Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

• Published in: Journal of mathematical imaging and vision Vol. 57; no. 1; pp. 1 - 25
• Main Authors: De los Reyes, J. C., Schönlieb, C.-B., Valkonen, T.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2017; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Image Processing and Computer Vision; Image reconstruction; Learning; Mathematical analysis; Mathematical Methods in Physics; Noise levels; Optimization; Parameters; Regularization; Signal,Image and Speech Processing; Bilevel optimisation; Total variation regularisers; Image quality measures
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-016-0662-8

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.