An Elastica-Driven Digital Curve Evolution Model for Image Segmentation
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An Elastica-Driven Digital Curve Evolution Model for Image Segmentation Daniel Antunes1 · Jacques-Olivier Lachaud1 · Hugues Talbot2 Received: 8 July 2019 / Accepted: 10 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Geometric priors have been shown to be useful in image segmentation to regularize the results. For example, the classical Mumford–Shah functional uses region perimeter as prior. This has inspired much research in the last few decades, with classical approaches like the Rudin–Osher–Fatemi and most graph-cut formulations, which all use a weighted or binary perimeter prior. It has been observed that this prior is not suitable in many applications, for example for segmenting thin objects or some textures, which may have high perimeter/surface ratio. Mumford observed that an interesting prior for natural objects is the Euler elastical model, which involves the squared curvature. In other areas of science, researchers have noticed that some physical binarization processes, like emulsion unmixing, can be well-approximated by curvature-related flow like the Willmore flow. However, curvature-related flows are not easy to compute because curvature is difficult to estimate accurately, and the underlying optimization processes are not convex. In this article, we propose to formulate a digital flow that approximates an Elastica-related flow using a multigrid-convergent curvature estimator, within a discrete variational framework. We also present an application of this model as a post-processing step to a segmentation framework. Keywords Multigrid convergence · Digital estimator · Curvature · Shape optimization · Image segmentation
1 Introduction Geometric quantities are particularly useful as regularizers in low-level image analysis, especially when little prior information is known about the shape of interest. Length penalization is a well-behaved, general-purpose regularizer, and many models in the literature make use of it, from active contours [8] to level-set formulations [26,27]. Discrete graph-based variational models have been particularly
This work has been partially funded by CoMeDiC ANR-15-CE40-0006 research grant. B Hugues Talbot [email protected] Daniel Antunes [email protected] Jacques-Olivier Lachaud [email protected] 1
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
2
Université Paris-Saclay, CentraleSupelec, Inria, 9 rue Joliot-Curie, 91190 Gif-sur-Yvette, France
successful to incorporate length penalization as a penalizer while keeping the ability to extract a global optimum [2,5]. However, length regularization shows limitations when segmenting small or thin and elongated objects, as it tends to shrink solutions or yields disconnected solutions. Such drawbacks can be problematic in image segmentation, image restoration or image inpainting. It is thus rather natural to consider curvature (especially squared curvature) as a potential regularizer. Energies involving both
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