Tensor Convolution-Like Low-Rank Dictionary for High-Dimensional Image Representation

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 underlying data. To these problem, we propose a high-dimensional image representation framework based on Tensor Convolution-like 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.