Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

Yunho Kim; Luminita A. Vese

doi:10.3934/ipi.2009.3.43

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2009, Volume 3, Issue 1: 43-68. Doi: 10.3934/ipi.2009.3.43

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Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability

Yunho Kim^1, and
Luminita A. Vese^1,

1.
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555

Received: August 31, 2008

Revised: October 31, 2008

Published: January 2009

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Abstract

In this work we wish to recover an unknown image from a blurry, or noisy-blurry version. We solve this inverse problem by energy minimization and regularization. We seek a solution of the form $u + v$, where $u$ is a function of bounded variation (cartoon component), while $v$ is an oscillatory component (texture), modeled by a Sobolev function with negative degree of differentiability. We give several results of existence and characterization of minimizers of the proposed optimization problem. Experimental results show that this cartoon + texture model better recovers textured details in natural images, by comparison with the more standard models where the unknown is restricted only to the space of functions of bounded variation.

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Mathematics Subject Classification: Primary: 35J85, 49J40, 65M06; Secondary: 65M32.

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