基于学习字典的图像去噪：稀疏与冗余表示法

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"《基于学习字典的稀疏和冗余表示图像去噪》是一篇发表在2006年IEEE Transactions on Image Processing第15卷第12期的研究论文。作者Michael Elad和Michal Aharon探讨了图像去噪问题，主要关注的是从含有零均值、均匀的高斯白噪声的给定图像中去除噪声。他们采用的方法是利用训练有素的字典来捕捉图像内容的稀疏和冗余特性。论文的核心内容是利用K-SVD（交替最小二乘与稀疏表示算法）来生成一个能够有效地描述图像内容的字典。有两种训练策略：一是直接使用被噪声污染的图像，另一种是在高质量图像数据库上进行训练。由于K-SVD在处理小图像块时存在局限性，论文提出了一种全局图像先验方法，通过定义一种强制各位置图像块保持稀疏的规则，从而将这一算法扩展到任意大小的图像。文章引入了贝叶斯重建思想，将图像看作是由多个局部区域的稀疏表示组合而成的，这使得去噪过程变得简单且高效。这种方法展示了出类拔萃的图像去噪性能，有时甚至超越了当时已发表的一些领先去噪算法。研究的关键术语包括贝叶斯重构和字典方法。这篇论文对图像去噪领域的稀疏和冗余表示理论进行了深入探索，展示了如何通过学习字典和贝叶斯分析有效地解决实际的图像去噪问题，对于理解并改进后续的图像处理技术具有重要意义。"

3736 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 12, DECEMBER 2006

Image Denoising Via Sparse and Redundant

Representations Over Learned Dictionaries

Michael Elad and Michal Aharon

Abstract—We address the image denoising problem, where

zero-mean white and homogeneous Gaussian additive noise is to

be removed from a given image. The approach taken is based

on sparse and redundant representations over trained dictio-

naries. Using the K-SVD algorithm, we obtain a dictionary that

describes the image content effectively. Two training options are

considered: using the corrupted image itself, or training on a

corpus of high-quality image database. Since the K-SVD is limited

in handling small image patches, we extend its deployment to

arbitrary image sizes by deﬁning a global image prior that forces

sparsity over patches in every location in the image. We show how

such Bayesian treatment leads to a simple and effective denoising

algorithm. This leads to a state-of-the-art denoising performance,

equivalent and sometimes surpassing recently published leading

alternative denoising methods.

Index Terms—Bayesian reconstruction, dictionary learning, dis-

crete cosine transform (DCT), image denoising, K-SVD, matching

pursuit, maximum a posteriori (MAP) estimation, redundancy,

sparse representations.

I. INTRODUCTION

N THIS paper, we address the classic image denoising

problem: An ideal image

is measured in the presence of

an additive zero-mean white and homogeneous Gaussian noise,

, with standard deviation . The measured image is, thus

(1)

We desire to design an algorithm that can remove the noise from

, getting as close as possible to the original image, .

The image denoising problem is important, not only because

of the evident applications it serves. Being the simplest possible

inverse problem, it provides a convenient platform over which

image processing ideas and techniques can be assessed. Indeed,

numerous contributions in the past 50 years or so addressed

this problem from many and diverse points of view. Statistical

estimators of all sorts, spatial adaptive ﬁlters, stochastic anal-

ysis, partial differential equations, transform-domain methods,

splines and other approximation theory methods, morphological

analysis, order statistics, and more, are some of the many direc-

tions explored in studying this problem. In this paper, we have

no intention to provide a survey of this vast activity. Instead,

Manuscript received December 18, 2005; revised May 3, 2006. The associate

editor coordinating the review of this manuscript and approving it for publica-

tion was Dr. Tamas Sziranyi.

The authors are with the Department of Computer Science, The Technion—

Israel Institute of Technology, Haifa 32000, Israel (e-mail: elad@cs.technion.

ac.il; michalo@cs.technion.ac.il).

Digital Object Identiﬁer 10.1109/TIP.2006.881969

we intend to concentrate on one speciﬁc approach towards the

image denoising problem that we ﬁnd to be highly effective and

promising: the use of

sparse and redundant representations over

trained dictionaries.

Using redundant representations and sparsity as driving

forces for denoising of signals has drawn a lot of research

attention in the past decade or so. At ﬁrst, sparsity of the unitary

wavelet coefﬁcients was considered, leading to the celebrated

shrinkage algorithm [1]–[9]. One reason to turn to redundant

representations was the desire to have the shift invariance

property [10]. Also, with the growing realization that regular

separable 1-D wavelets are inappropriate for handling images,

several new tailored multiscale and directional redundant

transforms were introduced, including the curvelet [11], [12],

contourlet [13], [14], wedgelet [15], bandlet [16], [17], and the

steerable wavelet [18], [19]. In parallel, the introduction of the

matching pursuit [20], [21] and the basis pursuit denoising [22]

gave rise to the ability to address the image denoising problem

as a direct sparse decomposition technique over redundant

dictionaries. All these lead to what is considered today as some

of the best available image denoising methods (see [23]–[26]

for few representative works).

While the work reported here is also built on the very same

sparsity and redundancy concepts, it is adopting a different

point of view, drawing from yet another recent line of work

that studies example-based restoration. In addressing general

inverse problems in image processing using the Bayesian

approach, an image prior is necessary. Traditionally, this has

been handled by choosing a prior based on some simplifying

assumptions, such as spatial smoothness, low/max-entropy,

or sparsity in some transform domain. While these common

approaches lean on a guess of a mathematical expression for the

image prior, the example-based techniques suggest to learn the

prior from images somehow. For example, assuming a spatial

smoothness-based Markov random ﬁeld prior of a speciﬁc

structure, one can still question (and, thus, train) the derivative

ﬁlters to apply on the image, and the robust function to use in

weighting these ﬁlters’ outcome [27]–[29].

When this prior-learning idea is merged with sparsity and re-

dundancy, it is the dictionary to be used that we target as the

learned set of parameters. Instead of the deployment of a pre-

chosen set of basis functions as the curvelet or contourlet would

do, we propose to learn the dictionary from examples. In this

work we consider two training options: 1) training the dictio-

nary using patches from the corrupted image itself or 2) training

on a corpus of patches taken from a high-quality set of images.

This idea of learning a dictionary that yields sparse represen-

tations for a set of training image-patches has been studied in

a sequence of works [30]–[37]. In this paper, we propose the

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