Indexing visual representations through the complexity map

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. Note that the map defining such curves is unknown, as is the proper way to represent them. We propose a formal complexity theory of curves appropriate for computational vision in general, and for problems like separating straight lines from jumbles in particular. The theory is applied to the problem of perceptual grouping