A Topological Perspective On The Structure Of Images

A.Peter Blicher

doi:10.1117/12.944704

24 June 1988 A Topological Perspective On The Structure Of Images

A.Peter Blicher

Proceedings Volume 0901, Image Processing, Analysis, Measurement, and Quality; (1988) https://doi.org/10.1117/12.944704
Event: 1988 Los Angeles Symposium: O-E/LASE '88, 1988, Los Angeles, CA, United States

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.

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A.Peter Blicher "A Topological Perspective On The Structure Of Images", Proc. SPIE 0901, Image Processing, Analysis, Measurement, and Quality, (24 June 1988); https://doi.org/10.1117/12.944704

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KEYWORDS

Eye

Image processing

Distortion

Image analysis

Image quality

Quality measurement

Convolution

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