Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci - G. Sundaramoorthi - A. Yezzi

Sobolev Active Contours

created by mennucci on 20 Dec 2006
modified on 13 Feb 2011


Published Paper

Inserted: 20 dec 2006
Last Updated: 13 feb 2011

Journal: International Journal of Computer Vision
Volume: 73
Number: 3
Year: 2007

The original publication is available at http:/dx.doi.org10.1007s11263-006-0635-2


All previous geometric active contour models that have been formulated as gradient flows of various energies use the same $L^2$-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and region-based active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.

Keywords: space of curves, sobolev active contours