A Combined First and Second Order Variational Approach for Image Reconstruction

K. Papafitsoros, C. B. Schönlieb

Research output: Contribution to journalArticlepeer-review

175 Scopus citations


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. ©Springer Science+Business Media New York 2013.
Original languageEnglish (US)
Pages (from-to)308-338
Number of pages31
JournalJournal of Mathematical Imaging and Vision
Issue number2
StatePublished - May 10 2013
Externally publishedYes


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