Communications in Mathematical Sciences

Extrapolation of vector fields using the infinity Laplacian and with applications to image segmentation

Carole Le Guyader and Laurence Guillot

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In this paper, we investigate a new Gradient-Vector-Flow (GVF)-inspired static external force field for active contour models, deriving from the edge map of a given image and allowing to increase the capture range. Contrary to prior related works, we reduce the number of unknowns to a single one v by assuming that the expected vector field is the gradient field of a scalar function. The model is phrased in terms of a functional minimization problem comprising a data fidelity term and a regularizer based on the supremum norm of $Dv$.

The minimization is achieved by solving a second order singular degenerate parabolic equation. A comparison principle as well as the existence/uniqueness of a viscosity solution together with regularity results are established. Experimental results for image segmentation with details of the algorithm are also presented.

Article information

Commun. Math. Sci., Volume 7, Number 2 (2009), 423-452.

First available in Project Euclid: 27 May 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer 68U10: Image processing 49L25: Viscosity solutions 35G25: Initial value problems for nonlinear higher-order equations 35D05 35D10 74G65: Energy minimization

Gradient Vector Flow infinity Laplacian AMLE partial differential equations viscosity solutions segmentation


Le Guyader, Carole; Guillot, Laurence. Extrapolation of vector fields using the infinity Laplacian and with applications to image segmentation. Commun. Math. Sci. 7 (2009), no. 2, 423--452.

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