B. Dubuc, S.W. Zucker In differential geometry curves are characterized as mappings from an interval to the plane. In topology curves are characterized as a Hausdorff space with certain countability properties. Neither of these definitions captures the role that curves play in vision, however, in which curves can denote simple objects (such as a straight line), or complicated objects (such as a jumble of string). The difference between these situations is in part a measure of their complexity, and in part a measure of their dimensionality. We seek a formal theory of curves appropriate for computational vision in general, and for problems like separating straight lines from jumbles in particular (see Fig. 7). We propose a theory of the complexity of curves that is sufficient to separate those which extend along their length (e.g., in one dimension) from those that cover an area (e.g., 2-D) and those that are isolated (e.g. 0-D), such as discontinuities. The theory is based on original results in geometric measure theory, and is applied to the problems of perceptual grouping. Most importantly, it provides the basis of a classification scheme for curve-like sets as those encountered in edge/line detection.
Figure 7: Segmenting and grouping Paolina edges. (a) original image of a statue, (b) discrete tangent map (edges), (c)-(f) segmented tangent map using normal and tangential complexities and a partitioning scheme for the complexity space; (c) low normal, low tangential: dust (d) low normal, high tangential: curves (e) high normal, low tangential: turbulence (f) high normal, high tangential: flow.