We introduce a structured low rank matrix completion algorithm to recover a series of images from their under-sampled measurements, where the signal along the parameter dimension at every pixel is described by a linear combination of exponentials. We exploit the exponential behavior of the signal at every pixel, along with the spatial smoothness of the exponential parameters to derive an annihilation relation in the Fourier domain. This relation translates to a low-rank property on a structured matrix constructed from the Fourier samples. We enforce the low rank property of the structured matrix as a regularization prior to recover the images. Since the direct use of current low rank matrix recovery schemes to this problem is associated with high computational complexity and memory demand, we adopt an iterative re-weighted least squares (IRLS) algorithm, which facilitates the exploitation of the convolutional structure of the matrix. Novel approximations involving two dimensional Fast Fourier Transforms (FFT) are introduced to drastically reduce the memory demand and computational complexity, which facilitates the extension of structured low rank methods to large scale three dimensional problems. We demonstrate our algorithm in the MR parameter mapping setting and show improvement over the state-of-the-art methods.
Deeply Learned Robust Matrix Completion for Large-scale Low-rank Data Recovery
School of Data, Mathematical, and Statistical Sciences and the Department of Computer Science, University of Central Florida（数据、数学与统计科学学院和计算机科学系，中央佛罗里达大学）；School of Data, Mathematical, and Statistical Sciences, University of Central Florida（数据、数学与统计科学学院，中央佛罗里达大学）；Damo Academy, Alibaba US（阿里云美国研究院）
Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Peng Cheng Laboratory（鹏城实验室）；The University of Hong Kong（香港大学）；RIKEN Center on Advanced Intelligence Project（理化学研究所先进智能项目中心）
Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Framework
The University of British Columbia（不列颠哥伦比亚大学）；Memorial University of Newfoundland（纽芬兰纪念大学）；Qingdao Preschool Education College（青岛幼儿师范高等专科学校）；Zhejiang University of Technology（浙江工业大学）
Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guarantee
Virginia Tech（弗吉尼亚理工大学）
本研究的目的是通过将视频采集、视频恢复、背景/前景分离结合到一个框架中来解决具有缺失像素的背景/前景分离问题。为了实现这一点，平滑鲁棒张量完成（SRTC）模型被提出来恢复数据，并将其分解为静态背景和平滑前景，分别。具体地说，静态背景建模的低秩塔克分解和平滑的前景（运动物体）建模的时空连续性，这是由总变分正则化。基于张量近似交替最小化（tensor proximate alternative minimization，tenPAM）的有效算法实现求解所提出的模型，在非常温和的条件下具有全局收敛性保证。大量的实验表明，该方法显着优于国家的最先进的方法，背景/前景分离与丢失像素。
