Tensor block-block terms decomposition for matrix-valued imaging applications

Matrix-valued images appear in many applications, ranging from polarimetric remote sensing to medical imaging. Such images can be represented as 4th-order tensors, where the first two dimensions correspond to spatial variables and the last two encode the matrix feature in each pixel. To efficiently analyze, decompose, and process these images, this paper considers the block-block terms decomposition (2BTD), a versatile low-rank tensor decomposition model that extends bilinear matrix factorization to 4th-order tensors by representing the latter as the sum of outer products of low-rank matrix blocks. Low-rank assumptions allow for a significantly reduced number of parameters to be estimated and enable the enforcement of key physical constraints on matrix sources. We establish both necessary and sufficient conditions for the uniqueness of the 2BTD model. To enable the use of 2BTD in covariance matrix-valued imaging, we develop an optimization framework that allows efficient handling of non-negativity and symmetry constraints together with low-rank assumptions on matrix blocks. Numerical experiments on synthetic and real data from Diffusion Tensor Imaging (DTI) illustrate the potential of the 2BTD model in matrix-valued imaging, as well as its effectiveness in practical settings.