Robust Low Transformed Multi-Rank Tensor Methods for Image Alignment

Aligning a group of linearly correlated images is an important task in computer vision. In this paper, we propose a combination of transformed tensor nuclear norm and tensor \(\ell _1\) norm to deal with this image alignment problem, where the observed images, stacked into a third-order tensor, are deformed by unknown domain transformations and corrupted by sparse noise like impulse noise, partial occlusions, and illumination variation. The key advantage of the proposed method is that both spatial correlation and images variation can be captured by the use of transformed tensor nuclear norm. We show that when the underlying of correlated images is a low multi-rank tensor, an upper error bound of the estimator of the proposed model can be established and this bound can be better than the previous result. Besides the proposed convex transformed tensor model, the method can be further studied by incorporating nonconvex functions in the transformed tensor nuclear norm and the sparsity norm. Both the proposed convex and nonconvex optimization models are solved by generalized Gauss–Newton algorithms. Also the global convergence of the numerical methods for solving the subproblems of convex and nonconvex optimization models can be provided under very mild conditions. Extensive numerical experiments on real images with misalignment and sparse corruptions demonstrate the performance of our proposed methods is better than that of several state-of-the-art methods in terms of accuracy and efficiency.