Published Paper

Inserted: 23 dec 2021
Last Updated: 14 oct 2022

Journal: SIAM Journal on Imaging Sciences
Volume: 15
Year: 2022
Doi: 10.1137/21M1467328

Abstract:

In the context of image processing, we study a class of integral regularizers defined in terms of spatially inhomogeneous integrands that depend on general linear differential operators. Particularly, the spatial dependence is assumed to be only measurable. The setting is made rigorous by means of the theory of Radon measures and of suitable function spaces modeled on functions of bounded variation. We prove the lower semicontinuity of the functionals at stake and existence of minimizers for the corresponding variational problems. Then, we embed the latter into a bilevel scheme in order to automatically compute the regularization parameters. These parameters are considered to be spatially varying, thus allowing for good flexibility and preservation of details in the reconstructed image. After identifying a series of spatially inhomogeneous regularization functionals commonly used in image processing that are included in our framework, we substantiate its feasibility by performing numerical denoising examples in which the spatial dependence of the integrand is measurable. Specifically, we use Huber versions of the first and second order total variation (and their sum) with both the Huber and the regularization parameter being spatially varying. Notably, the spatially varying version of second order total variation produces high quality reconstructions when compared to regularizations of similar type, and the introduction of the low regularity spatially dependent Huber parameter leads to a further enhancement of the image details. We expect that our theoretical investigations and our numerical feasibility study will support future work on setting up schemes where general differential operators with spatially dependent coefficients will also be part of the optimization scheme.

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• PPRV-Bilevel-vDec21.pdf