Communications in Mathematical Sciences

Volume 20 (2022)

Number 4

Adaptive image processing: A bilevel structure learning approach for mixed-order total variation regularizers

Pages: 1117 – 1149

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n4.a8

Authors

Pan Liu (China National Clinical Research Center for Neurological Diseases, Beijing Tiantan Hospital, Capital Medical University, Beijing, China)

Xin Y. Lu (Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada)

Wan L. Zhu (China National Clinical Research Center for Neurological Diseases, Beijing Tiantan Hospital, Capital Medical University, Beijing, China)

Abstract

We propose a class of mixed-order PDE-constraint regularizer for image processing problem, generalizing the standard first-order total variation (TV). Then, we study the corresponding semi-supervised (bilevel) training scheme, which provides a simultaneous optimization with respect to parameters and new class of regularizers. Finally, by relying on the finite approximation method, we solve the global optimization problem on such training scheme, and analyze the resulting numerical results.

Keywords

image processing, optimal training scheme, higher-order differential operators, $\Gamma$-convergence

2010 Mathematics Subject Classification

26B30, 47J20, 94A08

Full Text (PDF format)

P. Liu acknowledges support from the EPSRC Centre Nr. EP/N014588/1 and the Leverhulme Trust project on Breaking the non-convexity barrier.

X.Y. Lu acknowledges the support of his NSERC Discovery Grant “Regularity of minimizers and pattern formation in geometric minimization problems”.

Received 27 April 2021

Received revised 30 October 2021

Accepted 30 October 2021

Published 11 April 2022