A Convex Variational Model for Learning Convolutional Image Atoms from Incomplete Data

A Chambolle,T Pock,M Holler

doi:10.1007/s10851-019-00919-7

Abstract

A variational model for learning convolutional image atoms from corrupted and/or incomplete data is introduced and analyzed both in function space and numerically. Building on lifting and relaxation strategies, the proposed approach is convex and allows for simultaneous image reconstruction and atom learning in a general, inverse problems context. Further, motivated by an improved numerical performance, also a semi-convex variant is included in the analysis and the experiments of the paper. For both settings, fundamental analytical properties allowing in particular to ensure well-posedness and stability results for inverse problems are proven in a continuous setting. Exploiting convexity, globally optimal solutions are further computed numerically for applications with incomplete, noisy and blurry data and numerical results are shown.

Highlights

An important task in image processing is to achieve an appropriate regularization or smoothing of images or image-related data
Doing so and starting from (1), we introduce a convex variational method for learning image atoms from noisy and/or incomplete data in an inverse problems context
We have introduced a variational approach for learning image atoms from corrupted and/or incomplete data

Summary

Introduction

An important task in image processing is to achieve an appropriate regularization or smoothing of images or image-related data. This is indispensable for most applicationdriven problems in the field, such as denoising, inpainting, reconstruction, segmentation, registration or classification. For general problem settings in the field of inverse problems, an appropriate regularization of. The Institute of Mathematics and Scientific Computing is a member of NAWI Graz (http://www.nawigraz.at) and BioTechMed Graz (http:// www.biotechmed.at)

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