Multi-Dimensional Sparse Models

Abstract

Traditional synthesis/analysis sparse representation models signals in a one dimensional (1D) way, in which a multidimensional (MD) signal is converted into a 1D vector. 1D modeling cannot sufficiently handle MD signals of high dimensionality in limited computational resources and memory usage, as breaking the data structure and inherently ignores the diversity of MD signals (tensors). We utilize the multilinearity of tensors to establish the redundant basis of the space of multi linear maps with the sparsity constraint, and further propose MD synthesis/analysis sparse models to effectively and efficiently represent MD signals in their original form. The dimensional features of MD signals are captured by a series of dictionaries simultaneously and collaboratively. The corresponding dictionary learning algorithms and unified MD signal restoration formulations are proposed. The effectiveness of the proposed models and dictionary learning algorithms is demonstrated through experiments on MD signals denoising, image super-resolution and texture classification. Experiments show that the proposed MD models outperform state-of-the-art 1D models in terms of signal representation quality, computational overhead, and memory storage. Moreover, our proposed MD sparse models generalize the 1D sparse models and are flexible and adaptive to both homogeneous and inhomogeneous properties of MD signals.

Nonlocal Tensor Sparse Representation and Low-Rank Regularization for Hyperspectral Image Compressive Sensing Reconstruction

“…In addition, as shown in Figure 2e, we give a detailed analysis for another notable representation form for the sparsity prior based on tensor sparse decomposition, which suggests that we can depict the structured sparsity of an HSI from the perspective of core tensor. Some pioneering works are presented in [42,43,[52][53][54]. Here, we draw attention to the structured sparsity formulation of an HSI under tensor sparse representation framework, thus each third-order tensor X p can be approximated by following problem:…”

“…where U1p, U2p, and U3p are factor matrices and S(G p ) is sparse constraint term, and we assume S(G p ) = G p 0 as suggested in [42,43,52]. However, the optimization problem based on l 0 constraint deduced by Equation 8is non-convex, the research in [53,54] further relaxes the l 0 -based core sparsity to l 1 case as S(G p ) = G p 1 . The convex optimization problem corresponding to l 1 case can be represented in Lagrangian form as following:…”

“…It can be solved by the Tensor-based Iterative Shrinkage Thresholding Algorithm (TISTA) in [53,54]. (c) M p sub-problem:…”

Color Image and Multispectral Image Denoising Using Block Diagonal Representation

“…For multiway filtering approaches described in (5), there are roughly two strategies to encourage the sparsity of linear approximation in the transform domain: threshold core tensor C (via L0-norm [70], Wiener filter [7,71], soft-or hardthresholding [7]), and threshold factor matrices via low rank prior [72]. Directly modeling each mode of 4D data with low rank prior raises two major concerns.…”

“…14(c) demonstrate the effectiveness of the twist implementation. Considering its efficiency, its performance may be improved by further modeling of tensor sparsity [70].…”