A constructive arbitrary-degree Kronecker product decomposition of tensors

We propose the tensor Kronecker product singular value decomposition~(TKPSVD) that decomposes a real $k$-way tensor $\mathcal{A}$ into a linear combination of tensor Kronecker products with an arbitrary number of $d$ factors $\mathcal{A} = \sum_{j=1}^R \sigma_j\, \mathcal{A}^{(d)}_j \otimes \cdots \otimes \mathcal{A}^{(1)}_j$. We generalize the matrix Kronecker product to tensors such that each factor $\mathcal{A}^{(i)}_j$ in the TKPSVD is a $k$-way tensor. The algorithm relies on reshaping and permuting the original tensor into a $d$-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many different structured tensors, the Kronecker product factors $\mathcal{A}^{(1)}_j,\ldots,\mathcal{A}^{(d)}_j$ are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor

Best Kronecker Product Approximation of The Blurring Operator in Three Dimensional Image Restoration Problems

In this paper, we propose a method to find the best Kronecker product approximationof the blurring operator which arises in three dimensional image restoration problems. We show thatthis problem can be reduced to a well known rank-1 approximation of the scaled three dimensionalpoint spread function (PSF) array, which is much smaller. This approximation can be used as apreconditioner in solving image restoration problems with iterative methods. The comparison ofthe approximation by the new scaled PSF array and approximation by the original PSF array that is used in [J. G. Nagy and M. E. Kilmer, IEEE Trans. Image Process., 15 (2006), pp. 604–613],confirms the performance of the new proposed approximation

Best Kronecker Product Approximation of The Blurring Operator in Three Dimensional Image Restoration Problems

In this paper, we propose a method to find the best Kronecker product approximationof the blurring operator which arises in three dimensional image restoration problems. We show thatthis problem can be reduced to a well known rank-1 approximation of the scaled three dimensionalpoint spread function (PSF) array, which is much smaller. This approximation can be used as apreconditioner in solving image restoration problems with iterative methods. The comparison ofthe approximation by the new scaled PSF array and approximation by the original PSF array that is used in [J. G. Nagy and M. E. Kilmer, IEEE Trans. Image Process., 15 (2006), pp. 604–613],confirms the performance of the new proposed approximation