The author presents three distinct but related branches of science in this book: digital geometry, mathematical morphology, and discrete optimization. They are united by a common mindset as well as by the many applications where they are useful. In addition to being useful, each of these relatively new branches of science is also intellectually challenging.The book contains a systematic study of inverses of mappings between ordered sets, and so offers a uniquely helpful organization in the approach to several phenomena related to duality.To prepare the ground for discrete convexity, there are chapters on convexity in real vector spaces in anticipation of the many challenging problems coming up in digital geometry. To prepare for the study of new topologies introduced to serve in discrete spaces, there is also a chapter on classical topology.The book is intended for general readers with a modest background in mathematics and for advanced undergraduate students as well as beginning graduate students.
Contents:
- Preface
- Acknowledgments
- List of Figures
- List of Tables
- Introduction
- Sets, Mappings, and Order Relations
- Morphological Operations: Set-Theoretical Duality
- Complete Lattices
- Inverses and Quotients of Mappings
- Structure Theorems for Mappings
- Digitization
- Digital Straightness and Digital Convexity
- Convexity in Vector Spaces
- Discrete Convexity
- Discrete Convexity in Two Dimensions
- Three Problems in Discrete Optimization
- Duality of Convolution Operators
- Topology
- The Khalimsky Topology
- Distance Transformations
- Skeletonizing
- Solutions
- Bibliography
- Author Index
- Subject Index
Readership: Advanced undergraduate and graduate students, researchers.
Key Features:
- The concepts of upper and lower inverse of a mapping between preordered sets are completely elementary ones. They unify and generalize many procedures used in several fields of knowledge, starting with the Galois connections in algebra two centuries ago and with significant later instances, maybe the most prominent being the Fenchel transformation in convexity theory
- In later chapters the inverses are used to define dualities for convolution equations. This is essential since there is no general associative convolution multiplication
- Difference operators are the simplest convolution operators and yet serve as a most useful tool in the study of digital straightness and digital convexity, also using the duality just mentioned
