Directions in Digital Topology
by
Azriel Rosenfeld
University of Maryland

This talk will broadly survey ways in which the basic concepts that underlie digital topology can be formulated and (possibly) generalized.

The discrete grid (of pixels or voxels) used in digital topology can be regarded as a "digitization" of (two or three-dimensional) Euclidean space; from this viewpoint, it is of interest to study conditions under which this digitization process preserves topological (or other geometric) properties. Alternatively, the grid can be regarded as an abstract discrete space; several approaches of this type have been extensively studied.

We usually assume that a two-valued function is defined on the grid, which is thus regarded as consisting of "black" and "white" elements ("1's" and "0's"). A ßofter" (and more general) approach is to use a function into [0, 1] (rather than {0, 1}); it can be regarded as measuring the "membership" of an element in the fuzzy set of "dark" elements. Many "geometrical" properties-including some topological properties-can be defined in this extended framework.

Various types of topological "properties" can be defined in digital topology; these include not only numerical-valued properties such as the Euler characteristic, but also "graph-valued" properties (e.g., the component containment tree) or "group-valued" properties (e.g., the fundamental group of a knot). Issues that arise in connection with all these types of properties are their "local computability" and the complexity of their computation.

A final problem of interest is that of characterizing topology-preserving operations; this issue can be raised for both mappings defined on the grid (e.g., distance-nonincreasing mappings) and mappings that modify the function defined on the grid (e.g., topology-preserving local operations).

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caaf-71.