Structured images are then analyzed in terms of their component parts such as subsets of image regions, local neighborhoods, regional topologies, nearness and remoteness of sets, local convex sets, and mappings between image structures. The book introduces topology and its many applications viewed within a framework that begins with metric spaces and deals with the usual topics such as continuity, open and closed sets, metric nearness, compactness and completeness and glides into topology, proximity and uniformity. In the topic of extensions of continuous functions from dense subspaces, there are at least three important results due to Bourbaki, Hahn and Taimanov. A few proofs of the results are given and references are given for others. With the introduction of an abstract measure of oscillation and using the techniques of nearness spaces, it is shown that all the above results are consequences of one single proposition. The combination of traditional image analysis strategies and the more recent topology of digital images leads to a more complete view of digital images. The basic approach is to decompose an planar region containing an image object shape into 2-simplexes in such a way that the filled triangles cover either part or all of a shape.
It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. Third, an approach to quantifying the nearness of digital images is shown using the Henry-Peters nearness measure. It includes a presentation of spatially near sets as well as descriptively near sets. These include results pertaining to Fell topology, bounded Vietoris topology and, moreover, many other new hypertopologies are constructed. Many exercises come from the current literature and some occur in simplified form in metric spaces. The contribution of this article is threefold.
The introduction to nerve complexes includes a recent extension of nerve complexes that includes nerve spokes. The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces. Application: grills in pattern recognition. A cell complex K on a space X is a finite collection of subsets of X called cells such as 0-cells vertices and 1-cells open arcs. The focus of this article is on defining descriptive approximations of sets or algebraic structures in proximal relator spaces. Remarkably, no weaker convergence results if uniform convergence is replaced by pointwise convergence in the last statement. Special cases include characterizations of regular p-spaces and topologically complete or Čech complete spaces both with G δ -diagonal, complete Moore spaces, complete metric and compact Hausdorff spaces.
Structures induced by a metric. Use of near and far is intuitive yet rigourous at the same time, which is rare in mathematics. Near sets are disjoint sets that resemble each other, especially resemblance defined within perceptual representative spaces a. But in this paper, in a more general setting that includes data mining, probe functions ϕ i would be defined to allow for non-numerical values, i. This Foreword is especially important, since it lucidly brings together the principal highlights of this book and it serves as a commemoration of the seminal work on topology by A. Application: proximal neighbourhoods in camouflage neighbourhood filters.
Application: hit and miss topologies in population dynamics. The second author and P. The focus of this chapter is on fiber bundles defined by projections on a set of fibers that are nerve complexes into a base space such as the set of descriptions of nerve complexes. This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising. Applications include extensions of continuous functions to completions and compactifications. A space X is called countably compact if sequence in X has a cluster point. Intelligent Systems Reference Library, vol 102.
Compactness of a Hausdorff space. Taimanov, who proved one of the most fundamental results in topology concerning extensions of continuous functions from dense subspaces. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. We investigate properties of type D namely D-complete regularity and D-normality, i. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.
Application: quotient topology in image analysis. This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising. A novel feature throughout this book is the use of near and far, discovered by F. In 2006, a formal approach to the descriptive nearness of objects was considered by J. We gratefully acknowledge the insightful Foreword by I.
The end result of this work is a proximal physical geometric approach to detecting and classifying image object shapes. A number of results are given for fiber bundles viewed in the context of proximity spaces. In 2007, descriptively near sets were introduced by J. This article introduces descriptive cellular homology on cell complexes, which is an extension of J. Application: proximal neighbourhoods in cell biology. Gap functional and closure of a set. Thus, the ovals in Figure 1 are closer more near to each other in terms of their descriptions than the ovals in Figure 2.
Riesz over 100 years ago. Resemblance is determined by consider-ing set descriptions defined by feature vectors n-dimensional vectors of numerical features that represent characteristics of objects such as digi-tal image pixels. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. In this paper it is shown that the locally finite topology on 2 X is the Hausdorff generalized uniform topology corresponding to a generalized uniformity on X which is Mozzocchi as well as local. This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising.