Beyond Low Rank + Sparse: A Multi-scale Low Rank Decomposition

We present a multi-scale version of low rank matrix decomposition. Our motivation comes from imaging applications, in which image sequences are correlated locally on several scales in space and time rather than globally. We model our data matrix as a sum of matrices, where each matrix has increasing scales of locally low-rank matrices. Using this multi-scale modeling, we can capture different scales of correlation in our data matrix and provide a more compact representation than conventional low rank methods. We solve the proposed decomposition via a convex formulation. INTRODUCTION In many image processing applications, the data matrix of interest exhibits low rank structures [1]. Often, we know in addition that entries in a neighborhood are more likely to be correlated. If the size of the locality is not known, these locally low rank structures cannot be exploited using traditional low rank methods and the matrix itself may even be considered as full rank. Inspired by low rank + sparse modeling [?], [2], [3], we present a way of representing these matrices compactly by decomposing the matrix into low rank blocks across multiple scales. Using this multi-scale modeling, we can capture different scales of correlation in our data matrix and provide a more compact representation than conventional low rank methods. Our motivation comes from Dynamic Contrast Enhanced (DCE) Imaging in Magnetic Resonance Imaging. In DCE imaging, different tissue contrasts evolve differently over time. When stacking each image frame as a column of the matrix, the resulting matrix is low rank with various block sizes [1], [3]. A small block size captures blood vessel dynamics better while a large block size captures background tissues better. Hence, a multi-scale low rank approach is desired to exploit all scales of correlations. SIGNAL MODEL AND PROBLEM FORMULATION We model our M ×N data matrix Y as a sum of matrices Y = ∑L−1 i=0 Xi, where each Xi has an increasing scale of locality. That is, each Xi is block low-rank with a different block size, with the smallest one being a sparse matrix and the largest one being a low rank matrix. Fig. 1 provides an illustration of the modeling. Low Rank Low Rank Low Rank Low Rank Low Rank Low Rank Low Rank