A variational model for capturing illusory contours using curvature

Wei Zhu*, Tony Chan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

30 Scopus citations


Illusory contours, such as the classical Kanizsa triangle and square [9], are intrinsic phenomena in human vision. These contours are not completely defined by real object boundaries, but also include illusory boundaries which are not explicitly present in the images. Therefore, the major computational challenge of capturing illusory contours is to complete the illusory boundaries. In this paper, we propose a level set based variational model to capture a typical class of illusory contours such as Kanizsa triangle. Our model completes missing boundaries in a smooth way via Euler's elastica, and also preserves corners by incorporating curvature information of object boundaries. Our model can capture illusory contours regardless of whether the missing boundaries are straight lines or curves. We compare the choice of the second order Euler's elastica used in our model and that of the first order Euler's elastica developed in Nitzberg-Mumford-Shiota's work on the problem of segmentation with depth [15, 16]. We also prove that with the incorporation of curvature information of objects boundaries our model can preserve corners as completely as one wants. Finally we present the numerical results by applying our model on some standard illusory contours.

Original languageEnglish (US)
Pages (from-to)29-40
Number of pages12
Issue number1
StatePublished - Jan 2007
Externally publishedYes


  • Euler's elastica
  • Illusory contours
  • Level set methods

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Condensed Matter Physics
  • Computer Vision and Pattern Recognition
  • Geometry and Topology
  • Applied Mathematics


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