The Topological Complexity of Spaces of Digital Jordan Curves

08/19/2019
by   Shelley Kandola, et al.
0

This research is motivated by studying image processing algorithms through a topological lens. The images we focus on here are those that have been segmented by digital Jordan curves as a means of image compression. The algorithms of interest are those that continuously morph one digital image into another digital image. Digital Jordan curves have been studied in a variety of forms for decades now. Our contribution to this field is interpreting the set of digital Jordan curves that can exist within a given digital plane as a finite topological space. Computing the topological complexity of this space determines the minimal number of continuous motion planning rules required to transform one image into another, and determining the motion planners associated to topological complexity provides the specific algorithms for doing so. The main result of Section 3 is that our space of digital Jordan curves is connected, hence, its topological complexity is finite. To build up to that, we use Section 2 to prove some results about paths and distance functions that are obvious in Hausdorff spaces, yet surprisingly elusive in T_0 spaces. We end with Section 4, in which we study applications of these results. In particular, we prove that our interpretation of the space of digital Jordan curves is the only topologically correct interpretation. This article is an adaptation of the author's Ph.D. dissertation.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/08/2023

Geodesic complexity of a cube

The topological (resp. geodesic) complexity of a topological (resp. metr...
research
07/06/2022

Value-Offset Bifiltrations for Digital Images

Persistent homology, an algebraic method for discerning structure in abs...
research
05/16/2013

Geometric primitive feature extraction - concepts, algorithms, and applications

This thesis presents important insights and concepts related to the topi...
research
08/23/2022

Digital topological groups

In this article, we develop the basic theory of digital topological grou...
research
12/15/2021

LTB curves with Lipschitz turn are par-regular

Preserving the topology during a digitization process is a requirement o...
research
09/30/2017

Unsupervised Classification of Intrusive Igneous Rock Thin Section Images using Edge Detection and Colour Analysis

Classification of rocks is one of the fundamental tasks in a geological ...
research
09/16/2021

0-Gaps on 3D Digital Curves

In Digital Geometry, gaps are some basic portion of a digital object tha...

Please sign up or login with your details

Forgot password? Click here to reset