结构保持的图像非局部分数阶数值模型与算法研究

Fractional-order variational models can preserve useful structures such as edges and textures well than integer-order variational models in image restoration. So this method becomes one of the popular techniques. However, fractional derivative has high computational complexity. How to build good models and design efficient algorithms is a significant research subject. In this project, based on integer-order variational PDE and fractional stability analysis,it mainly makes researches on the following issue:(1) Based on the theory of Meyer's oscillatory component modeling and combined with fractional-order derivative and function space of bounded variation，the fractional-order function space of bounded variation is proposed. Some fractional-order regularization models for image restoration are proposed;(2)Using the conceptions of non-local discontinuity of image and fractional differential,and by different examples of non-local regularization energy functional,non-local fractional variational models are proposed; (3) Internal relations among wavelet, curvelet and fractional-order scale space are studied and the advantages of fractional-order variational in transform domain are developed. The subject mainly proposes the fractional-order regularization model and gives some fast algorithms.In information science, our research will have wide applied prospect.

分数阶变分模型能很好地保持图像的边缘、纹理等重要结构特征，较整数阶变分模型在图像恢复时优越，逐渐成为信号和图像处理领域中崭新的研究方法。但分数阶的理论分析和算法设计较困难，如何构建好的优化模型和高效的数值算法是一个非常有意义的研究课题。本项目在前期整数阶变分PDE和分数阶稳定性分析的基础上拟对如下问题展开研究：(1)在Meyer的振荡分量建模的理论框架下，结合有界变差空间(BV)和分数阶导数，构建分数阶有界变差空间，设计各种分数阶正则的图像恢复模型；(2) 利用图像非局部不连续性测度的概念及分数阶微分理论，通过选择不同势函数的非局部正则化泛函，建立非局部的分数阶变分模型；(3) 研究小波、曲线波等稀疏表示工具和分数阶多尺度空间的内在联系，发挥变换域上分数阶变分问题的优势。本项目立足模型选择和设计高效的数值算法，具有一定的开创性和前沿性，相信会推动其在信息科学领域的应用与发展。

分数阶变分模型是一个非凸非光滑的优化问题，较整数阶变分模型在图像复原和重建时优越，逐渐成为信号和图像处理领域中最受欢迎的技术之一。如何构建好的分数阶优化模型和高效的数值算法是一个非常有意义的研究课题。我们以数学建模的思想和理论为指导，充分利用变分法和最优化理论，构建了分数阶模型空间域和变换域的桥梁，深刻探讨了分数阶变分问题的模型选择，给出了模型的性能分析，研究了直接求解非凸变分模型的光滑化算法以及把此类变分问题转化为不可微的分数阶导数问题，进一步给出快速的迭代算法，部分解决了这种非凸问题的多最优解问题。针对该问题课题组提出了若干个算法和解决思路，但苦于问题的复杂性，目前提出的算法收敛效果不理想、收敛速度不高。这些是我们课题组今后还要继续努力探讨的地方。另外，我们可以将研究成果推广到解决其它扩展形式的变分问题或其他图像处理的反问题中（例如图像分割，图像融合等）。以此项目为依托培养了研究生2名，他们以此课题内容展开的毕业论文，获得了较好的毕业成果。