Sobolev active contours

Ganesh Sundaramoorthi*, Anthony Yezzi, Andrea C. Mennucci

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

140 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)345-366
Number of pages22
JournalInternational Journal of Computer Vision
Issue number3
StatePublished - Jul 2007
Externally publishedYes

Bibliographical note

Funding Information:
The authors would like to acknowledge the support of the following grants: NSF CCR-0133736, NIH/NINDS R01-NS-037747 and an Airforce MURI grant.


  • Active contours
  • Global flows
  • Gradient flows
  • Shape optimization
  • Sobolev norm

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Artificial Intelligence


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