Communications in Mathematical Sciences

Volume 7 (2009)

Number 2

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

Pages: 423 – 452



Laurence Guillot

Carole Le Guyader


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.


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

2010 Mathematics Subject Classification

35G25, 49L25, 68U10, 74G65

Published 1 January 2009