The TVL1 Model: A Geometric Point of View

The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated Rudin-Osher-Fatemi image restoration model, yielding important advances in both f...

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Bibliographic Details
Published in:Multiscale modeling & simulation 2009-01, Vol.8 (1), p.154-189
Main Authors: Duval, Vincent, Aujol, Jean-François, Gousseau, Yann
Format: Article
Language:English
Subjects:
Algorithms
Byproducts
Computer Science
Computer simulation
Engineering Sciences
Exact solutions
Geometry
Image Processing
Image processing systems
Joints
Mathematical analysis
Mathematical models
Readers
Signal and Image processing
Studies
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Summary:The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated Rudin-Osher-Fatemi image restoration model, yielding important advances in both fields. We provide the reader with a geometrical characterization of the TVL1 model. We show that, in the convex case, exact solutions of the TVL1 problem are given by an opening followed by a simple test over the ratio perimeter/area. Shapes remain or suddenly vanish depending on this test. As a result of our theoretical study, we suggest a new and efficient numerical scheme to apply the model to digital images. As a by-product, we justify the use of the TVL1 model for image decomposition, by establishing a connection between the model and morphological granulometry. Eventually, we propose an extension of TVL1 into an adaptive framework, in which we derive some theoretical results. [PUBLICATION ABSTRACT]
ISSN:1540-3459
1540-3467
DOI:10.1137/090757083