The Topological Complexity of Spaces of Digital Images
2019-06
Loading...
View/Download File
Persistent link to this item
Statistics
View StatisticsJournal Title
Journal ISSN
Volume Title
Title
The Topological Complexity of Spaces of Digital Images
Authors
Published Date
2019-06
Publisher
Type
Thesis or Dissertation
Abstract
The motivation of this dissertation is to study 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. In Chapter 2, we develop tools for computing the topological complexity of finite spaces, with an emphasis on spheres, joins, and wedge sums. The main result of Chapter 4 is that our space of digital Jordan curves is connected, hence, its topological complexity is finite. To build up to that, we use Chapter 3 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 Chapter 5, 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.
Description
University of Minnesota Ph.D. dissertation. June 2019. Major: Mathematics. Advisor: Craig Westerland. 1 computer file (PDF); x, 98 pages.
Related to
Replaces
License
Collections
Series/Report Number
Funding information
Isbn identifier
Doi identifier
Previously Published Citation
Suggested citation
Kandola, Shelley Burrows. (2019). The Topological Complexity of Spaces of Digital Images. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/206253.
Content distributed via the University Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. By using these files, users agree to the Terms of Use. Materials in the UDC may contain content that is disturbing and/or harmful. For more information, please see our statement on harmful content in digital repositories.