# A topological perspective on the structure of images

## Abstract

Motivated by the correspondence problem in computer vision, we take a theoretical look at the topological invariants of images (i.e., in-variants under arbitrary distortion). For monochrome images, these are the intensity contour portraits, and they have a simple structure: isolated points, nested circles (closed curves topologically equivalent to circles) and (topological) figure-eights. The isolated points are maxima and minima, while the figure-eights are contours through saddle points. The circles are contours through points with nonzero gradient. The nesting of the contours is represented as a binary tree structure. As the picture is varied, whether by adding noise, blurring, or some other smooth change, the tree structure of the picture usually stays constant, but sometimes undergoes abrupt changes. These “bifurcations” are classified into only 2 basic types: saddle-node, and saddle connection. The results are based on genericity assumptions i.e., they are true for almost all pictures, thus excluding pathological cases. This leads to the conclusion that based on topological constraints alone, correspondence within a smooth object can be determined at best only at a sparse set of points. If we allow color images, the situation changes, and if there is enough color variation, correspondence can be determined for all points on a “generic” smooth object. © 1988, SPIE.