One of the main difficulties in tensor completion is the calculation of the tensor rank. Recently a tensor nuclear norm, which is equal to the weighted sum of matrix nuclear norms of all unfoldings of the tensor, was proposed to address this issue. However, in this approach, all the singular values are minimized simultaneously. Hence the tensor rank may not be well approximated. In addition, many existing algorithms ignore the structural information of the tensor. This paper presents a tensor completion algorithm based on the proposed tensor truncated nuclear norm, which is superior to the traditional tensor nuclear norm. Furthermore, to maintain the structural information, a sparse regularization term, defined in the transform domain, is added into the objective function. Experimental results showed that our proposed algorithm outperforms several state-of-the-art tensor completion schemes.