  Currently, low-rank tensor completion has gained cumulative attention in
recovering incomplete visual data whose partial elements are missing. By taking
a color image or video as a three-dimensional (3D) tensor, previous studies
have suggested several definitions of tensor nuclear norm. However, they have
limitations and may not properly approximate the real rank of a tensor.
Besides, they do not explicitly use the low-rank property in optimization. It
is proved that the recently proposed truncated nuclear norm (TNN) can replace
the traditional nuclear norm, as a better estimation to the rank of a matrix.
Thus, this paper presents a new method called the tensor truncated nuclear norm
(T-TNN), which proposes a new definition of tensor nuclear norm and extends the
truncated nuclear norm from the matrix case to the tensor case. Beneficial from
the low rankness of TNN, our approach improves the efficacy of tensor
completion. We exploit the previously proposed tensor singular value
decomposition and the alternating direction method of multipliers in
optimization. Extensive experiments on real-world videos and images demonstrate
that the performance of our approach is superior to those of existing methods.
