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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.