Journal of Mathematical Imaging and Vision

, Volume 48, Issue 2, pp 308–338 | Cite as

A Combined First and Second Order Variational Approach for Image Reconstruction



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.


Functions of bounded Hessian Split Bregman Denoising Deblurring Inpainting Staircasing 



The authors acknowledge the financial support provided by the Cambridge Centre for Analysis (CCA), the Royal Society International Exchanges Award IE110314 for the project “High-order Compressed Sensing for Medical Imaging”, the EPSRC/Isaac Newton Trust Small Grant “Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors” and the EPSRC first grant Nr. EP/J009539/1 “Sparse & Higher-order Image Restoration”. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). We thank Clarice Poon for providing us with the Euler’s elastica code. Finally, we would like to thank the referees for their very useful comments and suggestions that improved the presentation of the paper.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Cambridge Centre for Analysis, Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK

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