  Second order total variation (SOTV) models have advantages for image
reconstruction over their first order counterparts including their ability to
remove the staircase artefact in the reconstructed image, but they tend to blur
the reconstructed image. To overcome this drawback, we introduce a new Tensor
Weighted Second Order (TWSO) model for image reconstruction. Specifically, we
develop a novel regulariser for the SOTV model that uses the Frobenius norm of
the product of the SOTV Hessian matrix and the anisotropic tensor. We then
adapt the alternating direction method of multipliers (ADMM) to solve the
proposed model by breaking down the original problem into several subproblems.
All the subproblems have closed-forms and can thus be solved efficiently. The
proposed method is compared with a range of state-of-the-art approaches such as
tensor-based anisotropic diffusion, total generalised variation, Euler's
elastica, etc. Numerical experimental results of the method on both synthetic
and real images from the Berkeley database BSDS500 demonstrate that the
proposed method eliminates both the staircase and blurring effects and
outperforms the existing approaches for image inpainting and denoising
applications.
