From a Non-Local Ambrosio-Tortorelli Phase Field to a Randomized Part Hierarchy Tree

  • Sibel TariEmail author
  • Murat Genctav


In its most widespread imaging and vision applications, Ambrosio and Tortorelli (AT) phase field is a technical device for applying gradient descent to Mumford and Shah simultaneous segmentation and restoration functional or its extensions. As such, it forms a diffuse alternative to sharp interfaces or level sets and parametric techniques. The functionality of the AT field, however, is not limited to segmentation and restoration applications. We demonstrate the possibility of coding parts—features that are higher level than edges and boundaries—after incorporating higher level influences via distances and averages. The iteratively extracted parts using the level curves with double point singularities are organized as a proper binary tree. Inconsistencies due to non-generic configurations for level curves as well as due to visual changes such as occlusion are successfully handled once the tree is endowed with a probabilistic structure. As a proof of concept, we present (1) the most probable configurations from our randomized trees; and (2) correspondence matching results between illustrative shape pairs.

The work is a significant step towards establishing exponentially decaying diffuse distance fields as bridges between low level visual processing and shape computations.


Bridging low level and high level vision Shape computation Screened Poisson PDE Implicit representations Linear model for reaction-diffusion 



This work has been partially funded by TUBITAK grant 112E208, the Alexander von Humboldt Foundation, and TUBITAK-BIDEB fellowship.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Middle East Technical UniversityAnkaraTurkey

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