NDLI: The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory

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The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Cremers, Daniel Strekalovskiy, Evgeny Goldluecke, Bastian
Copyright Year	2012
Abstract	Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by $\text{TV}_J$, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that $\text{TV}_J$ can be derived from the generalized Jacobians from geometric measure theory. Thus, within the context of this theory, $\text{TV}_J$ is the most natural form of a vectorial total variation. As an important feature, we derive how $\text{TV}_J$ can be written as the support functional of a convex set in $\mathcal{L}^2$. This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist of an extensive experimental section, where we compare the performance of a number of provable convergent algorithms for inverse problems with our proposed regularizer. In particular, we show in experiments for denoising, deblurring, superresolution, and inpainting that its use leads to a significantly better restoration of color images, both visually and quantitatively. Source code for all algorithms employed in the experiments is provided online.
Starting Page	537
Ending Page	563
Page Count	27
File Format	PDF
DOI	10.1137/110823766
e-ISSN	19364954
Journal	SIAM Journal on Imaging Sciences (SJISBI)
Issue Number	2
Volume Number	5
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-04-12
Access Restriction	Subscribed
Subject Keyword	algorithms Optimization and variational techniques Image processing Methods involving duality vectorial total variation regularization color image restoration duality
Content Type	Text
Resource Type	Article

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