Laurence Boxer

A Classical Construction for the Digital Fundamental Group

AbstractA version of topologys fundamental group is developed for digital images in dimension at most 3 in [7] and [8]. In the latter paper, it is shown that such a digital image X ⊂

Properties of Digital Homotopy

\mathcal{Z}^k

Digital Products, Wedges, and Covering Spaces

, k ≤ 3, has a continuous analog C(X) ⊂ Rk such that X has digital fundamental group isomorphic to Π1(C(X)). However, the construction of the digital fundamental group in [7] and [8] does not greatly resemble the classical construction of the fundamental group of a topological space. In the current paper, we show how classical methods of algebraic topology may be used to construct the digital fundamental group. We construct the digital fundamental group based on the notions of digitally continuous functions presented in [10] and digital homotopy [3]. Our methods are very similar to those of [6], which uses different notions of digital topology. We show that the resulting theory of digital fundamental groups is related to that of [7] and [8] in that it yields isomorphic fundamental groups for the digital images considered in the latter papers (for certain connectedness types).

Homotopy Properties of Sphere-Like Digital Images

Abstract In (Rosenfeld, 1986), a natural analog of the usual ϵ-δ definition of a continuous function is stated for digital pictures, and several properties of such digitally continuous functions are examined. The current paper expands on the work of (Rosenfeld, 1986). In particular, we examine digital versions of several important classes of continuous functions, including homeomorphisms, retractions, and homotopies.

On Hausdorff-like metrics for fuzzy sets (poster session)

Several recent papers have adapted notions of geometric topology to the emerging field of ‘digital topology.’ An important notion is that of digital homotopy. In this paper, we study a variety of digitally-continuous functions that preserve homotopy types or homotopy-related properties such as the digital fundamental group.

Polygonal approximation by boundary reduction

Abstract Known measurements on planar figures that may be computed efficiently include the Hausdorff metric and the homotopy type. However, these allow two objects with radically different topological or geometric properties to be considered close. In this paper, we discuss measures for polygons that may be computed efficiently, and in which closeness implies similarity with respect to deviation from convexity.

Point set pattern matching in 3-D

The paper [9] introduces the important tool of the digital covering space for studying the digital fundamental group. From a classical construction of algebraic topology [16,17,19], we show the existence of digital universal covering spaces and their significance for the study of the digital fundamental group.

The Classification of Digital Covering Spaces

Recent papers have discussed digital versions of the classical fundamental group for digital images. It has been shown that for non-contractible digital simple closed curves, the digital fundamental group is isomorphic to the integers, in analogy with Euclidean simple closed curves. In this paper, we show that the digital fundamental groups of sphere-like digital images Sn, n > 1, are trivial, as are the fundamental groups of their Euclidean analogs Sn.