Aspect Graphs of Algebraic Surfaces.

Marie-Françoise Roy

Thierry Van Effelterre

IRMAR, University of Rennes

Abstract:

An Aspect graph is a representation of 3-d objects that is used in the field of Computer Vision for recognition in 2-d images. The viewspace around the object is tesselated in a finite number of cells by the semi-algebraic Visual Events locus. The topology of the image contour remains stable in each cell and may only change on the Visual Events locus. An Aspect Graph represents a 3-d object whose surface boundary is algebraic or semi-algebraic by the finite number of different topological aspects of its image contour and by the visual events that make a stable aspect switch to another one. We show here that the number of different topological aspects of an algebraic surface of degree d is upper-bounded by a for orthographic projection and for perspective projection. This result is a generalisation of the upper-bound of we obtained in [RV] for surfaces of revolution under perspective projection and improves the most recent upper-bounds of for orthographic projection and for perspective projection.([R4]). We also show how to compute the equations of the Visual Events locus with Gröbner bases systems and Hermite's method.

• Aspect Graphs of 3-d objects.
• Upper-bound for the degree of the Visual Events Locus.
  • Visual Events for Compact surfaces.
    • Perspective projection.
    • Orthographic projection.
  • Singularities at Infinity for Non Compact surfaces.
    • Perspective projection.
    • Orthographic projection.
• Number of nodes of the Aspect Graph.
• Description of the global Visual Events locus.
• Summary.
• About this document ...