基于分析算子正则化模型的惩罚函数学习研究

This proposal aims to simultaneously train linear filters and penalty functions in an analysis prior based regularization model. In this proposal, we will focus on the study of flexible training framework for this task, and finally obtain analysis prior models, which can achieve better performance for classic image restoration problems than existing analysis models. This proposal can be divided into three categories. Firstly, existing analysis prior models typically make use of functions like f(x)=x^0.5, which are unable to adapt to different linear filters and limit the performance of these models. To solve this problem, we first introduce a parameterized nonconvex function with a single minimum at zero, which can be optimized in a supervised training scheme, together with the linear filters. Therefore, the trained filters will correspond to different penalty functions. Secondly, it should be noted that as penalty functions with a single minimum at zero will always impose certain penalty for any filter response, they will always lead to suppression of the filter response (i.e., pure image smoothing) in a minimization framework. Therefore, the second goal of this proposal is to exploit penalty functions, which can result in adaptive image smoothing and sharpening. To this end, we introduce parameterized functions with multiple extrema of similar shape to the negative Mexican hat function. For this type of penalty function, once the magnitude of the filter response is larger than certain thresholding, the penalty function encourages to increase the magnitude of the filter response, alluding to an image sharpening operation. Therefore, the analysis prior model controlled by this type of penalty function can adaptively switch between image smoothing and sharpening. Thirdly, we propose to train the influence functions with arbitrary shape for adapting to different trained filters to the utmost. This is the third category of this proposal.

该项目旨在研究如何同时学习分析算子正则化模型中的滤波器和惩罚函数，提出可行的参数训练框架，用于更好地解决经典的图像复原问题。我们的研究内容分成以下三类。首先，目前的分析模型通常采用形如f(x)=x^0.5的固定形状单极值非凸惩罚函数，不能根据不同滤波器进行自适应调整。因此，该项目的第一个研究内容是：提出一种参数化的单极值非凸惩罚函数并有监督地训练这些参数，使得惩罚函数可以根据不同滤波器进行自适应调整。其次，单极值非凸函数仅在零处有一极小值，无论滤波器响应值大小，对应的图像特征都会被削弱。为解决该问题，我们提出基于双峰型惩罚函数的分析模型。当滤波器响应超过某个阈值时，该类特征不会被削弱反而被加强，因此该模型可以起到图像自适应光滑/锐化的效果。这是第二个研究内容。最后，我们提出任意形式的惩罚函数学习，极大地释放了惩罚函数的自由度，以便更好地适应于不同的滤波器响应。这是该项目的第三个研究内容。

本项目针对图像复原问题，做出了较为突出的贡献，包括：（1）系统地提出了多尺度扩散模型的概念、模型与优化方法，以及在图像复原中的应用；（2）提出一种可以克服梯度弥散的扩散模型，成功应用在乘性噪声和泊松噪声的去噪中。从研究成果来看，通过本项目的支持，共发表论文6篇，其中第一作者论文3篇，包括IEEE Trans. Cybernetics和SIAM Journal on Imaging Sciences等本领域顶级与权威期刊。