Simultaneous Rectification and Alignment via Robust Recovery of Low-rank Tensors

In this work, we propose a general method for recovering low-rank three-order tensors, in which the data can be deformed by some unknown transformation and corrupted by arbitrary sparse errors. Since the unfolding matrices of a tensor are interdependent, we introduce auxiliary variables and relax the hard equality constraints by the augmented Lagrange multiplier method. To improve the computational efficiency, we introduce a proximal gradient step to the alternating direction minimization method. We have provided proof for the convergence of the linearized version of the problem which is the inner loop of the overall algorithm. Both simulations and experiments show that our methods are more efficient and effective than previous work. The proposed method can be easily applied to simultaneously rectify and align multiple images or videos frames. In this context, the state-of-the-art algorithms "RASL" and "TILT" can be viewed as two special cases of our work, and yet each only performs part of the function of our method.