$

Full text

K****eywords Speckle noise  $\cdot$ Despeckling  $\cdot$ Telegraph diffusion equation  $\cdot$ Coupled System  $\cdot$ Well-posedness  $\cdot$ Schauder fixed point theorem.

1 Introduction

Beginning with the Perona-Malik model [23], non-linear partial differential equations (PDEs) are extensively used to develop noise reduction models. Due to the availability of well established numerical schemes and theoretical properties, PDE based image processing is an exciting research area for real-life application purpose as well as for the theoretical study. In the real situation, images are often degraded by different types of noises, e.g., additive, multiplicative, or mixed nature. Hence the noise extraction is a very initial stage for high-level image analysis. In this work, we only consider the multiplicative speckle [22] noise removal process. A Mathematical representation for a degraded image affected by speckle noise[7] can be expressed as

[TABLE]

where $J$ is the noisy image, $I$ is the noise-free image, and $\eta$ signifies the speckle-noise process.

In general, the speckle noise process $\eta$ is Gamma$(L,L)$ distributed, where $L\in{\rm I\!N}$ is the the number of looks corresponding to the noise level in the corrupted images [2, 10, 19]. To remove speckle based noise in the images, different types of PDE based models are proposed and resulted in significant momentum both in the development of theoretical as well as numerical aspects of the problems. Most popular PDE based approaches are anisotropic diffusion-based methods [13, 14, 16, 26, 31, 33, 34], and variational methods [3, 6, 11, 15, 17, 18, 20, 25, 27]. Most of the above models take the generalized form

[TABLE]

with the appropriate initial and boundary conditions. Here $\Omega$ is the domain of the original image $I$ and the observed noisy image $J,$ $T$ is a specified time, $\lambda$ is the weight parameter, div and $\nabla$ represents the divergence and gradient operator respectively. The source term $f(J,I)$ is derived from the variational model approach [3, 6, 18, 25]. In (1.1), $g(t,x)$ signifies the degree of denoising which preserves the image characteristics, e.g., textures and edges in the noise removal process. All the above-discussed PDE based models are parabolic type.

Later, V. Ratner, and Y. Zeevi [24] introduces the idea of hyperbolic PDE for additive noise removal process. By considering the image as an elastic sheet, the authors in [24] suggest the following telegraph diffusion equation (TDE) based model

[TABLE]

where $\gamma$ is the damping parameter and $K$ is a threshold constant. Even though the TDE model can effectively preserve the sharp edges but failed to produce satisfactory smoothing in the presence of a large level of noise. To overcome this issue, several non-linear telegraph diffusion-based method have been proposed [4, 12, 28, 30, 32]. However, in spite of their impressive applications in the field of additive noise removal process, hyperbolic PDE based approaches have not successfully used for speckle noise removal process. Recently Sudeb et al. suggest a couple of hyperbolic PDE based models [21, 22] for speckle noise removal process. The authors in [21] developed a model in a telegraph total variation framework as

[TABLE]

where $\theta$ is the fuzzy edge detector function [5]. In [22], the authors developed a model in a telegraph diffusion based framework of the form

[TABLE]

with the similar initial and boundary conditions as in [21], where the diffusion control function $g$ is given by

[TABLE]

In the above, $I_{\xi}=G_{\xi}\ast I$, $M^{I}_{\xi}=\underset{x\in\Omega}{\text{max}}|I_{\xi}(t,x)|,$ $\nu\geq 1,$ $\xi>0,$ and $``\ast"$ represents convolution in $x$ only and $G_{\xi}$ is a two dimensional Gaussian kernel.

To the best of our knowledge, most of the PDE based models for speckle noise removal are single and parabolic types. Inspired by the ideas of [14, 22], we propose the following improved nonlinear and coupled hyperbolic-parabolic model

[TABLE]

with $s:=\frac{|I_{\xi}|}{M_{\xi}^{I}}\in[0,1]$. In the above, $I_{0}$ is the observed noisy image, $\alpha\geq 1,\beta\geq 1,\gamma>0,\nu>0,$ and $\iota>0$ are constants. $\Delta$ is Laplace operator, $h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a bounded, Lipschitz continuous function. Moreover, $u$ represents the edge strength at each scale. Here we utilize an extra equation to calculate the edge variable, which improves the present model over our previous model [22]. For simulation purpose, we opt an explicit numerical method to solve the present model and then apply it on different types of gray level test images. A comparison study regarding the quality of the despeckled image is carried out with recently developed models [22, 26]. Moreover, we compare the quantitative and qualitative results at different noise levels.

The rest of the paper is organized as follows. In section 2, we study the wellposedness of the proposed model. Section 3 describes the numerical implementation and despeckling performance of the proposed model. We conclude the paper in Section 4 with a scope on future work.

2 Existence and Uniqueness of weak solution

This section is devoted to the wellposedness result of the proposed system (1.2)-(1.4). Due to the nonlinearity in the system (1.2)-(1.4), we first consider the associated linearized problem and then use Schauder’s fixed-point theorem [8] to complete the proof. Without loss of generality, we assume that $\gamma=1,\iota=1,$ and $\nu=1$ in the equations (1.2) and (1.3).

2.1 Technical framework and statement of the main result

Throughout this article, we consider $C>0$ as a generic constant. By $(L^{p},\|\cdot\|_{L^{p}})$ with $1\leq p\leq\infty$, we denote the standard spaces of $p$-th order integrable functions on $\Omega$. Moreover, for $r\in\mathbb{N}$ we write $(H^{r},\|\cdot\|_{H^{r}})$ as the usual Sobolev spaces on $\Omega$, and $(H^{1})^{\prime}$ as the dual space of $H^{1}$. We consider the solution space $W(0,T)$ for the underlying problem (1.2)-(1.4) as $W(0,T)=W_{1}(0,T)\times W_{2}(0,T)$, where

[TABLE]

Definition 2.1** (Weak solution)**

A pair $(I,u)$ is said to be a weak solution of (1.2)-(1.4), if

• a)

$I\in W_{1}(0,T),~{}u\in W_{2}(0,T)$* and (1.4) holds.*

• b)

For all $\phi\in H^{1}$ and a.e $t\in(0,T)$, there hold

[TABLE]

Theorem 2.1

The system (1.2)-(1.4) admits a unique weak solution $(I,u)\in W$ in the sense of Definition 2.1, provided the following two conditions hold:

1. A.1

$I_{0}\in H^{2}$* satisfying $0<\rho:=\underset{x\in\Omega}{\inf}I_{0}(x)$.* 2. A.2

$h:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$* is a bounded, Lipschitz continuous function with Lipschitz constant $c_{h}$ such that*

[TABLE]

2.2 Linearized problem $\&$ its Well-posedness

For any positive constants $M_{1},M_{2}>0$, define the convex set

[TABLE]

For any fixed $(\bar{I},\bar{u})\in\mathcal{B}_{M_{1},M_{2}}$, consider the linearized problem:

[TABLE]

with the condition (1.4), where the function $\bar{g}$ is given by

[TABLE]

Since $(\bar{I},\bar{u})\in\mathcal{B}_{M_{1},M_{2}}$, a similar argument as in the proof of [22, Claim $2.1$] revels that

[TABLE]

where $\kappa,C>0$ are constants depending only on $G_{\xi},I_{0},M_{1},M_{2},\beta$, $\alpha$ and $\rho$. Hence, thanks to the classical Galerkin method [8], one can show that there exists a unique weak solution $(I,u)\in W(0,T)$ of the linearized problem (2.1)-(2.2) with the condition (1.4).

Lemma 2.2

The unique solution $(I,u)\in W(0,T)$ of the linearized problem (2.1)-(2.2) with the condition (1.4) satisfies the following:

• a)

$\|I\|_{L^{\infty}(0,T;H^{1})}+\|I_{t}\|_{L^{\infty}(0,T;L^{2})}\leq C\|I_{0}\|_{H^{1}}$*, *

• b)

$\int_{0}^{T}\|I_{tt}\|_{(H^{1})^{\prime}}^{2}\,dt\leq CT\|I_{0}\|_{H^{1}}^{2}$*, *

• c)

$\|u\|_{L^{\infty}(0,T;H^{1})}+\|u_{t}\|_{L^{\infty}(0,T;L^{2})}\leq C\|I_{0}\|_{H^{1}}$,

where $C>0$ is a constant, depends only on $G_{\xi},I_{0},h,M_{1},M_{2},\alpha,\beta$ and $\rho$.

Proof: Since $\|\bar{u}_{t}\|_{L^{\infty}(0,T;L^{2})}\leq C\|I_{0}\|_{H^{1}}$, by following computations as in Sudeb at el. [22, Lemma 3.2], one can show the validation of the estimates ${\rm a)}$ and ${\rm b)}$ of Lemma 2.2. To prove ${\rm c)}$, we proceed as follows: multiply (2.2) by $u_{t}$, integrate by parts over $\Omega$, use Cauchy-Schwarz and Young’s inequalities, and then integrate w.r.t time between [math] to $t$. We have, for a.e. $t\in(0,T)$

[TABLE]

Moreover, since $u_{0}\in H^{2}$ and $\bar{h}_{t}\in L^{\infty}(0,\mathcal{T};L^{2})$, by regularity theory [8], $u_{t}\in L^{\infty}(0,\mathcal{T};L^{2})$ with

[TABLE]

Hence ${\rm c)}$ of Lemma 2.2 follows from (2.4).

2.3 Proof of Theorem 2.1

As mentioned earlier, we show the well-posedness of the system (1.2)-(1.4) via Schauder’s fixed-point theorem. To do so, we introduce a non-empty, convex and weakly compact subset $W_{0}$ of $W(0,T)$ defined by

[TABLE]

Consider a mapping

[TABLE]

If we show that the mapping $\mathcal{P}:(w,v)\rightarrow(I_{w},u_{v})$ is weakly continuous from $W_{0}$ into $W_{0}$, then by Schauder’s fixed-point theorem, there exists $(w,v)\in W_{0}$ such that $(w,v)=\mathcal{P}(w,v)$ . In other words, the coupled system (1.2)-(1.3) has a weak solution. In order to prove weak continuity of $\mathcal{P}$, let $(w_{k},v_{k})$ be a sequence that converges weakly to some $(w,v)$ in $W_{0}$ and let (I_{k},u_{k})=\big{(}I_{w_{k}},u_{v_{k}}\big{)}. We have to show that $\mathcal{P}(w_{k},v_{k}):=(I_{k},u_{k})$ converges weakly to $\mathcal{P}(w,v):=(I_{w},u_{v})$.

Thanks to Lemma 2.2, one can use classical results of compact inclusion in Sobolev spaces [1] to extract subsequences $\{w_{k_{n}}\}$ of $\{w_{k}\}$, $\{v_{k_{n}}\}$ of $\{v_{k}\}$, $\{I_{k_{n}}\}$ of $\{I_{k}\}$ and $\{u_{k_{n}}\}$ of $\{u_{k}\}$, still denoted by same sequences $\{w_{k}\}$, $\{v_{k}\}$, $\{I_{k}\}$ and $\{u_{k}\}$, such that for some $(I,u)\in W_{0}$, the followings hold as $k\rightarrow\infty$:

[TABLE]

In view of the above convergences, one can pass to the limit in (2.1)-(2.2) and obtain $(I,u)=\mathcal{P}(w,v)$. Moreover, since the solution of (2.1)-(2.2) is unique, the whole sequence $(I_{k},u_{k})=\mathcal{P}(w_{k},v_{k})$ converges weakly in $W_{0}$ to $(I,u)=\mathcal{P}(w,v)$. Hence $\mathcal{P}$ is weakly continuous. Therefore, the problem (1.2)-(1.4) admits a weak solution.

Uniqueness of weak solution: To prove the uniqueness of weak solutions of the underlying problem (1.2)-(1.4), we use here a standard methodology [8]. Let $(I_{1},u_{1})$ and $(I_{2},u_{2})$ be two weak solutions of (1.2)-(1.4). Then, we have

[TABLE]

where $I=I_{1}-I_{2}$ and $u=u_{1}-u_{2}$. It suffices to show that $(I,u)\equiv(0,0)$ . To verify this, fix $0<s<T$, and set for $i=1,2$,

[TABLE]

Note that, for $t\in(0,T)$,

[TABLE]

Set $v=v_{1}-v_{2}$. Then $v(s,\cdot)=0$. Now one can follow the same argumentation as in [22, Section $3.3$] to arrive at

[TABLE]

Like in (2.3), there exist positive constants $\kappa_{1},C_{1}>0$ such that

[TABLE]

Moreover, by using property of convolution and the positive lower bound $\rho$ of the solutions $I_{i}$, we get

[TABLE]

Thus we have, from (2.3)

[TABLE]

where in the last inequality, we have used the fact that $\alpha\geq 1$. Set

[TABLE]

Then, by using a similar argument as in [22, Section $3.3$], we obtain

[TABLE]

Choose $T_{1}$ sufficiently small such that $C-\tilde{C}T_{1}>0$. Then, for $0<s\leq T_{1},$ we have

[TABLE]

Now, by multiplying (2.6) by $u$ and integrating over $\Omega$, we have

[TABLE]

Since $h$ is Lipschitz continuous, by using Young’s inequality for convolution, we see that

[TABLE]

Thus, we have, for $0<s\leq T_{1}$,

[TABLE]

Adding (2.11) and (2.12), we finally get , for $0<s\leq T_{1}$,

[TABLE]

Hence by Gronwall’s lemma, we see that $(I,u)\equiv(0,0)$ on $[0,T_{1}]$. We repeatedly use the above argument on the intervals $(T_{1},2T_{1}]$, $(2T_{1},3T_{1}],\ldots$ step by step, and arrive at the conclusion that $I_{1}=I_{2}$ and $u_{1}=u_{2}$ on $(0,T)$. This completes the proof of Theorem 2.1.

For any weak solution $(I,u)$ of (1.2)-(1.4), we next show the boundedness of $I$ under the assumption that initial image $I_{0}$ has a finite upper bound, whose proof follows from the proof of [22, Lemma $3.3$].

Lemma 2.3

Let $(I,u)$ be a weak solution of the system (1.2)-(1.4), and $\varrho:=\underset{x\in\Omega}{\sup}I_{0}(x)<\infty$. Then

[TABLE]

3 Numerical method and Experimental Results

In this section, we show the image despeckling performance of the suggested model over two existing approaches [22, 26]. To solve the model (1.2)-(1.4) numerically, we opt an explicit finite difference scheme. We replace the derivative terms in the model (1.2)-(1.4) using the following finite difference formulas:

[TABLE]

In the above, $\tau$ resp. $\tilde{h}$ denotes the time step size resp. the spatial step size. $I^{n}_{i,j}=I(t_{n},x_{i},y_{j})$ where $x_{i}=i\tilde{h}~{}(i=0,1,2...,N)$, $y_{j}=j\tilde{h}~{}(j=0,1,2...,M)$, $t_{n}=n\tau~{}(n=0,1,2,\ldots)$ where $n$ is the number of iterations and $M\times N$ is the image dimension. Then, the discrete form of the equation (1.2) could be written as

[TABLE]

where

[TABLE]

Moreover, $u(t_{n},x_{i},y_{j})=u^{n}_{i,j}$ is calculated from the discretized equation of (1.3) as follows

[TABLE]

where h_{i,j}^{n}=h\big{(}|\nabla(G_{\xi}\ast I^{n}_{i,j})|\big{)}. We choose the function $h$ as $h(\theta)=\epsilon+\text{min}\{\theta^{2},K\}$ for numerical experiments, where $K$ is square of the maximum gray level value of the image $I$ and $\epsilon>0$ is a very small number. The boundary and initial conditions are given as follows:

[TABLE]

We choose similar boundary conditions for edge variable $u$, and $u_{i,j}^{0}=G_{\xi}\ast|\nabla I^{0}_{i,j}|^{2}.$ We start the simulation with the initial value $I_{0}$ and utilize the equations (3.1) and (3.2) repeatedly to find a sequence of values of ${I(t,x)};t>0$, which represents the filtered versions of $I_{0}$. We made a stopping criterion for the noise elimination process when the best PSNR [9] value for the restored image $I(t,x)$ is reached.

We perform all the experiments on three standard test images 1, which are initially degraded with different level of speckle noise. All the numerical experiments are performed under windows $7$ and MATLAB version $R2018b$ running on a desktop with an Intel Core $i5$ dual-core CPU at $2.53$ GHz with $4$ GB of memory. In this process, we use same numerical scheme as done it for the proposed model to discretize the considered existing models. We choose an uniform time step size $\tau=0.2$, spatial step size $\tilde{h}=1$, and $\xi=1$ for each models.

To compare the quantitative results, we compute the values of the two standard parameters PSNR[9] and MSSIM [29]. A higher numerical value of MSSIM and PSNR suggests that the despeckled image is closer to the noise-free image. Apart from the despeckled image, for qualitative observations, we compute the 2D contour plot, 3D surface plot for the better visualization of the computational results.

In figures 2-4, we represent the restored results of a Texture image which is contaminated by multiplicative speckle noise with $L=\{1,3,5\}$. From figure 2(b), we can see that the Shan model failed to preserve the fine edges for very high noise level. TDM model works better than the Shan model, but the present model preserves the fine edges better than Shan and TDM models.

In figures 5-7, we represent the reconstructed results of a Peppers image (Natural Image) which is corrupted by speckle noise with $L=\{1,3,5\}$. From figure 5, we see that the present model leave less speckle than the other two models.

To check the more reconstruction ability of the present model in figures 8- 12 illustrate the qualitative results of a Circle image (Synthetic Image) which is corrupted by speckle noise with $L=\{1,3,5\}$.

In the figures 8- 10 we demonstrate the despeckled images, and in the figures 11-12 we illustrate the contour maps and 3D surface plots when the image is corrupted by $L=5$. One can observe that from the contour maps, and 3D surface plots, Shan and TDM models left some speckles in the homogeneous regions, but the present model produces fewer artifacts with better edge preservation.

Computational Values of PSNR and MSSIM are presented in the Table 1. The highest values of PSNR and MSSIM for each noise level clearly shows that the suggested model is better than the other two models.

Conclusively, summarize the quantitative and qualitative results, we can confirm that the present model performance better than the other discussed models.

4 Conclusion

In this work, we present a non-linear hyperbolic-parabolic coupled system applied to image despeckling. Such a improve method preserves the image characteristics in the noise removal process. To the best of our knowledge, coupled hyperbolic-parabolic PDE based model has not been used before for image speckle reduction. Moreover, we establish the well-posedness of the present model, show the boundedness of the weak solution. We compare the experimental results with two recently developed models and arrive at the conclusion that the proposed model well recovered the corrupted images without introducing undesired artifacts than that of existing models.