Hyperspectral image super-resolution by fusing high-resolution multispectral image (HR-MSI) and low-resolution hyperspectral image (LR-HSI) aims at reconstructing high resolution spatial-spectral information of the scene. Existing methods mostly based on spectral unmixing and sparse representation are often developed from a low-level vision task perspective, they cannot sufficiently make use of the spatial and spectral priors available from higher-level analysis. To this issue, this paper proposes a novel HSI super-resolution method that fully considers the spatial/spectral subspace low-rank relationships between available HR-MSI/LR-HSI and latent HSI. Specifically, it relies on a new subspace clustering method named “structured sparse low-rank representation” (SSLRR), to represent the data samples as linear combinations of the bases in a given dictionary, where the sparse structure is induced by low-rank factorization for the affinity matrix. Then we exploit the proposed SSLRR model to learn the SSLRR along spatial/spectral domain from the MSI/HSI inputs. By using the learned spatial and spectral low-rank structures, we formulate the proposed HSI super-resolution model as a variational optimization problem, which can be readily solved by the ADMM algorithm. Compared with state-of-the-art hyperspectral super-resolution methods, the proposed method shows better performance on three benchmark datasets in terms of both visual and quantitative evaluatiaon.

Existing methods for tensor completion (TC) have limited ability for characterizing low-rank (LR) structures. To depict the complex hierarchical knowledge with implicit sparsity attributes hidden in a tensor, we propose a new multilayer sparsity-based tensor decomposition (MLSTD) for the low-rank tensor completion (LRTC). The method encodes the structured sparsity of a tensor by the multiple-layer representation. Specifically, we use the CANDECOMP/PARAFAC (CP) model to decompose a tensor into an ensemble of the sum of rank-1 tensors, and the number of rank-1 components is easily interpreted as the first-layer sparsity measure. Presumably, the factor matrices are smooth since local piecewise property exists in within-mode correlation. In subspace, the local smoothness can be regarded as the second-layer sparsity. To describe the refined structures of factor/subspace sparsity, we introduce a new sparsity insight of subspace smoothness: a self-adaptive low-rank matrix factorization (LRMF) scheme, called the third-layer sparsity. By the progressive description of the sparsity structure, we formulate an MLSTD model and embed it into the LRTC problem. Then, an effective alternating direction method of multipliers (ADMM) algorithm is designed for the MLSTD minimization problem. Various experiments in RGB images, hyperspectral images (HSIs), and videos substantiate that the proposed LRTC methods are superior to state-of-the-art methods.

Recently, tensor sparsity modeling has achieved great success in the tensor completion (TC) problem. In real applications, the sparsity of a tensor can be rationally measured by low-rank tensor decomposition. However, existing methods either suffer from limited modeling power in estimating accurate rank or have difficulty in depicting hierarchical structure underlying such data ensembles. To address these issues, we propose a parametric tensor sparsity measure model, which encodes the sparsity for a general tensor by Laplacian scale mixture (LSM) modeling based on three-layer transform (TLT) for factor subspace prior with Tucker decomposition. Specifically, the sparsity of a tensor is first transformed into factor subspace, and then factor sparsity in the gradient domain is used to express the local similarity in within-mode. To further refine the sparsity, we adopt LSM by the transform learning scheme to self-adaptively depict deeper layer structured sparsity, in which the transformed sparse matrices in the sense of a statistical model can be modeled as the product of a Laplacian vector and a hidden positive scalar multiplier. We call the method as parametric tensor sparsity delivered by LSM-TLT. By a progressive transformation operator, we formulate the LSM-TLT model and use it to address the TC problem, and then the alternating direction method of multipliers-based optimization algorithm is designed to solve the problem. The experimental results on RGB images, hyperspectral images (HSIs), and videos demonstrate the proposed method outperforms state of the arts.

Conventional tensor completion (TC) methods generally assume that the sparsity of tensor-valued data lies in the global subspace. The so-called global sparsity prior is measured by the tensor nuclear norm. Such assumption is not reliable in recovering low-rank (LR) tensor data, especially when considerable elements of data are missing. To mitigate this weakness, this article presents an enhanced sparsity prior model for LRTC using both local and global sparsity information in a latent LR tensor. In specific, we adopt a doubly weighted strategy for nuclear norm along each mode to characterize global sparsity prior of tensor. Different from traditional tensor-based local sparsity description, the proposed factor gradient sparsity prior in the Tucker decomposition model describes the underlying subspace local smoothness in real-world tensor objects, which simultaneously characterizes local piecewise structure over all dimensions. Moreover, there is no need to minimize the rank of a tensor for the proposed local sparsity prior. Extensive experiments on synthetic data, real-world hyperspectral images, and face modeling data demonstrate that the proposed model outperforms state-of-the-art techniques in terms of prediction capability and efficiency.

High-dimensional image representation is a challenging task since data has the intrinsic low-dimensional and shift-invariant characteristics. Currently, popular methods, such as tensor-Singular Value Decomposition (t-SVD), have limited ability in expressing shift-invariant subspace knowledge under-lying data. To these problem, we propose a high-dimensional image representation framework based on Tensor Convolutionlike Low-Rank Dictionary (TCLRD), which considers the shift-invariant low-dimensional structure of a tensor-valued data by convolution-like low-rank dictionary learning and coefficient coding, to promote the high-dimensional image representation ability. To be specific, we first define the TCLRD framework with low-rank constraint for dictionary and coefficient, in which tensor factorization and tensor-tensor product over frequency domain can be understood as convolution-like operation when describing shift-invariant. Then, the tensor Schatten-p norm is introduced to verify that TCLRD has rational mathematical interpretation. We study the TCLRD minimization problem in tensor completion with the ADMM-based optimization algorithm. The efficient solving scheme with TCLRD is extendable to various low-rank models like tensor robust principal component analysis and subspace clustering, and prove their theoretical guarantees based on generalization error. Extensive experimental results demonstrate the proposed TCLRD methods are beyond state-of-the-arts in typical tasks, including image denoising, HSI completion and image clustering.

We propose a novel graph Laplacian-guided coupled tensor decomposition (gLGCTD) model for fusion of hyperspectral image (HSI) and multispectral image (MSI) for spatial and spectral resolution enhancements. The coupled Tucker decomposition is employed to capture the global interdependencies across the different modes to fully exploit the intrinsic global spatial-spectral information. To preserve local characteristics, the complementary submanifold structures embedded in high-resolution (HR)-HSI are encoded by the graph Laplacian regularizations. The global spatial-spectral information captured by the coupled Tucker decomposition and the local submanifold structures are incorporated into a unified framework. The gLGCTD fusion framework is solved by a hybrid framework between the proximal alternating optimization (PAO) and the alternating direction method of multipliers (ADMM). Experimental results on both synthetic and real data sets demonstrate that the gLGCTD fusion method is superior to state-of-the-art fusion methods with a more accurate reconstruction of the HR-HSI.