Digital Geometry in Image Processing

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Features

  • Provides comprehensive coverage of digital metric spaces in arbitrary dimensions
  • Brings together significant research results from the last three decades on digital distances and the analysis of digital curves and straight lines
  • Emphasizes shape analysis and object representation using concepts and theories from digital geometry
  • Offers algorithmic solutions of several challenging problems in computer graphics and image processing

Summary

Exploring theories and applications developed during the last 30 years, Digital Geometry in Image Processing presents a mathematical treatment of the properties of digital metric spaces and their relevance in analyzing shapes in two and three dimensions. Unlike similar books, this one connects the two areas of image processing and digital geometry, highlighting important results of digital geometry that are currently used in image analysis and processing.

The book discusses different digital geometries in multi-dimensional integral coordinate spaces. It also describes interesting properties of the geometries, including metric and topological properties, shapes of circles and spheres, proximity to Euclidean norms, and number theoretic representations of geometric objects such as straight lines and circles. The authors—all active researchers in image processing and digital geometry—demonstrate how these concepts and properties are useful in various techniques for image processing and analysis. In particular, the book covers applications in object representation and shape analysis.

With many figures (some in color) and end-of-chapter exercises, this book provides an in-depth, unified account of digital metrics, the characterization of digital curves and straight lines, and their uses in shape analysis. It gives you insight on the latest two- and three-dimensional image processing applications.

Table of Contents

Digital Topology: Fundamentals
Tessellation of a Continuous Space
Digital Grid
Grid Topology
Topology Preserving Operations
The Euler Characteristics

Distance Functions in Digital Geometry
Mathematical Definitions and Notation
Neighborhoods, Paths and Distances
Neighborhood Distances
Path-Dependent Neighborhoods and Distances
Hyperspheres of Digital Distances
Error Estimation and Approximation of Euclidean Distance

Digitization of Straight Lines and Planes
2-D Discrete Straight Line Segments
Iterative Refinement: An Algebraic Characterization
3-D Digital Straight Line Segments
Digital Plane Segments

Digital Straightness and Polygonal Approximation
Digital Straightness
Approximate Straightness
Polygonal Approximation
Approximation on Gray-Scale Images
Examples

Parametric Curve Estimation and Reconstruction
Digital Conics in Canonical Form
Circles and Parabolas in Canonical Form
Estimation of Major and Minor Axes of an Ellipse in Canonical Form
Reconstruction of Hyperbola in Canonical Form
A Restricted Class of Digitized Planar Curves

Medial Axis Transform
Distance Transform
Medial Axis Transform (MAT)
Skeletonization using MAT
Geometric Transformation
Computation of Normals at Boundary Points of 2-D Objects
Computation of Cross-Sections of 3-D Objects
Shading of 3D Objects

Modeling of Voxelated Surface
Voxelation and Approximation of 3-D Surface
Voxelation of Surface of Revolution

Bibliography

Index

A Summary appears at the end of each chapter.

Author Bio(s)

Editorial Reviews

" ... there is a lot in this book to like: it covers a great deal of material ... . I hope to see a second edition--this book deserves one, and the material deserves plenty of readers."
--Alasdair McAndrew, Computing Reviews, December 2013