Tensor Based Second Order Variational Model for Image Reconstruction

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.