Peters J.F. Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces

Springer, 2014. — 414 p.This book carries forward recent work on visual patterns and structures in digital images and introduces a near set-based topology of digital images. Visual patterns arise naturally in digital images viewed as sets of non-abstract points (also called picture points1 or places) endowed with some form of proximity (nearness) relation. Proximity relations make it possible to construct uniform topologies on the sets of points that constitute a digital image. A uniform topology on a digital image is constructed by finding all sets of points that are near each given set in an image. By stepping back and taking a look at any given set in an image, one finds that each selected set of points serves as a visual pattern generator.
The discovery of visual patterns in digital images starts by viewing each given set of image points as a pattern generator and then finding all sets of points that are near the generator.
In keeping with an interest in gaining an understanding of digital images themselves as a rich source of patterns, this book introduces the basics of digital images from a computer vision perspective. In parallel with a computer vision perspective on digital images, this book also introduces the basics of proximity spaces. Not only the traditional view of spatial proximity relations but also the more recent descriptive proximity relations are considered. Spatial proximity of sets of points in images is important to consider in gaining an understanding of how visual set patterns fit together in a mosaic of overlapping and adjacent geometric structures.
This study leads to the discovery of various digital image structures such as neighbourhoods of points, neighbourhoods of sets, boundary points, interior points, clusters and covers as well as many other structures that can be found in an image. A byproduct to the structured set view of digital images is the discovery of a number of different, useful forms of visual patterns in image structures. The study of digital image patterns leads to a deeper perception and understanding of the underlying meanings in images.
A generous provision of MatLAB and Mathematica scripts are used in this book to lay bare the fabric and essential features of digital images for those who are interested in finding visual patterns in images. The combination of computer vision techniques and topological methods lead to a deep understanding of images. Some of the shorter MatLAB and Mathematica scripts are included in some of the chapters as illustrations of what can be done. Otherwise, the remaining MatLAB and Mathematica scripts used to produce the chapter examples can be found in the book Appendix. The Appendix sections correspond to the book chapters where the scripts are referenced.Topology of Digital Images: Basic Ingredients
Structures Arising from Sets of Pixels
Visualisations and Covers
Linear Filtering and Visual Patterns in Images
Edges, Lines, Ridges, and Nearness Structures
Corners and Symmetric Proximity
Separation of Image Regions and Set Patterns
Descriptive Raster Spaces
Component Analysis and Uniform Spaces
Shapes and Shape Set Patterns
Texture and Texture Set Patterns
Pattern-Based Picture Classification
A Appendix: MatLAB and Mathematica Scripts
B Notes and Further Readings