ℓ0TV-PADMM：新方法对抗图像脉冲噪声恢复

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"L0TV是一种新的图像恢复方法，专门针对脉冲噪声环境下图像处理的问题。该方法由Ganzhao Yuan和Bernard Ghanem在CVPR2015上提出，结合了低阶范数（ℓ0-norm）和全变分（Total Variation, TV）理论，旨在提高在脉冲噪声中的图像恢复质量。传统的TV模型虽然在图像平滑和去噪方面表现出色，但在处理脉冲噪声时效果并不理想。AOP（Adaptive Outlier Pursuit）是基于TV和ℓ02-norm数据保真度的先进方法，但其性能仍有提升空间。论文中，作者提出的新方法——ℓ0TV-PADMM，利用ℓ0-norm数据保真度解决基于TV的图像恢复问题，并通过等价的数学规划问题（Mathematical Programming with Equilibrium Constraints, MPEC）形式化非凸非光滑优化问题，以更有效地处理这一挑战。" 在图像处理领域，全变分（TV）模型被广泛用作正则化的先验模型，因为它能够保持图像边缘的清晰并去除平滑噪声。然而，当图像受到脉冲噪声的影响时，TV模型的效果会大打折扣。脉冲噪声通常由于传感器故障、模拟到数字转换器错误等原因在数据获取和传输过程中产生，其特点是随机且强度显著的像素值异常。去除这种噪声是图像恢复的关键任务。现有的方法，如AOP，尝试通过结合TV和ℓ02-norm数据保真度来处理这个问题，但其性能仍存在不足。在新提出的ℓ0TV-PADMM方法中，研究人员利用了ℓ0-norm，它能够更好地捕捉图像中的稀疏特性，尤其适合处理脉冲噪声。因为脉冲噪声通常导致图像中少数像素的异常，而ℓ0-norm可以有效地识别和处理这些离群值。为了求解这个非凸非光滑的优化问题，作者将其转化为一个等价的数学规划问题，即MPEC，这是一种处理混合整数非线性优化问题的有效工具。通过这种方法，他们能够设计出一种迭代算法，逐步逼近问题的全局最优解，从而在去除脉冲噪声的同时保持图像细节。 "L0TV_ A new method for image restoration in the presence of impulse noise" 提出了一个创新的图像恢复策略，将低阶范数与全变分相结合，以应对脉冲噪声造成的挑战，为图像处理领域的噪声去除提供了新的解决方案。这一工作不仅在理论上具有重要价值，也为实际应用提供了有效的工具。

TV: A New Method for Image Restoration in the Presence of

Impulse Noise

Ganzhao Yuan

and Bernard Ghanem

South China University of Technology (SCUT), P.R. China

King Abdullah University of Science and Technology (KAUST), Saudi Arabia

yuanganzhao@gmail.com, bernard.ghanem@kaust.edu.sa

Abstract

Total Variation (TV) is an eﬀective and popular

prior model in the ﬁeld of regularization-based im-

age processing. This paper focuses on TV for image

restoration in the presence of impulse noise. This

type of noise frequently arises in data acquisition and

transmission due to many reasons, e.g. a faulty sen-

sor or analog-to-digital converter errors. Removing

this noise is an important task in image restora-

tion. State-of-the-art methods such as Adaptive Out-

lier Pursuit(AOP) [

], which is based on TV with

-norm data ﬁdelity, only give sub-optimal perfor-

mance. In this paper, we propose a new method,

called

T V

-PADMM, which solves the TV-based

restoration problem with

-norm data ﬁdelity. To

eﬀectively deal with the resulting non-convex non-

smooth optimization problem, we ﬁrst reformulate it

as an equivalent MPEC (Mathematical Program with

Equilibrium Constraints), and then solve it using a

proximal Alternating Direction Method of Multipli-

ers (PADMM). Our

T V

-PADMM method ﬁnds a

desirable solution to the original

-norm optimiza-

tion problem and is proven to be convergent under

mild conditions. We apply

T V

-PADMM to the

problems of image denoising and deblurring in the

presence of impulse noise. Our extensive experi-

ments demonstrate that

T V

-PADMM outperforms

state-of-the-art image restoration methods.

1. Introduction

Image restoration is an inverse problem, which

aims at estimating the original clean image

from a

blurry and/or noisy observation

. Mathematically,

this problem is formulated as:

b = (Ku  ε

) + ε

, (1)

where

is a linear operator,

and

are the noise

vectors, and



denotes an elementwise product. Let

and

be column vectors of all entries equal to one

and zero, respectively. When

and

(or

and

), Eq (1) corresponds to the addi-

tive (or multiplicative) noise model. For convenience,

we adopt the vector representation for images, where

a 2D

M × N

image is column-wise stacked into a

vector

u ∈ R

M×N

. So, for completeness, we have

1, 0, b, u, ε

, ε

∈ R

, and K ∈ R

n×n

In general image restoration problems,

rep-

resents a certain linear operator, e.g. convolution,

wavelet transform, etc., and recovering

from

known as image deconvolution or image deblurring.

When

is the identity operator, estimating

from

is referred to as image denoising [

]. The prob-

lem of estimating

from

is called a linear inverse

problem which, for most scenarios of practical in-

terest, is ill-posed due to the singularity and/or the

ill-conditioning of

. Therefore, in order to stabilize

the recovery of

, it is necessary to incorporate prior-

enforcing regularization on the solution. Therefore,

image restoration can be modelled globally as the

following optimization problem:

min

`(Ku, b) + λ Ω(∇

u, ∇

u) (2)

where

(

Ku, b

) measures the data ﬁdelity between

and the observation

and

∇

∈ R

n×n

and

∇

∈ R

n×n

are two suitable linear transformation

matrices such that

∇

u ∈ R

and

∇

u ∈ R

com-

pute the discrete gradients of the image

along

the

-axis and

-axis, respectively

, Ω(

∇

u, ∇

)

is the regularizer on

∇

and

∇

, and

is a

In practice, one does not need to compute and store

the matrices

∇

and

∇

explicitly. Since the adjoint of the

gradient operator

∇

is the negative divergence operator

−div

i.e.,

hr, ∇

h−div

r, ui, hs, ∇

h−div

s, ui

for any

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