A texture model based on a concentration of measure

Hayden Schaeffer; John Garnett; Luminita A. Vese

doi:10.3934/ipi.2013.7.927

Article Contents

2013, Volume 7, Issue 3: 927-946. Doi: 10.3934/ipi.2013.7.927

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A texture model based on a concentration of measure

1.
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095
2.
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555

Received: June 30, 2012

Revised: April 30, 2013

Published: July 2013

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Abstract

Cartoon-texture regularization is a technique for reconstructing fine-scale details from ill-posed imaging problems, which are often ill-posed because of some non-invertible convolution kernel. Here we propose a cartoon-texture regularization for deblurring problems with semi-known kernel. The cartoon component is modeled by a function of bounded variation, while the texture component is measured by an approximate duality with Lipschitz functions ($W^{1,\infty}$). To approximate the dual Lipschitz norm, which is difficult to calculate, we propose an approach using concentration of measure. This provides an accurate and differentiable expression. We also present numerical results for our cartoon-texture decomposition, both in the case of a semi-known deblurring kernel and the case of a known kernel. The texture norm enables one to numerically reconstruct fine scale details that are typically difficult to recover from blurred images, and for semi-known deblurring the method quickly leads to the correct kernel, even when that kernel contains noise.

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Mathematics Subject Classification: 49M27, 28E99, 35K65.

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