nach oben

2016 | Buch

Kapitel lesen Erstes Kapitel lesen

Computational Proximity

Excursions in the Topology of Digital Images

verfasst von: James F. Peters

Verlag: Springer International Publishing

Buchreihe : Intelligent Systems Reference Library

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book introduces computational proximity (CP) as an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. Typically in computational proximity, the book starts with some form of proximity space (topological space equipped with a proximity relation) that has an inherent geometry. In CP, two types of near sets are considered, namely, spatially near sets and descriptivelynear sets. It is shown that connectedness, boundedness, mesh nerves, convexity, shapes and shape theory are principal topics in the study of nearness and separation of physical aswell as abstract sets. CP has a hefty visual content. Applications of CP in computer vision, multimedia, brain activity, biology, social networks, and cosmology are included. The book has been derived from the lectures of the author in a graduate course on the topology of digital images taught over the past several years. Many of the students have provided important insights and valuable suggestions. The topics in this monograph introduce many forms of proximities with a computational flavour (especially, what has become known as the strong contact relation), many nuances of topological spaces, and point-free geometry.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Computational Proximity

Abstract

This chapter introduces computational proximity. Basically, computational proximity (CP) is an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. The methods used by CP to find either near sets or remote sets result from the study of structures called proximity spaces.

James F. Peters

Chapter 2. Proximities Revisited

Abstract

This chapter takes another look at the very rich proximity landscape. An overview of the proximity landscape is given in the life and work of S.A. Naimpally (Som) (Beer et al., Topol Appl 188:97–109, 2015), [1]. This is a remarkable story of a mathematician who began studying proximity space theory after he completed his Ph.D. as a result of a chance meeting at the University of Michigan between Som and a visitor from Cambridge University Press, who invited him to write a monograph on proximity.

James F. Peters

Chapter 3. Distance and Proximally Continuous

Abstract

This chapter introduces distance functions called metrics and continuous functions.

James F. Peters

Chapter 4. Image Geometry and Nearness Expressions for Image and Scene Analysis

Abstract

This chapter suggests an approach to image and scene analysis based on Dirichlet (also called Voronoï) tessellations and Delaunay triangulation of selected seed (generating or site) points in a digital image. A visual scene is a collection of objects in a visual field that captures our attention. In human vision, a visual field is the total area in which objects can be seen. A normal visual field is about 60\(^\circ \) from the vertical meridian of each eye and about 60\(^\circ \) above and 75\(^\circ \) below the horizontal meridian.

James F. Peters

Chapter 5. Homotopic Maps, Shapes and Borsuk–Ulam Theorem

Abstract

This chapter introduces object spaces, where objects are located in a visual field.

James F. Peters

Chapter 6. Visibility, Hausdorffness, Algebra and Separation Spaces

Abstract

This chapter introduces visibility and separation spaces, useful in the study of set patterns. The notion of visibility stems from our daily experience in being able to seg our vision. In a sense, visibility is the opposite of separation of sets. In topology, disjoint sets are separated.

James F. Peters

Chapter 7. Strongly Near Sets and Overlapping Dirichlet Tessellation Regions

Abstract

This chapter introduces various forms of sites (seed or generating points, including hybrid generating points) used in tessellating digital images. A natural outcome of an image tessellation is an approximate image segmentation. The segments in this case are the byproduct of various forms of Voronoï and Delaunay mesh cells containing all of those image pixels nearest to each mesh generating point.

James F. Peters

Chapter 8. Proximal Manifolds

Abstract

A topological manifold is a Hausdorff space with a countable basis where each point has a neighbourhood homeomorphic to some Euclidean space (Milnor, Topological manifolds and smooth manifolds, 1962, [1]).

James F. Peters

Chapter 9. Watershed, Smirnov Measure, Fuzzy Proximity and Sorted Near Sets

Abstract

This chapter introduces proximal watershed image segments and Voronoï diagrams. The watershed image segmentation method is a centerpiece in mathematical morphology (NM).

James F. Peters

Chapter 10. Strong Connectedness Revisited

Abstract

This chapter takes another look at connectedness. The traditional view of connected spaces is augmented with the introduction of strongly near proximity, which ushers in new forms of connectedness, namely.

James F. Peters

Chapter 11. Helly’s Theorem and Strongly Proximal Helly Theorem

Abstract

This chapter introduces a strongly proximal version of Helly’s theorem for convex sets and convex bodies. The collection of labelled Voronoï regions in the mesh nerve on the apple surface in Fig. 11.1 is an example of a Helly family of convex bodies.

James F. Peters

Chapter 12. Nerves and Strongly Near Nerves

Abstract

This chapter introduces various forms of geometric nerves, which are usually collections of what known as simplical complexes in a normed linear space. This chapter also introduces geometric nerves called polyform nerves that vary from the usual view of simplical complexes. In general, a nerve is a simplical complex (Aequationes Mathematicae 2(2–3):400–401, 1969, [1]).

James F. Peters

Chapter 13. Connnectedness Patterns

Abstract

This chapter introduces the study of connectedness patterns, which are collections of sets that exhibit a repetition of strongly connected polygons in a tiling of the plane.

James F. Peters

Chapter 14. Nerve Patterns

Abstract

This chapter introduces mesh nerve patterns. A nucleus nerve pattern is an arrangement of mesh nerves based on a partial ordering of the nuclei of the nerves. A nerve nucleus in a Voronoï diagram is an ngon that is strongly connected to polygons along the border of the nucleus. The partial ordering relation \(\le \) on either the number of polygon sides or the areas of the nuclei leads to the arrangement of the nerves in a mesh nerve pattern.

James F. Peters

Backmatter

Titel: Computational Proximity
verfasst von: James F. Peters
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-30262-1
Print ISBN: 978-3-319-30260-7
DOI: https://doi.org/10.1007/978-3-319-30262-1