泊松噪声下的图像反卷积：SURE-LET方法

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"这篇研究论文探讨了如何在泊松噪声环境下使用SURE-LET（Stein's Unbiased Risk Estimate with Laplacian Wavelet）方法进行图像反卷积。SURE-LET算法最初是为处理高斯噪声设计的，但作者们证明了它同样适用于泊松噪声图像，并提出了一种有效的算法实现。" 在图像处理领域，图像反卷积是一个关键问题，它涉及到恢复图像的原始细节，尤其是在噪声污染严重的情况下。泊松噪声通常出现在低光照条件下的成像系统中，例如天文摄影或生物显微镜等，此时每个像素的亮度遵循泊松分布。与高斯噪声不同，泊松噪声与图像强度成比例，这使得处理起来更具挑战性。该研究提出的SURE-LET方法基于Stein's Unbiased Risk Estimate（斯特恩无偏风险估计），这是一种用于选择平滑参数的技术，可以避免过拟合或欠拟合的问题。在高斯噪声中，SURE是一个无偏估计，可以用来优化重构图像的质量。然而，在泊松噪声环境中，SURE的计算需要知道噪声的方差，这是个复杂的问题，因为泊松噪声的方差等于其均值。作者发现通过中值绝对差（MAD）方法估计的噪声方差与最佳反卷积性能（以均方误差MSE衡量）之间存在简单直接的关联。MAD是一种统计量，用于估计数据的离散程度，特别适合于非正态分布的噪声，如泊松噪声。实验表明，使用这种优化的噪声方差估计，对于自然图像的广泛范围，都能得到满意的反卷积效果。关键词：图像反卷积、SURE-LET、泊松噪声、噪声方差、中值绝对差 1. 引言图像反卷积的挑战在于在噪声存在下重建清晰图像。SURE-LET算法的扩展应用展示了其在处理泊松噪声图像时的有效性，这为低光照成像环境中的图像恢复提供了新的解决方案。通过MAD估计噪声方差，该方法能够在不完全了解噪声特性的情况下，提供可靠的反卷积性能，从而对图像恢复技术做出重要贡献。

Image Deconvolution under Poisson Noise

using SURE-LET Approach

Feng Xue, Jiaqi Liu, Gang Meng, Jing Yan and Min Zhao

National Key Laboratory of Science and Technology on Test Physics and

Numerical Mathematics, Beijing, 100076, P. R. China

ABSTRACT

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. By minimizing

Stein’s unbiased risk estimate (SURE), the SURE-LET method was ﬁrstly proposed to deal with Gaussian noise

corruption. Our key contribution is to demonstrate that the SURE-LET algorithm is also applicable for Poisson

noisy image and proposed an eﬃcient algorithm.

The formulation of SURE requires knowledge of Gaussian noise variance. We experimentally found a simple

and direct link between the noise variance estimated by median absolute diﬀerence (MAD) method and the

optimal one that leads to the best deconvolution performance in terms of mean squared error (MSE). Extensive

experiments show that this optimal noise variance works satisfactorily for a wide range of natural images.

Keywords: Image deconvolution, SURE-LET, Poisson noise, noise variance, MAD

1. INTRODUCTION

Deconvolution is a longstanding problem in many areas of signal and image processing, e.g. biomedical imag-

ing

and astronomy.

2, 3

In presence of Poisson noise, several deconvolution methods have been proposed such

as Tikhonov-Miller inverse ﬁlter and Richardson-Lucy (RL) algorithms,

4, 5

and also been comprehensively re-

viewed.

2, 6

The RL has been extensively used in many applications, but as it tends to amplify the noise after a

few iterations, several extension have been proposed. For example, wavelet-regularized RL algorithm has been

proposed by several authors.

2, 7

Recently, it is increasingly recognized that most of those algorithms proposed for Gaussian noise are ap-

plicable and eﬀective for Poisson noise.

8, 9

Both problems with diﬀerent noise distributions share very similar

deconvolution methods, e.g. sparsity-promoting regularization,

total variation (TV) regularization and the

related optimization algorithms, e.g. ADMM and augmented Lagrangian algorithms.

Note that the recently proposed SURE-LET algorithm achieves the most state-of-the-art deconvolution per-

formance under Gaussian noise assumption.

By minimizing Stein’s unbiased risk estimate (SURE),

12, 13

SURE-

LET method linearly parametrizes the deconvolution process and simpliﬁes the deconvolution as solution to a

low-order linear system of equations.

This paper is to develop a new algorithm to deal with Poisson noise,

based on the SURE-LET framework.

The key contribution of this work is to propose a simple method to

optimize the SURE as objective functional.

The paper is organized as follows. Section 2 is devoted to review the SURE-LET approach. In Section 3, we

propose a novel algorithm for Poisson noise within SURE-LET framework. Section 4 reports the experimental

results for discussion. Some concluding remarks are ﬁnally given in Section 5.

Throughout this paper, we use boldface lowercase letters, e.g. x ∈ R

, to denote N-dimensional real vectors,

where N is typically the number of pixels in an image. The n-th element of x is written as x

. The linear

transformations (matrices) R

→ R

are denoted by boldface uppercase letters, e.g. H ∈ R

M×N

. H

∈ R

N×M

denotes the transpose of matrix H.

This work was supported by the National Natural Science Foundation of China under Grant No. 61401013.

Further author information: (Send correspondence to Feng Xue)

Feng Xue: E-mail: fxue2012@gmail.com, Telephone: +86 10-88520038

Jiaqi Liu: ljq006@vip.sina.com; Gang Meng: mgang2012@yeah.net;

Jing Yan: jyan1998@126.com; Min Zhao: mzhao2012@yeah.net

AOPC 2015: Image Processing and Analysis, edited by Chunhua Shen, Weiping Yang, Honghai Liu,

Proc. of SPIE Vol. 9675 96750B-1

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