A combined first and second order variational approach for image reconstruction

Details A combined first and second order variational approach for image reconstruction A combined first and second order variational approach for image reconstruction

2012-02-27

arxiv.org/abs/1202.6341v1

Mathematics

Numerical Analysis

33 pages, 33 figures

Scientific paper

In this paper we formulate and study a variational problem in the space of functions of bounded Hessian $BH(\Omega)$. This forms a higher order extension of the well known ROF functional (total variation minimisation) which is widely used for image reconstruction. Our functional involves convex functions of the total variation and the total variation of the first derivatives. We prove existence and uniqueness of minimisers via the method of relaxation. We use the split Bregman method in order to solve numerically the corresponding discretised problem and we prove some convergence results. We apply our model to image denoising (Gaussian and impulse noise) and deblurring as well as image inpainting. The numerical results show that our model avoids the creation of undesirable artifacts and blocky-like structures in the reconstructed images which is a disadvantage of the ROF model.

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