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Various fields of image analysis, computer vision, and artificial intelligence require a description of shape in grey-level images, but the automatic performance of these tasks can only succeed if a suitable description of shape is available.
This book presents a comprehensive overview of the mathematical description of shape from the viewpoint of a broad range of disciplines. The emphasis is on the methodology and theunderlying mathematics. The concept of shape is explored in terms of such disciplines as digital topology and geometry, categorial shape theory, mathematical morphology, wavelets, differential geometry, graph and hierarchical representation methods, and evolutionary systems.
The Khalimsky Line as a Foundation for Digital Topology.- Topological Foundations of Shape Analysis.- A New Concept for Digital Geometry.- Theoretical Approaches to N-Dimensional Digital Objects.- On Boundaries and Boundary Crack-Codes of Multidimensional Digital Images.- Studying Shape Through Size Functions.- to Categorical Shape Theory, with Applications in Mathematical Morphology.- Shape Theory: an ANR-Sequence Approach.- Can Categorical Shape Theory Handle Grey-level Images?.- Mathematical Morphology as a Tool for Shape Description.- On Information Contained in the Erosion Curve.- Morphological Area Openings and Closings for Grey-scale Images.- Manifold Shape: from Differential Geometry to Mathematical Morphology.- On Negative Shape.- An Overview of the Theory and Applications of Wavelets.- Fractal Surfaces, Multiresolution Analyses, and Wavelet Transforms.- Interpolation in Multiscale Representations.- Discrete Shastic Growth Models for Two-Dimensional Shapes.- Classical and Fuzzy Differential Methods in Shape Analysis.- Elements of a Fuzzy Geometry for Visual Space.- On the Relationship Between Surface Covariance and Differential Geometry.- Image Representation Using Affine Covariant Coordinates.- Equivariant Dynamical Systems: a Formal Model for the Generation of Arbitrary Shapes.- Neural Processing of Overlapping Shapes.- Contour Texture and Frame Curves for the Recognition of Non-Rigid Objects.- Conic Primitives for Projectively Invariant Representation of Planar Curves.- Blind Approximation of Planar Convex Shapes.- Recognition of Affine Planar Curves Using Geometric Properties.- Recognizing 3-D Curves from a Stereo Pair of Images: a Semi-differential Approach.- Statistical Shape Methodology in Image Analysis.- Recognition of Shapes from a Finite Series of Plane Figures.- Polygonal Harmonic Shape Characterization.- Shape Description and Classification Using the Interrelationship of Structures at Multiple Scales.- Learning Shape Classes.- Inference of Shastic Graph Models for 2-D and 3-D Shapes.- Hierarchical Shape Analysis in Grey-level Images.- Irregular Curve Pyramids.- Multiresolution Shape Description by Corners.- Model-based Bottom-Up Grouping of Geometric Image Primitives.- Hierarchical Shape Representation for Image Analysis.- Scale-Space for N-dimensional Discrete Signals.- Scale-Space Behaviour and Invariance Properties of Differential Singularities.- Exploring the Shape Manifold: the Role of Conservation Laws.- Performance in Noise of a Diffusion-based Shape Descriptor.- Towards a Morphological Scale-Space Theory.- Geometry-based Image Segmentation Using Anisotropic Diffusion.- Images: Regular Tempered Distributions.- Local and Multilocal Scale-Space Description.- List of Authors.