Aspect graphs of bodies of revolution with algorithms of real algebraic geometry

Abstract

When we look at the image of a 3-D object onto a “retinal” plane under perspective projection, the abrupt changes of intensity in the image plane may have many different physical causes: the geometry of the object, the ambient illumination, texture, markings,etc. If we restrict the investigation to “geometric” image contours, we have to take two types of curves that lie on the object’s surface into account: the occluding contours, which are the points where the tangent plane contains a line of view (these curves change with the viewpoint), and the edges, where the normal vector’s direction is discontinuous (which are fixed). The image contour is the union of the projections of the occluding contours and of the edges onto the image plane. Eventually, the visible contour is the subset of the image contour which is really visible that is obtained after the removal of the occluded segments. A 3-D object has in general an infinite number of different images. However, an object whose surface boundaries are piecewise algebraic (as it is the case of any “CAD” object) has only a finite number of different visible contours under topological equivalence. From there comes the idea of representing a 3-D object with an aspect graph whose nodes are its qualitatively different visible contours (its aspects) and whose arcs are the transitions between aspects (its visual events) ([Kl], [K2]). This type of viewer-centered representation of 3-D objects could be relevant for visual object recognition ([V2]).KeywordsSymmetry AxisVisual EventAlgebraic SurfacePerspective ProjectionImage ContourThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.