Collaborative Block-Term Tensor Decomposition and Its Application to Image Processing

The rank-$(L,M,N)$ block-term tensor decomposition (termed as BTD for convenience), which decomposes a tensor into a summation of multiple block-terms, has shown promise in multidimensional signal processing. However, in real-world applications, different components of a tensor usually have common and individual structures (e.g., shared dictionaries and individual coefficients), while BTD neglects the common structures across different components. Moreover, the corresponding algorithms for BTD (e.g., alternating least squares algorithm (BTD-ALS)) are computationally expensive for large-scale real data. To address these problems, we propose a collaborative block-term tensor decomposition (termed as CBTD), which decomposes a tensor into a summation of different block-terms multiplied by shared matrices. Compared with BTD, CBTD can better capture the structure of the data (i.e., the shared dictionaries and individual coefficients of multiple components), which is important especially in the challenging noisy/missing cases; and CBTD essentially projects the original tensor into a low-dimensional latent tensor by the shared factor matrices and then performs BTD on the latent tensor, which allows us to design more efficient algorithms. Moreover, we discuss the connection between BTD and CBTD and establish the essential uniqueness for CBTD. Empowered with CBTD, we propose the multidimensional signal reconstruction model and develop the corresponding proximal alternating minimization-based algorithm (CBTD-PAM) with the convergence guarantee. Extensive experiments on multispectral images and gray videos under Gaussian/sparse noise removal and tensor completion tasks demonstrate that the CBTD-PAM improves the reconstruction performance over competing methods in most cases. Especially, compared to the BTD-ALS, CBTD-PAM not only delivers higher quality results but also achieves substantial speedup.