Boundaries, shading, and border ownership: A cusp at their interaction

Matthew Lawlor, Daniel Holtmann-Rice, Patrick Huggins, Ohad Ben-Shahar, Steven W. Zucker

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


The association of borders with "figure" rather than "background" provides a topological organizing principle for early vision. Such global influences have recently been shown to have local effects, with neuronal activity modulated by stimulus properties from well outside the classical receptive field. We extend the theoretical analysis of such phenomena by developing the geometry of interaction between shading, boundaries, and boundary ownership for smooth surfaces. The purely exterior edges of smooth objects enjoy a fold-type relationship between shading and boundary, due to foreshortening, while the background is cut off transversely. However, at cusp points in the image mapping the exterior boundary ends abruptly. Since such singular points are notoriously unstable, we conjecture that this process is regularized by a natural quantization of suggestive contours due to physiological boundary-detection mechanisms. The result extends a theorem about how contours must end to one that characterizes surface (Gaussian) curvature in the neighborhood of where they appear to end. Apparent contours and their interaction with local shading thus provide important monocular shape cues.

Original languageEnglish
Pages (from-to)18-36
Number of pages19
JournalJournal of Physiology Paris
Issue number1-2
StatePublished - 1 Jan 2009


  • Co-circularity
  • Cusp
  • Differential geometry
  • Edge analysis
  • Long-range horizontal connections
  • Shading analysis


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