We consider the problem of low-rank tensor decomposition of incomplete tensors that has applications in many data analysis problems, such as recommender systems, signal processing, machine learning, and image inpainting. In this paper, we focus on nonnegative tensor completion via low-rank Tucker decomposition for dealing with it. The specialty of our model is that the ranks of nonnegative Tucker decomposition are no longer constants, while they all become a part of the decisions to be optimized. Our solving approach is based on the penalty method and blocks coordinate descent method with prox-linear updates for regularized multiconvex optimization. We demonstrate the convergence of our algorithm. The numerical results on the three image datasets show that the proposed algorithm offers competitive performance compared with other existing algorithms even though the data is highly sparse.