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This volume discusses the basic geometric contents of an image and presents a tree data structure to handle those contents efficiently. The nodes of the tree are derived from connected components of level sets of the intensity, while the edges represent inclusion information. Grain filters, morphological operators simplifying these geometric contents, are analyzed and several applications to image comparison and registration, and to edge and corner detection, are presented. The mathematically inclined reader may be most interested in Chapters 2 to 6, which generalize the topological Morse description to continuous or semicontinuous functions, while mathematical morphologists may more closely consider grain filters in Chapter 3. Computer scientists will find algorithmic considerations in Chapters 6 and 7, the full justification of which may be found in Chapters 2 and 4 respectively. Lastly, all readers can learn more about the motivation for this work in the image processing applications presented in Chapter 8.

Image processing --- Applied Physics --- Computer Science --- Engineering & Applied Sciences --- Mathematical models --- Geometric analysis. --- Topographic maps. --- Mathematical models. --- Hypsographic maps --- Hypsometric maps --- Maps, Topographic --- Relief maps --- Topo maps --- Topographical maps --- Geometric analysis PDEs (Geometric partial differential equations) --- Pictorial data processing --- Picture processing --- Processing, Image --- Computer science. --- Coding theory. --- Computers. --- Information theory. --- Topology. --- Discrete mathematics. --- Combinatorics. --- Computer Science. --- Coding and Information Theory. --- Discrete Mathematics. --- Information and Communication, Circuits. --- Computing Methodologies. --- Combinatorics --- Algebra --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Communication theory --- Communication --- Cybernetics --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Machine theory --- Calculators --- Cyberspace --- Data compression (Telecommunication) --- Digital electronics --- Information theory --- Signal theory (Telecommunication) --- Computer programming --- Informatics --- Science --- Mathematics --- Maps --- Imaging systems --- Optical data processing --- Mathematics. --- Artificial intelligence. --- Artificial Intelligence. --- AI (Artificial intelligence) --- Artificial thinking --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Logic machines --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Math --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis