A combined first and second order variational approach for image reconstruction

TL;DR

This paper introduces a higher-order variational model for image reconstruction that extends the ROF functional with a non-smooth second order regulariser, demonstrating improved artifact reduction and computational efficiency.

Contribution

It proposes a novel combined first and second order variational model in the space of functions of bounded Hessian, with proven existence, uniqueness, and efficient numerical solution methods.

Findings

The model effectively reduces artifacts compared to traditional TV-based methods.

Numerical results show competitive performance with TGV, infimal convolution, and Euler's elastica.

The approach is simple, efficient, and avoids blocky artifacts in reconstructed images.

Abstract

In this paper we study a variational problem in the space of functions of bounded Hessian. Our model constitutes a straightforward higher-order extension of the well known ROF functional (total variation minimisation) to which we add a non-smooth second order regulariser. It combines convex functions of the total variation and the total variation of the first derivatives. In what follows, we prove existence and uniqueness of minimisers of the combined model and present the numerical solution of the corresponding discretised problem by employing the split Bregman method. The paper is furnished with applications of our model to image denoising, deblurring as well as image inpainting. The obtained numerical results are compared with results obtained from total generalised variation (TGV), infimal convolution and Euler's elastica, three other state of the art higher-order models. The numerical discussion confirms that the proposed higher-order model competes with models of its kind in avoiding the creation of undesirable artifacts and blocky-like structures in the reconstructed images -- a known disadvantage of the ROF model -- while being simple and efficiently numerically solvable.