  Recently, there has been a lot of research into tensor singular value
decomposition (t-SVD) by using discrete Fourier transform (DFT) matrix. The
main aims of this paper are to propose and study tensor singular value
decomposition based on the discrete cosine transform (DCT) matrix. The
advantages of using DCT are that (i) the complex arithmetic is not involved in
the cosine transform based tensor singular value decomposition, so the
computational cost required can be saved; (ii) the intrinsic reflexive boundary
condition along the tubes in the third dimension of tensors is employed, so its
performance would be better than that by using the periodic boundary condition
in DFT. We demonstrate that the tensor product between two tensors by using DCT
can be equivalent to the multiplication between a block Toeplitz-plus-Hankel
matrix and a block vector. Numerical examples of low-rank tensor completion are
further given to illustrate that the efficiency by using DCT is two times
faster than that by using DFT and also the errors of video and multispectral
image completion by using DCT are smaller than those by using DFT.

Computer Vision and Pattern Recognition