  Robust tensor principal component analysis (RTPCA) can separate the low-rank
component and sparse component from multidimensional data, which has been used
successfully in several image applications. Its performance varies with
different kinds of tensor decompositions, and the tensor singular value
decomposition (t-SVD) is a popularly selected one. The standard t-SVD takes the
discrete Fourier transform to exploit the residual in the 3rd mode in the
decomposition. When minimizing the tensor nuclear norm related to t-SVD, all
the frontal slices in frequency domain are optimized equally. In this paper, we
incorporate frequency component analysis into t-SVD to enhance the RTPCA
performance. Specially, different frequency bands are unequally weighted with
respect to the corresponding physical meanings, and the frequency-weighted
tensor nuclear norm can be obtained. Accordingly we rigorously deduce the
frequency-weighted tensor singular value threshold operator, and apply it for
low rank approximation subproblem in RTPCA. The newly obtained
frequency-weighted RTPCA can be solved by alternating direction method of
multipliers, and it is the first time that frequency analysis is taken in
tensor principal component analysis. Numerical experiments on synthetic 3D
data, color image denoising and background modeling verify that the proposed
method outperforms the state-of-the-art algorithms both in accuracy and
computational complexity.
