基于分形变换理论与正则化的图像复原方法研究

Recently, the fractal transforms technique and the non-local means (NLM) method have been investigated in an unified way under the NL self-similar framework. Compared with NLM, which is one of the hottest techniques in the field of image restoration, the research on fractal transforms-based method is still in its initial stage and much work should be done. This project will mainly conduct an in-depth study of image denoising (and deblurring) from classic fractal transforms, multi-parent generalized fractal transforms (MPGFT), and their regularizations. Specifically, we will concentrate on building various robust optimization models for predicting fractal codes under the condition of different noise and designing their corresponding algorithms; formulating the related theory on the contractivity of the MPGFT operator; constructing new similarity measure functions and weight functions compatible with MPGFT-iterated function systems under different noisy conditions and developing collaborative techniques by combining with other effective methods; establishing effective regularization models and algorithms which can update fractal codes gradually through iterations; introducing new additional regularization terms for enhancing the performance of the models. With these findings, we plan to further explore the practical applications of fractal transforms in image restoration. This project will not only develop a series of new high performance fractal transforms-based algorithms for image denoising (and deblurring), but also the related theories and conclusions can be extended to the study of image inpainting and image super-resolution reconstruction.

近期，分形变换技术与非局部平均(NLM)方法被统一至非局部自相似框架下进行研究。比较当前图像复原领域热点技术NLM，基于分形变换的图像复原研究尚处起步阶段，很多问题值得深入研究。本项目拟重点从经典分形变换、多父块广义分形变换(MPGFT)、及其正则化融合等几方面对图像去噪(和去模糊)展开研究。具体研究针对不同噪声的鲁棒性分形码预测优化模型的建立及算法设计；研究MPGFT算子的收敛性相关理论；研究不同噪声情形下与MPGFT迭代函数系统相容的相似性测度函数及权重函数的构造，并考虑与其他方法结合的协同技术；研究构建可迭代更新分形码的正则化模型和算法；研究通过辅助方法加入新的正则项以提升模型性能；最后在取得的成果基础上研究分形图像复原的实际应用。本项目的研究将不仅实现一系列基于分形变换的高性能图像去噪(和去模糊)算法，其相关结论还可拓展至用于研究图像修复和图像超分辨率重建。

本项目围绕经典分形变换、多父块广义分形变换(MPGFT)、基于正则化和字典学习技术、以及在这些工作基础上构建可迭代更新分形码模型等方面对图像去噪（和去模糊）开展研究，取得了一系列的研究成果，其主要工作为：在经典分形变换框架下，构建了鲁棒性分形码预测的带约束优化模型，提出使用一种Primal-Dual求解算法，实现了去噪效果明显优于传统算法的新方法；研究了多父块广义分形变换相关理论和图像复原方法，给出了多父块广义分形变换的收敛性条件，研究了选取相似性多父块的具体方法，导出了利用噪声图像预测无噪MPGFT分形参数的具体公式；研究了基于TV正则化、tight framelet、小波L0正则化、字典学习等的图像复原新方法，包括提出了一系列分别针对Speckle噪声、脉冲噪声、Rician噪声、高斯-脉冲混合噪声等的图像复原新模型和对应的求解算法；借助和融合正则化和字典学习的思想和方法，构建了可迭代更新分形码的分形图像复原新模型并设计相应求解算法；同时将这些提出的新方法应用于模拟的和实际的SAR和医学图像数据；此外，深入研究了分形和动力系统有关理论，等。本项目的研究不仅实现了一系列高性能图像复原方法，将为图像复原的研究注入新的活力，其相关的理论和方法还可拓展至用于研究图像修复和图像超分辨率重建。