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

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February 2009, 3(1): 43-68. doi: 10.3934/ipi.2009.3.43

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

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

Received September 2008 Revised November 2008 Published February 2009

Abstract
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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.

Keywords: Sobolev spaces, oscillatory functions., Image restoration, bounded variation, variational models.

Mathematics Subject Classification: Primary: 35J85, 49J40, 65M06; Secondary: 65M3.

Citation: Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43

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