Well-posedness of a Variable-Exponent Telegraph Equation Applied to Image Despeckling

TL;DR

The existence and uniqueness of weak solutions of the telegraph diffusion model with variable exponents for image despeckling are rigorously proved and the model outperforms the latter at speckle elimination and edge preservation.

Abstract

In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.

Figures (19)

• Figure 1: (a) Clean Image (f) Noisy Image ($L=10$). First row: (b)-(e) Restored Images using \ref{['diff_power_constant']} with $\nu=0$ and different values of $p$. (b) PSNR=40.53, (c) PSNR=42.98, (d) PSNR=42.72, (e) PSNR=42.08. Second row: (g)-(j) Restored Images using \ref{['diff_power_constant']} with $\nu=1$ and different values of $p$. (g) PSNR=40.76, (h) PSNR=43.47, (i) PSNR=43.22, (j) PSNR=42.58.
• Figure 2: (a) Clean Image. (f) Noisy Image ($L=10$). First row: (b)-(e) Restored Images using \ref{['diff_power_constant']} with $\nu=0$ and different values of $p$. (b) PSNR=24.51, (c) PSNR=24.52, (d) PSNR=24.43, (e) PSNR=24.30. Second row: (g)-(j) Restored Images using \ref{['diff_power_constant']} with $\nu=2$ and different values of $p$. (g) PSNR=25.20, (h) PSNR=25.26, (i) PSNR=24.90, (j) PSNR=24.35.
• Figure 3: (a) Clean Image. (f) Noisy Image ($L=10$). First row: (b)-(e) Restored Images using \ref{['diff_power_constant']} with $\nu=0$ and different values of $p$. (b) PSNR=25.17 (c) PSNR=25.59, (d) PSNR=25.58 (e) PSNR=25.40. Second row: (g)-(j) Restored Images using \ref{['diff_power_constant']} with $\nu=2$ and different values of $p$. (g) PSNR=25.23, (h) PSNR=25.61, (i) PSNR=25.56, (j) PSNR=25.42.
• Figure 4: Restored images using \ref{['diff_power_constant']} with variable exponents (Clean Image as in Figure \ref{['circle_clean1']}, Noisy Image as in Figure \ref{['circle_noisy101']}). First row ($p_0=2, \nu=0$): (a) $p_0-p_1$ ($K=0.1$) (b) $p_0-p_2$ ($K=0.1, \alpha=2$) (c) $p_0-p_3$ ($K=0.1, k=2$). Second row ($\nu=0$): (d) $p_0-p_1$ ($p_0=2.2, K=0.1$) (e) $p_0-p_2$ ($p_0=2.6, K=0.2, \alpha=2$) (f) $p_0-p_3$ ($p_0=1.9, K=0.1, k=2$). Third row ($\nu=1$): (g) $p_0-p_1$ ($p_0=2.2, K=0.1$) (h) $p_0-p_2$ ($p_0=2.6, K=0.2, \alpha=2$) (i) $p_0-p_3$ ($p_0=1.9, K=0.1, k=2$).
• Figure 5: Restored images using \ref{['diff_power_constant']} with variable exponents (Clean Image as in Figure \ref{['lake11_clean']}, Noisy Image as in Figure \ref{['lake11_noisy10']}). First row ($p_0=2, \nu=0$): (a) $p_0-p_1$ ($K=1$) (b) $p_0-p_2$ ($K=0.2, \alpha=2$) (c) $p_0-p_3$ ($K=4, k=2$). Second row ($\nu=0$): (d) $p_0-p_1$ ($p_0=2.0, K=1$) (e) $p_0-p_2$ ($p_0=1.8, K=0.2, \alpha=2$) (f) $p_0-p_3$ ($p_0=2, K=4, k=2$). Third row ($\nu=2$): (g) $p_0-p_1$ ($p_0=1.85, K=0.40$) (h) $p_0-p_2$ ($p_0=2.0, K=1, \alpha=2$) (i) $p_0-p_3$ ($p_0=2.0, K=4, k=2$).

Theorems & Definitions (8)

• Definition 3.1: Weak solution
• Theorem 3.1
• Corollary 3.2
• Definition 3.2: Weak solution
• Theorem 3.3
• Lemma 3.4
• Corollary 3.5
• Lemma 3.6