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基于一、二阶导数的图像恢复变分模型与Split Bregman算法
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"这篇研究论文提出了一种基于一阶和二阶导数的变分图像恢复模型,并结合了分裂Bregman算法,旨在解决图像去噪中的边缘保持和光滑性问题。" 在图像处理领域,变分模型常用于图像去噪,尤其是基于一阶导数的扩散模型,它们能有效地去除图像噪声同时保留边缘细节。然而,这些模型的一个主要缺点是可能会产生“阶梯效应”(staircase effect),即图像在平滑区域呈现出不连续的台阶状。为了解决这个问题,研究者们引入了包含一阶和二阶导数的混合正则化器。这样的混合正则化可以减小阶梯效应,但实现起来复杂,且计算效率较低。 本文提出的变分模型通过一阶和二阶导数的凸组合构建,旨在同时实现图像的边缘保持和光滑性。这种方法创新地结合了一阶和二阶导数的特性,优化了去噪效果,减少了实现难度并提高了计算效率。论文中还介绍了与该模型配套的快速分裂Bregman算法,这是一种优化工具,能有效地求解复杂的变分问题,进一步提升了模型的实用性。 此外,作者将这个模型扩展到了彩色图像去噪问题上,展示了其在处理多通道图像噪声时的有效性。论文最后通过对比分析,证明了所提模型相对于仅使用一阶导数的模型在去噪质量上的提升,以及分裂Bregman算法相对于其他方法在运算速度上的优势。 这篇研究论文为图像恢复提供了新的理论基础和实用方法,对于改善图像去噪效果,特别是在保留图像细节和减少计算复杂性方面,具有重要的理论价值和实际应用潜力。
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978-1-4673-0174-9/12/$31.00 ©2012 IEEE ICALIP2012
860
A Variational Model of Image Restoration Based on First and Second Order
Derivatives and Its Split Bregman Algorithm
Shixiu Zheng, Zhenkuan Pan,Guodong Wang,Xu Yan
(College of Information Engineering, Qingdao University, Qingdao, 266071)
Zhengwang_001@163.com
Abstract
The variational models of diffusion using first order
derivatives can efficiently remove the noises of images
with edge preserving property, but they usually lead to
staircase effects. This problem can be overcome via
mixed regularizers using first order and second order
derivatives, but it is complex to implement and the
computation efficiency is low. In this paper, a
variational model via convex combination of
regularizers based on first and second derivatives to
realize image denoising with edge and smoothness
preserving is proposed along with its fast Split
Bregman algorithm. They are then extended to the
problems of color image denoising. Finally, the
denoising quality of the proposed model and the
models using first order derivative is compared and
the efficiency between the Split Bregman algorithm
and the method based on gradient descent equations is
compared also.
1. Introduction
An observed scalar value image
f x , x
with
additive noise can be considered as the sum of clear
image
u
and noise
:
fu
. Restoration of
u
is a
typical ill-posed inverse problem in image processing
because there are two variables in one equation.
According to the theory of solving an ill-posed
inverse problem proposed by of Tikchonov
[1], the
above mentioned image restoration task can be
modeled as the following energy minimization
problem
2
2
1
22
u
Min E u u f dx u dx
(1)
Where, the first term denotes the fidelity between the
original image and the clear image, the second term
constrains the restored image to be smooth, which is
called regularizer, and
is a penalty parameter to
control the smoothness, which is in fact a scale
parameter in image diffusion. Unfortunately, (1) leads
to smeared edges. In 1992, Rudin, Osher and Fatemi[2]
proposed a new variational model using total
variational regularizer instead of Tikhonov regularizer
in (1) to realize edge preserving along with image
diffusion. It is named TV(Total Variation) model or
ROF model usually and the energy functional is
2
1
2
E u u f dx u dx
(2)
In 1997, Aubert and Vese[3] extended it to a
general nonlinear diffusion variational model for
image denoising with a general regularizer, which is
2
1
2
E u u f dx u dx
(3)
Where,
u
is a general regularizer corresponding
to TV, PM(Perona and Malik) model[4].etc. (3) is
equivalent with the nonlinear image diffusion model
proposed by Alvarez, Lions, Morel[5] which is a
dynamic partial differential equation. Due to the
complex of property analysis of a general regularizer
for 2D cases, Pan, Wei and Zhang[6] made a
conclusion for 1d image diffusion
0
0
0
''
x
''
x
''
x
First order forward diffusion u
No diffusion u
First order backward diffusion u
(4)
Which can be extended to 2D cases directly, although
it is not exact from the view of theoretic analysis.
Usually, the models as (3) using first order
derivative can be designed to remove noise with edge
preserving, but the restored images have staircasing
effects. In order to overcome this problem, Blomgren
et al.[7], Chen et al.[8] proposed some adaptive
nonlinear diffusion model based on (1), (2) and
adaptive varying exponentials as follows
2
1
2
Qu
E u u f dx u dx
(5)
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