  In this paper, we study the problem of a batch of linearly correlated image
alignment, where the observed images are deformed by some unknown domain
transformations, and corrupted by additive Gaussian noise and sparse noise
simultaneously. By stacking these images as the frontal slices of a third-order
tensor, we propose to utilize the tensor factorization method via transformed
tensor-tensor product to explore the low-rankness of the underlying tensor,
which is factorized into the product of two smaller tensors via transformed
tensor-tensor product under any unitary transformation. The main advantage of
transformed tensor-tensor product is that its computational complexity is lower
compared with the existing literature based on transformed tensor nuclear norm.
Moreover, the tensor $\ell_p$ $(0<p<1)$ norm is employed to characterize the
sparsity of sparse noise and the tensor Frobenius norm is adopted to model
additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to
solve the resulting model by linearizing the domain transformations and a
proximal Gauss-Seidel algorithm is developed to solve the corresponding
subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm
is established, whose convergence rate is also analyzed based on the
Kurdyka-$\L$ojasiewicz property. Extensive numerical experiments on real-world
image datasets are carried out to demonstrate the superior performance of the
proposed method as compared to several state-of-the-art methods in both
accuracy and computational time.
