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.
This paper considers the (unrectified and misaligned) 
      tensor recovery problem, which is an interesting problem. 
      Originality: This work extends the matrix case in [1] to the 
      tensor case. Specifically, it replaces the matrix nuclear norm in Eq. (3) 
      of [1] by the tensor nuclear norm. I am afraid that this type of extension 
      may not be treated as a substantial contribution any more since similar 
      extensions have appeared in many early papers like [10-14, etc]. The new 
      staff beyond this extension seems to consider the misalignment issue in 
      tensors. The technique used to deal with unrectification and misalignment 
      actually follows [1] as well. This paper just simply splits the tensor 
      data into matrices and uses the technique proposed in [1] to deal with 
      uncertification and misalignment in matrices. Although this paper claims 
      that the proposed method align across images unlike [1], that is just due 
      to the tensor nuclear norm rather than any new alignment technique. 
      A variant of the ADMM method is used to solve a key step of the 
      proposed formulation. This paper claims ADMM is not efficient to solve 
      (12-1) --- the first subproblem in (12) because (12-1) does not give a 
      closed form. To my experiences, if properly introduce new variables into 
      (11), it can be formulated into a structure good for ADMM (that is, all 
      subproblems have closed form solutions): you can try duplicate \L and \E 
      in (11). The proof for Theorem 1 seems unnecessary. The 
      convergence for many variants for ADMM, including using the proximal 
      gradient descent (used in this paper) to replace the exactly solving, has 
      been shown in many optimization papers already, for example, [A], [B]. 
      Overall, this paper is not of sufficient contribution in term of 
      originality. My novelty judgement is only based on the aspects of modeling 
      and algorithm. Contributions in numerical results may be ignored. 
      Clarity: Some equations are not clear enough, and some 
      notations are abused apparently in section 3.1. Readers like me have to 
      read [1] to understand those undefined and unclear notations and 
      equations. Authors should seriously treat the following issues. - (5) 
      reuses "\circ" which has been used in (2). Two ``\circ'' actually mean 
      totally different things. Readers that unfamiliar with [1] may be misled 
      in (5). Moreover, the followed explanation for "A\circ \Tau" is also 
      unclear. ``Applying the transformation \tau_i to each matrix A(:,:,i)'' 
      sounds like the matrix multiplication \tau_i A(:,:,i). It actually means 
      A(\tau_x, \tau_y, i) from [1]. - What is the value of ``p'' in two 
      lines before (5)? - (8) is suddenly relaxed into (9) by using a 
      ``popular'' way. What is the ``popular'' way? No explanation, no 
      citations. Readers have to figure out it from other papers like [1]. - 
      The last line of Section 3.1: what is the value of "n". This notation "n" 
      appears for multiple times before this line. Apparently, they denote 
      different things. [A] On the Linear Convergence of the Alternating 
      Direction Method of Multipliers, Mingyi Hong, Zhi-Quan Luo, 2012. [B] 
      Linearized Alternating Direction Method of Multipliers for Constrained 
      Linear Least-Squares Problem, Raymond H. Chan1, Min Tao, Xiaoming Yuan, 
      2012.