基于低秩张量近似的图像估计

Low rank tensor structure widely exists in many multi-dimensional image applications. However, the classical image processing methods based on vector and matrix operations can not fully exploit this prior information. In this project, the low rank tensor approximation will be applied to the multi-dimensional image estimation (image reconstruction, image denoising, etc). First, the classical tensor completion with known positions of the missing components will be generalized to that with unknown positions. A new data fitting constraint will be developed based on the statistical model of outliers to solve this generalized tensor completion problem. Second, the additive uncertainty and multiplicative uncertainty will be incorporated in to the multi-dimensional measurement model, and a robust image estimation will be obtained by a nonconvex low rank tensor constraint and a generalized data fitting constraint with the measurement models of both signal and noise models. Third, the prior information of the tensor approximation will be generalized from a tensor’s low rank to multiple data structures. The multi-criterion optimization will be used to simultaneously exploit all the available data structures’ constraints, and further enhance the performance of multi-dimensional image estimation. To prove the proposed methods’ effectiveness, we will theoretically analyze the estimation bound, convergence and computational complexity, and give the numerically experimental evaluation based on real multi-dimensional image data.

低秩张量结构是高维图像应用中的常见的一个结构特征。但是传统基于向量和矩阵运算的图像处理不能充分利用这样的先验信息。本项目将低秩张量近似应用到高维图像估计应用，例如图像重构、图像降噪等。第一，将传统的缺失成分位置已知的张量补全问题推广到缺失成分位置未知的广义张量补全问题。通过野值的统计建模，开发新的数据拟合约束完成张量的补全。第二，将传统的高维图像测量模型推广到同时包括加性噪声和乘性噪声的测量模型。通过非凸低秩约束、数据拟合约束中信号和噪声建模，获得更稳健的图像估计方法。第三，将张量近似的先验结构信息从单个低秩结构推广到更广义的多个简单数据结构，通过多标准优化的方式，联合使用这些先验信息来提高基于张量近似的高维图像估计性能。为了检验所提方法的有效性，理论上，我们将分析各个算法的估计误差界、收敛性、计算复杂度等；而实际高维图像的数值实验将给出一些基于应用的性能指标。

针对传统的矩阵运算在图像估计中不能充分利用图像先验结构的问题，本项目开展低秩张量近似的基础理论及其在图像估计的应用。针对广义张量缺失成分估计问题，项目组提出张量全变分、光滑的张量数分解，并应用到动态磁共振成像、高维视觉图像恢复、高光谱成像。针对各种不同的噪声类型，开展了稳健的低秩张量近似研究，包括量化误差的稳健图像恢复、最佳尺度下的稳健低秩张量近似、基于两层张量奇异值分解的低秩张量近似，并应用到视觉图像降噪、光照归一化、背景提取等。项目组还从理论和实验两方面分析了各种高维图像的数据结构，联合开发对应的代数约束，应用到图像估计问题。在以上基础上，还拓展了张量回归、张量融合在图像估计中的应用。