﻿ 基于Cahn-Hilliard方程的二值图像修复方法

# 基于Cahn-Hilliard方程的二值图像修复方法Binary Image Inpainting Method Based on Cahn-Hilliard Equation

Abstract: As a branch of image processing, image inpainting is widely used in computer vision, astronomy, biology and other fields. In this paper, the modified Cahn-Hilliard equation is used for binary image inpainting. The second-order finite difference method is used to discretize the equation with nonlinear term in space, and the Crank-Nicolson method is used to discretize it in time. The fast discrete cosine transform combined with fixed point iteration method is used to solve the equations in the fully discrete scheme. The numerical method of image inpainting based on this model has the advantages of few parameters, small storage and high computational efficiency. Finally, numerical experiments are given to verify the effectiveness of the proposed method.

1. 引言

2. 数值方法

$\frac{\partial u}{\partial t}=\Delta \left(-\epsilon \Delta u+{\epsilon }^{-1}{W}^{\prime }\left(u\right)\right)+\lambda \left(f-u\right),\text{ }\left(x,y,t\right)\in \Omega ×\left(0,T\right].$ (1)

$\frac{\partial u}{\partial n}=\frac{\partial }{\partial n}\left(-\epsilon \Delta u+{\epsilon }^{-1}{W}^{\prime }\left(u\right)\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,y,t\right)\in \partial \Omega ×\left(0,T\right],$ (2)

$\frac{\text{d}\text{ }\mathcal{U}}{\text{d}t}=-\epsilon {K}^{2}\mathcal{U}+{\epsilon }^{-1}K{\mathcal{W}}^{\prime }\left(\mathcal{U}\right)+\lambda \left(\mathcal{F}-\mathcal{U}\right),$ (3)

$K=-\left({I}_{{N}_{y}}\otimes \Delta {x}^{-2}{B}_{{N}_{x}}+\Delta {y}^{-2}{B}_{{N}_{y}}\otimes {I}_{{N}_{x}}\right),$

${K}^{2}=\left({I}_{{N}_{y}}\otimes \Delta {x}^{-4}{B}_{{N}_{x}}^{2}+2\Delta {x}^{-2}\Delta {y}^{-2}{B}_{{N}_{y}}\otimes {B}_{{N}_{x}}+\Delta {y}^{-4}{B}_{{N}_{y}}^{2}\otimes {I}_{{N}_{x}}\right).$

${B}_{{N}_{x}}={C}_{{N}_{x}}^{\text{T}}{\Lambda }_{{N}_{x}}{C}_{{N}_{x}}={C}_{{N}_{x}}^{-\text{1}}{\Lambda }_{{N}_{x}}{C}_{{N}_{x}},$ (4)

$K=-{\left({C}_{{N}_{y}}\otimes {C}_{{N}_{x}}\right)}^{\text{T}}\Lambda \left({C}_{{N}_{y}}\otimes {C}_{{N}_{x}}\right),$ (5)

(6)

3. 数值实验

Figure 1. Repair of truncated bar image

Figure 2. Restoration of graffiti in images

Figure 3. Watermark restoration in image

4. 结论

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