Line-drawing interpretation: A mathematical framework

摘要

We report progress toward a mathematical theory of line-drawing interpretation. A working framework is developed, and a variety of tools and techniques are demonstrated within it. We begin with a concise review of related work. Then, we detail our assumptions—the imaging geometry is well approximated by orthographic or perspective projection, the viewpoint is general, the world is composed of piecewise C3 surfaces, and continuous-surface-normal depth discontinuities are the only viewpoint-dependent edges. This is followed by the presentation of a projective mapping which we expect will find considerable use in the extension of results derived for orthographic projection to perspective projection. Next, Whitney's definition of an excellent mapping is reviewed and it is shown that both orthographic and perspective projections of piecewise C3 surfaces are excellent under general viewpoint. It is thus demonstrated that there are basically only two different types of local singularities—“folds” and “cusps”—associated with the projection of a C3 surface. The loci of the singular values are C3 curves in the line drawing, except for cusps at their nonjunction terminations. All previous work based on Whitney's results on excellent mappings has needed to assume stability under perturbation of the viewed surface. This assumption, which is unverifiable by a remote observer, is rendered unnecessary by our results. After characterizing viewpoint-dependent edges as folds and cusps, we turn our attention to the modes of interaction between viewpoint-independent and viewpoint-dependent edges. It is shown that only isolated points on viewpoint-dependent edges may lie on surface-patch boundaries, and that viewpoint-independent and viewpoint-dependent edges are always cotangent at their nonocclusion junctions in line drawings, but have different curvatures. Finally, we investigate continuity constraints imposed on the scene by its line drawing. We show that a viewpoint-independent edge in a line drawing is C3 if and only if the edge is C3 in space, and that a viewpoint-dependent edge in a line drawing is C3 if and only if the surface on which it lies in space is C3 along the edge.