非高斯噪声背景下的图像恢复研究

Non-Gaussian noise removal is a very challenging problem since this kind of noise may contaminate images in a totally different way from Gaussian noise. Image recovery under non-Gaussian noise has attracted wide attention in the field of applied mathematics and information. This research proposal focuses on the development of image recovery models for multiplicative noise and impulse noise denoising and deblurring. The importance of multiplicative noise relies on many aspects. As is well-known, they are commonly found many image processing applications in the real world, such as laser images, microscope images, medical ultrasonic images, and SAR images,etc. Unlike additive noise, these kinds of noise are much more difficult to be removed from the corrupted image, mainly not only because of their multiplicative nature, but also because of their distributions which are generally not Gaussian. Impulse noise is often found in digital storage and transmission. In this proposal, we propose a variational approach based on the theory of image recovery, optimization and statistic. Indeed, we will first examine a general method to deal with the non-convex model revoked by multiplicative noise and the non-differential model revoked by impulse noise. We then derive new variational methods for blurred image corrupted by multiplicative noise or impulse noise and study how to efficiently solve them. We also study the blind deconvolution problem and convex relaxation problem. Finally, we study sparse representation in image recovery and the statistical analysis of dictionary learning.

非高斯噪声的去除是一个很有挑战性的问题，因为这种噪声的特性与通常研究的高斯噪声不同。非高斯噪声背景下图像恢复问题在应用数学领域及信息领域引起了广泛的关注。本项目将集中研究乘性噪声和脉冲噪声背景下的图像恢复问题：去噪和去模糊。乘性噪声存在于很多图像处理应用领域。例如激光图片、显微镜图片、医学超声图片及雷达图片。不同于加性噪声的是，乘性噪声很难去除。原因主要是乘性噪声污染图片的方式完全不同于高斯噪声。脉冲噪声常见于信息的存储和传输中。本项目将基于图像恢复、优化及统计理论来研究用于乘性噪声和脉冲噪声图像恢复的变分模型。首先我们研究用于乘性噪声去除的非凸模型及用于脉冲噪声去除的不可微模型，并设计有效的算法来求解这些模型。然后提出新的算法来解决乘性噪声背景下的图像去模糊问题。接着我们将研究非高斯噪声的盲反卷积问题和乘性噪声去除模型的凸松弛问题，最后我们将研究基于稀疏表示图像恢复及字典学习的统计分析。

非高斯噪声的去除是一个很有挑战性的问题，因为这种噪声的特性与通常研究的高斯噪声不同。非高斯噪声背景下图像恢复问题在应用数学领域及信息领域引起了广泛的关注。本项目集中研究乘性噪声和脉冲噪声背景下的图像恢复问题：去噪和去模糊。乘性噪声存在于很多图像处理应用领域。例如激光图片、显微镜图片、医学超声图片及雷达图片。不同于加性噪声的是，乘性噪声很难去除。原因主要是乘性噪声污染图片的方式完全不同于高斯噪声。本项目中我们基于图像恢复、优化及统计理论研究用于乘性噪声和脉冲噪声图像恢复的变分模型。首先我们研究了用于乘性噪声去除的非凸模型及用于脉冲噪声去除的不可微模型，并设计有效的算法来求解这些模型。然后提出了新的算法来解决乘性噪声背景下的图像去模糊问题。接着我们研究了非高斯噪声的盲反卷积问题和乘性噪声去除模型的凸松弛问题，最后我们研究了基于稀疏表示图像恢复及字典学习的统计分析。 本项目按计划顺利地完成了任务。主要成果为: 1). 研究了multiplicative噪音的凸化问题 2) 研究了字典学习方法对Poisson噪音的恢复问题 3) 研究了multiplicative噪音的二步法问题 4) 研究了Poisson与multiplicative噪音下的图像分割问题 5) 研究了字典学习去JPEG压缩图像中的artifact 6) 研究了单幅图像变分去雾问题 7) 研究了Rician噪音下的图像恢复问题 8) 研究了加权差图像恢复问题 9) 研究了Cauchy噪音下的图像恢复问题 10) 研究了基于稀疏表示的图像填充问题.