纹理是图像中非常重要的视觉线索，在医学影像诊断、遥感监测等领域起着重要作用，但是传统的图像复原正则化方法存在损失纹理的缺陷。本课题旨在揭示Hardy空间和BMO空间混合范数在基于纹理保护的图像复原反问题中的正则化机理，探索将调和分析中Hardy空间与BMO空间的相关理论应用于图像复原的建模、分析与求解。具体包括：引入Hardy空间和BMO空间混合范数用于图像复原组稀疏建模，结合TV正则化、稀疏正则化以及混合范数组稀疏正则化的优势，建立保护纹理的图像复原模型；通过研究模型能量泛函以及函数空间的性质，研究模型解的存在性和唯一性；研究Hardy空间与BMO空间的小波框架刻画，为模型在小波域中快速求解提供有力工具，并以凸分析和优化理论为基础研究模型的快速求解算法以及算法的收敛性和稳定性。本课题的研究成果对于基于边缘和纹理保护的图像复原具有重要意义，可以为纹理和噪声的区分这一开放问题新的思路。

Texture is an important visual cue, and plays an important role in medical image diagnosis. However, traditional regularization models for image restoration have the drawback of losing textures. This project is aimed at revealing the regularization mechanism of Hardy space and BMO space mixed norms in texture preserving image restoration inverse problems , and exploring how to apply the theories about Hardy space and BMO space in harmonic analysis into image restoration modeling and analysis. The research work mainly includes the following contents. We introduce Hardy space and BMO space for image restoration group sparsity modeling. Combing the total variation (TV) regularization method with the group sparse regularization method, we propose a TV-Hardy regularization model for image restoration; Through studying the properties of relevant functional of the model and regularization function spaces, we discuss the existence and uniqueness of the solution; We study the wavelet frame characterization of Hardy space and BMO space, which can provide powerful tool for solving the model fast in wavelet domain. On the basis of convex analysis and optimization theory, we will study the numerical algorithm for solving the model and discuss the convergence and stability of the algorithm. This project has great signification for image restoration aiming at edge and texture preservation, and can provide a new thought in the open problem of separating texture and noise.