基于全广义变差和小波变换的图像卡通-纹理分解研究

To overcome the defects of “staircasing effect” and higher computational complexity in image decomposition, caused by the traditional total variation regularization models and numerical algorithms. This project will adopt the total generalized variation (TGV) to depict the measure of image, and propose several improved variational PDE models for cartoon-texture image decomposition. Meanwhile, we will present more efficient numerical algorithms by improving the loop structure of classical algorithms, combining multiple algorithms together or algorithms parallelization. Our major innovative contributions are listed as follows. Firstly, we will investigate the anisotropic second-order TGV regularization and Gabor function fidelity based image decomposition model and fast algorithms. Secondly, the multiscale texture image decomposition model and fast algorithms based on adaptive second-order TGV and H^{-s} norm in negative Hilbert-Sobolev space will be researched. Moreover, taking advantage of the merits of TGV semi-norm and wavelet transform in image processing adequately, we construct several hybrid regularized models by employing the second-order TGV and wavelet basis to characterize the cartoon and sparse texture components respectively, and propose efficient numerical algorithms based on wavelet domain for super resolution image decomposition and restoration. Thus, the proposed more reasonable variational PDE models and efficient numerical methods will dramatically overcome “staircasing effect” and preserve texture details, and simultaneously improve the speed of calculation noticeably.

为了克服传统的全变差正则化模型和数值算法在图像分解时出现“阶梯效应”和计算复杂度高等缺点，本项目拟采用全广义变差（TGV）来刻画图像的测度，建立几类改进的变分PDE图像卡通-纹理分解模型；并通过改进经典算法的循环结构、多算法有机结合或算法并行化等方式设计出更加高效的数值算法。主要创新工作包括：建立各向异性二阶TGV正则化和Gabor函数保真项的图像分解模型及快速算法；建立自适应二阶TGV正则化和负Hilbert-Sobolev空间H^{-s}范数的多尺度纹理图像分解模型及快速算法；充分利用TGV半范数和小波变换在图像处理中的优势，分别采用二阶TGV和小波基来刻画图像的卡通和稀疏纹理部分，进而建立若干双正则化的超分辨率图像分解与复原模型以及基于小波域的高效数值算法。通过构建合理的变分PDE模型和高效数值算法，拟实现在克服结构图像“阶梯效应”并保护纹理细节的同时，大幅度地提高数值计算的速率。

本项目采用全广义变差（TGV）来刻画图像的测度，建立了几类改进的变分PDE图像复原与分解模型；并通过改进经典算法的循环结构、多算法有机结合等方式设计出更加高效的数值算法。取得的主要成果包括：提出了空间自适应二阶TGV正则化的保边高斯去噪模型和改进的快速分裂Bregman迭代算法；建立了二阶TGV正则化的泊松化图像复原模型和高效增广拉格朗日算法；研究了基于二阶TGV和H^{-1}范数的多尺度纹理图像分解模型及快速算法；充分利用TGV半范数和小波变换在图像处理中的优势，建立了二阶TGV和1-紧框架小波双正则化的高斯去噪、盐椒去噪模型，以及二阶TGV和剪切波双正则化的泊松去噪模型并提出了快速的交替最小化算法。数值实验表明，本项目构建的变分PDE模型和高效数值算法在有效克服平滑区域“阶梯效应”并保护边缘细节的同时，显著地提高了数值计算的速率。