Digital geometry - geometric methods for digital picture analysis

Introduction. Grids and Digitization. Metrics. Adjacency Graphs. Incidence Pseudographs. Topology: Basics. Curves and Surfaces: Topology. Curves and Surfaces: Geometry. Straightness. Arc Length and Curvature. 3D Straightness and Planarity. Surface and Area Curvature. Hulls and Diagrams. Transformations. Morphological Operations. Deformations. Other Properties and Relations. Bibliography.

[1]  V. A. Kovalevsky New definition and fast recognition of digital straight segments and arcs , 1990, [1990] Proceedings. 10th International Conference on Pattern Recognition.

[2]  Edouard Thiel,et al.  Computing 3D Medial Axis for Chamfer Distances , 2000, DGCI.

[3]  Azriel Rosenfeld,et al.  Parallel Image Processing Using Cellular Arrays , 1983, Computer.

[4]  John Mylopoulos,et al.  On the Topological Properties of Quantized Spaces, II. Connectivity and Order of Connectivity , 1971, JACM.

[5]  Yasuhito Suenaga,et al.  A High-Speed Algorithm for the Generation of Straight Lines and Circular Arcs , 1979, IEEE Transactions on Computers.

[6]  Pengfei Zhu,et al.  On Critical Point Detection of Digital Shapes , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  Pong C. Yuen,et al.  A novel method for parameter estimation of digital arc , 1996, Pattern Recognit. Lett..

[8]  D. A. O'Handley,et al.  Recent developments in digital image processing at the image processing laboratory at the jet propulsion laboratory , 1972 .

[9]  Godfried T. Toussaint,et al.  A note on linear expected time algorithms for finding convex hulls , 2005, Computing.

[10]  Fred Galvin,et al.  Distance functions and topologies , 1991 .

[11]  H. Sagan Space-filling curves , 1994 .

[12]  Richard Neumann,et al.  Extraction of dominant points by estimation of the contour fluctuations , 2002, Pattern Recognit..

[13]  Filippo Mignosi,et al.  On the Number of Factors of Sturmian Words , 1991, Theor. Comput. Sci..

[14]  K. Deguchi Multi-scale curvatures for contour feature extraction , 1988, [1988 Proceedings] 9th International Conference on Pattern Recognition.

[15]  Alfred M. Bruckstein,et al.  On Recursive, O(N) Partitioning of a Digitized Curve into Digital Straight Segments , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  A. Rényi,et al.  über die konvexe Hülle von n zufällig gewählten Punkten , 1963 .

[17]  C. Min Ma A 3D fully parallel thinning algorithm for generating medial faces , 1995, Pattern Recognit. Lett..

[18]  Stéphane Ubéda A Parallel Thinning Algorithm Using the Bounding Boxes Techniques , 1993, Int. J. Pattern Recognit. Artif. Intell..

[19]  W. Scherrer Die Einlagerung eines regulären Vielecks in ein Gitter. , 1946 .

[20]  Ingemar Ragnemalm Neighborhoods for distance transformations using ordered propagation , 1992, CVGIP Image Underst..

[21]  Partha Pratim Das,et al.  A note on "distance transformations in arbitrary dimensions" , 1988, Comput. Vis. Graph. Image Process..

[22]  Olivier Teytaud,et al.  Segmentation and Length Estimation of 3D Discrete Curves , 2000, Digital and Image Geometry.

[23]  Reinhard Klette,et al.  Algorithms for testing convexity of digital polygons , 1981 .

[24]  Patrick Shen-Pei Wang,et al.  A Fast and Flexible Thinning Algorithm , 1989, IEEE Trans. Computers.

[25]  Isabelle Debled-Rennesson,et al.  A Linear Algorithm for Segmentation of Digital Curves , 1995, Int. J. Pattern Recognit. Artif. Intell..

[26]  Son Pham Parallel, overlapped, and intersected digital straight lines , 2005, The Visual Computer.

[27]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[28]  Christian Ronse,et al.  A Bibliography on Digital and Computational Convexity (1961-1988) , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  Julia Jean Robinson,et al.  Line Symmetry of Convex Digital Regions , 1996, Comput. Vis. Image Underst..

[30]  D. Struik A Source Book in Mathematics , 1986 .

[31]  A. Rosenfeld,et al.  Digital geometry on graphs , 1989 .

[32]  P. P. Das An algorithm for computing the number of the minimal paths in digital images , 1989, Pattern Recognit. Lett..

[33]  Gabriella Sanniti di Baja,et al.  (3, 4)-weighted Skeleton Decomposition for Pattern Representation and Description , 1994, Pattern Recognit..

[34]  Pritimoy Bhattacharyya,et al.  Analysis of template matching thinning algorithms , 1992, Pattern Recognit..

[35]  Christian Ronse,et al.  Book-Review - Connected Components in Binary Images - the Detection Problem , 1984 .

[36]  Jean-Pierre Reveillès Géométrie discrète, calcul en nombres entiers et algorithmique , 1991 .

[37]  Dennis F. Kibler,et al.  A Theory of Nonuniformly Digitized Binary Pictures , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[38]  Edouard Thiel,et al.  Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2D or 3D , 2002, Pattern Recognit. Lett..

[39]  P. P. Das Counting minimal paths in digital geometry , 1991, Pattern Recognit. Lett..

[40]  Reinhard Klette On the approximation of convex hulls of finite grid point sets , 1983, Pattern Recognit. Lett..

[41]  Deepak Bhatnagar,et al.  Pseudo one pass thinning algorithm , 1991, Pattern Recognit. Lett..

[42]  Jack Koplowitz,et al.  On the Performance of Chain Codes for Quantization of Line Drawings , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[43]  J. Sossa An improved parallel algorithm for thinning digital patterns , 1989 .

[44]  Yazid M. Sharaiha,et al.  Digital straightness and the skeleton property , 1995, Pattern Recognit. Lett..

[45]  Zvi Galil,et al.  Truly alphabet-independent two-dimensional pattern matching , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[46]  Giovanni B Reggiori,et al.  Digital Computer Transformations for Irregular Line-Drawings , 1972 .

[47]  D. Proffitt,et al.  Metrication errors and coding efficiency of chain-encoding schemes for the representation of lines and edges , 1979 .

[48]  A. W. Roscoe,et al.  Concepts of digital topology , 1992 .

[49]  P. V. Sankar Grid intersect quantization schemes for solid object digitization , 1978 .

[50]  Søren Forchhammer,et al.  Digital plane and grid point segments , 1989, Comput. Vis. Graph. Image Process..

[51]  H. Weber,et al.  Enzyklopadie der Elementarmathematik. I. Arithmetik, Algebra und Analysis , 1935 .

[52]  Du-Ming Tsai,et al.  Boundary-based corner detection using eigenvalues of covariance matrices , 1999, Pattern Recognit. Lett..

[53]  A. Bonaert Introduction to the theory of Fuzzy subsets , 1977, Proceedings of the IEEE.

[54]  Peter Gritzmann,et al.  Discrete Tomography: Determination of Finite Sets by X-Rays , 1995, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[55]  Jerome Feder,et al.  Languages of Encoded Line Patterns , 1968, Inf. Control..

[56]  Laure Tougne,et al.  Discrete Curvature Based on Osculating Circle Estimation , 2001, IWVF.

[57]  Ron Shonkwiler Computing the Hausdorff Set Distance in Linear Time for Any L_p Point Distance , 1991, Inf. Process. Lett..

[58]  Maurice Maes Digitization of straight line segments closeness and convexity , 1990, Comput. Vis. Graph. Image Process..

[59]  L. Schläfli Theorie der vielfachen Kontinuität , 1901 .

[60]  James T. Smith Methods of Geometry , 2000 .

[61]  Driss Aboutajdine,et al.  Application of Adaptive Hypergraph Model to Impulsive Noise Detection , 2001, CAIP.

[62]  Azriel Rosenfeld,et al.  Digital Geometry: Introduction and Bibliography , 1997 .

[63]  J. Whitehead,et al.  On the Realizability of Homotopy Groups , 1949 .

[64]  Azriel Rosenfeld,et al.  A Medial Axis Transformation for Grayscale Pictures , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[65]  Frank Y. Shih,et al.  A skeletonization algorithm by maxima tracking on Euclidean distance transform , 1995, Pattern Recognit..

[66]  Arnold W. M. Smeulders,et al.  Length estimators for digitized contours , 1987, Comput. Vis. Graph. Image Process..

[67]  KUO-CHIN FAN,et al.  Skeletonization of binary images with nonuniform width via block decomposition and contour vector matching , 1998, Pattern Recognit..

[68]  G. Matheron,et al.  THE BIRTH OF MATHEMATICAL MORPHOLOGY , 2002 .

[69]  Stefan Mazurkiewicz,et al.  Sur les lignes de Jordan , 1920 .

[70]  Klaus Voss,et al.  Digitalisierungseffekte in der Automatischen Bildverarbeitung , 1975, J. Inf. Process. Cybern..

[71]  E. R. Davies,et al.  Thinning algorithms: A critique and a new methodology , 1981, Pattern Recognit..

[72]  G. E. Martin Transformation Geometry: An Introduction to Symmetry , 1982 .

[73]  H. Ryser Combinatorial Properties of Matrices of Zeros and Ones , 1957, Canadian Journal of Mathematics.

[74]  E. V. Krishnamurthy,et al.  On the compactness of subsets of digital pictures , 1978 .

[75]  Atsushi Imiya,et al.  Discrete combinatorial geometry , 1997, Pattern Recognit..

[76]  Arnold W. M. Smeulders,et al.  Best Linear Unbiased Estimators for Properties of Digitized Straight Lines , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[77]  Azriel Rosenfeld,et al.  Degree of adjacency or surroundedness , 1984, Pattern Recognit..

[78]  Michael Lindenbaum,et al.  Compression of chain codes using digital straight line sequences , 1988, Pattern Recognit. Lett..

[79]  Jack Sklansky,et al.  Recognition of convex blobs , 1970, Pattern Recognit..

[80]  Azriel Rosenfeld,et al.  On connectivity properties of grayscale pictures , 1981, Pattern Recognit..

[81]  Amitava Datta,et al.  A robust parallel thinning algorithm for binary images , 1994, Pattern Recognit..

[82]  Robert P. W. Duin,et al.  Spirograph Theory: A Framework for Calculations on Digitized Straight Lines , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[83]  Christian Ronse A strong chord property for 4-connected convex digital sets , 1986 .

[84]  Longin Jan Latecki Discrete Representation of Spatial Objects in Computer Vision , 1998, Computational Imaging and Vision.

[85]  S. H. Unger,et al.  A Computer Oriented toward Spatial Problems , 1958 .

[86]  C. Jordan Remarques sur les intégrales définies , .

[87]  Azriel Rosenfeld,et al.  A Pyramid Framework for Early Vision , 1994 .

[88]  Rémy Malgouyres There is no Local Characterization of Separating and Thin Objects in Z³ , 1996, Theor. Comput. Sci..

[89]  A. Rosenfeld,et al.  Digital topology: a comparison of the graph-based and topological approaches , 1991 .

[90]  Phalguni Gupta,et al.  Finding connected components in digital images by aggressive reuse of labels , 2002, Image Vis. Comput..

[91]  Jack Sklansky,et al.  Minimum-Perimeter Polygons of Digitized Silhouettes , 1972, IEEE Transactions on Computers.

[92]  Per-Erik Danielsson,et al.  Finding the Minimal Set of Maximum Disks for Binary Objects , 1997, CVGIP Graph. Model. Image Process..

[93]  Pascal Lienhardt,et al.  Topological models for boundary representation: a comparison with n-dimensional generalized maps , 1991, Comput. Aided Des..

[94]  T. Pavlidis A thinning algorithm for discrete binary images , 1980 .

[95]  Partha Pratim Das,et al.  On approximating Euclidean metrics by digital distances in 2D and 3D , 2000, Pattern Recognit. Lett..

[96]  Jean-Michel Jolion Stochastic pyramid revisited , 2003, Pattern Recognit. Lett..

[97]  Norman Levitt,et al.  The euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes , 1992, Discret. Comput. Geom..

[98]  Azriel Rosenfeld,et al.  Local deformations of digital curves , 1997, Pattern Recognit. Lett..

[99]  A. Tucker,et al.  An Abstract Approach to Manifolds , 1933 .

[100]  Longin Jan Latecki,et al.  An Algorithm for a 3D Simplicity Test , 1996, Comput. Vis. Image Underst..

[101]  Cristina Urdiales,et al.  Corner detection by means of contour local vectors , 2002 .

[102]  Masafumi Yamashita,et al.  Distance functions defined by variable neighborhood sequences , 1984, Pattern Recognit..

[103]  Azriel Rosenfeld,et al.  Representation of orthogonal regions by vertices , 1991, CVGIP Graph. Model. Image Process..

[104]  Cornelius W. A. M. van Overveld,et al.  Line rasterization algorithms that satisfy the subset line property , 1988, Comput. Vis. Graph. Image Process..

[105]  Atsushi Imiya,et al.  Boundary Extraction of Discrete Objects , 1998, Comput. Vis. Image Underst..

[106]  Jean-Pierre Reveilles,et al.  Combinatorial pieces in digital lines and planes , 1995, Optics & Photonics.

[107]  Hongchi Shi Image algebra techniques for binary image component labeling with local operators , 2005, Journal of Mathematical Imaging and Vision.

[108]  Azriel Rosenfeld,et al.  Discrete, nonlinear curvature-dependent contour evolution , 1998, Pattern Recognit..

[109]  Brian Randell,et al.  The Origins of Digital Computers , 1973, Springer Berlin Heidelberg.

[110]  Reinhard Klette,et al.  Cell complexes through time , 2000, SPIE Optics + Photonics.

[111]  Azriel Rosenfeld,et al.  A Converse to the Jordan Curve Theorem for Digital Curves , 1975, Inf. Control..

[112]  Azriel Rosenfeld,et al.  Topology-Preserving Deformations of Two-Valued Digital Pictures , 1998, Graph. Model. Image Process..

[113]  Stéphane Marchand-Maillet,et al.  Euclidean Ordering via Chamfer Distance Calculations , 1999, Comput. Vis. Image Underst..

[114]  Tien-Tsin Wong,et al.  Multiresolution Isosurface Extraction with Adaptive Skeleton Climbing , 1998, Comput. Graph. Forum.

[115]  Bidyut Baran Chaudhuri,et al.  Detection of 3-D Simple Points for Topology Preserving Transformations with Application to Thinning , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[116]  Partha Pratim Das,et al.  On connectivity issues of ESPTA , 1990, Pattern Recognit. Lett..

[117]  Alan Davies,et al.  An Introduction to Computational Geometry for Curves and Surfaces , 1996 .

[118]  Azriel Rosenfeld Clusters in Digital Pictures , 1978, Inf. Control..

[119]  D. W. Woodard On two-dimensional analysis situs with special reference to the Jordan curve-theorem , 1929 .

[120]  Azriel Rosenfeld Compact Figures in Digital Pictures , 1974, IEEE Trans. Syst. Man Cybern..

[121]  Antoine Manzanera,et al.  Ultra-Fast Skeleton Based on an Isotropic Fully Parallel Algorithm , 1999, DGCI.

[122]  Trevor J. Davis Fast Decomposition of Digital Curves into Polygons Using the Haar Transform , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[123]  George R. Cross,et al.  On representation of a shape's skeleton , 1991, Pattern Recognit. Lett..

[124]  P. P. Das A note on "Distance functions in digital geometry" , 1992, Inf. Sci..

[125]  King-Sun Fu,et al.  Stochastic skeleton modeling of objects , 1984, Comput. Vis. Graph. Image Process..

[126]  Deborah Walters,et al.  General Ribbon-Based Thinning Algorithms for Stylus-Generated Images , 1999, Comput. Vis. Image Underst..

[127]  Chris Pudney,et al.  Distance-Ordered Homotopic Thinning: A Skeletonization Algorithm for 3D Digital Images , 1998, Comput. Vis. Image Underst..

[128]  Gérard G. Medioni,et al.  Corner detection and curve representation using cubic B-splines , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[129]  G. Peano Sur une courbe, qui remplit toute une aire plane , 1890 .

[130]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[131]  Reinhard Klette,et al.  Digital Geometry — The Birth of a New Discipline , 2001 .

[132]  S. Shafer Shadows and Silhouettes in Computer Vision , 1985 .

[133]  Frank Rhodes On the metrics of Chaudhuri, Murthy and Chaudhuri , 1995, Pattern Recognit..

[134]  Yazid M. Sharaiha,et al.  A compact chord property for digital arcs , 1993, Pattern Recognit..

[135]  Reinhard Klette The m-dimensional grid point space , 1985, Comput. Vis. Graph. Image Process..

[136]  Azriel Rosenfeld,et al.  The perimeter of a fuzzy set , 1985, Pattern Recognit..

[137]  Azriel Rosenfeld,et al.  Geodesic Convexity in Discrete Spaces , 1994, Inf. Sci..

[138]  Byung Soo Moon A representation of digitized patterns and an edge tracking thinning method , 2001, Pattern Recognit..

[139]  Marcel Worring,et al.  Digitized Circular Arcs: Characterization and Parameter Estimation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[140]  Cherng Min Ma,et al.  On topology preservation in 3D thinning , 1994 .

[141]  Dimitrios Kalles,et al.  A novel fast and reliable thinning algorithm , 1993, Image Vis. Comput..

[142]  Ingemar Ragnemalm,et al.  The Euclidean distance transform in arbitrary dimensions , 1992, Pattern Recognit. Lett..

[143]  Alan M. McIvor A Cubic Facet Based Method for Estimating the Principal Quadric , 1998, IVCNZ.

[144]  Reinhard Klette A Parallel Computer for Digital Image Processing , 1979, J. Inf. Process. Cybern..

[145]  Herbert Freeman,et al.  Computer Processing of Line-Drawing Images , 1974, CSUR.

[146]  Lynn Arthur Steen,et al.  Counterexamples in Topology , 1970 .

[147]  Gabriella Sanniti di Baja,et al.  A new shape descriptor for surfaces in 3D images , 2002, Pattern Recognit. Lett..

[148]  Azriel Rosenfeld,et al.  Two simply connected sets that have the same area are IP-equivalent , 2002, Pattern Recognit..

[149]  Lyle Noakes,et al.  More-or-less-uniform sampling and lengths of curves , 2003 .

[150]  G. DIMAURO,et al.  A New Thinning Algorithm Based on Controlled Deletion of Edge Regions , 1993, Int. J. Pattern Recognit. Artif. Intell..

[151]  Reinhard Klette Switches may solve adjacency problems , 2002, Object recognition supported by user interaction for service robots.

[152]  Geoff Wyvill,et al.  Data structure forsoft objects , 1986, The Visual Computer.

[153]  F. Harary New directions in the theory of graphs , 1973 .

[154]  Ronald Lumia,et al.  A new three dimensional connected components algorithm , 1982, Comput. Graph. Image Process..

[155]  Jean Françon,et al.  Polyhedrization of the Boundary of a Voxel Object , 1999, DGCI.

[156]  Pierre Soille,et al.  Order independent homotopic thinning for binary and grey tone anchored skeletons , 2002, Pattern Recognit. Lett..

[157]  Serge Miguet,et al.  Discrete Geometry for Computer Imagery , 1996, Lecture Notes in Computer Science.

[158]  B. Rosenberg,et al.  Computing Dominant Points on Simple Shapes , 1974, Int. J. Man Mach. Stud..

[159]  Ezzatollah Salari,et al.  The ridge-seeking method for obtaining the skeleton of digital images , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[160]  James Turner,et al.  A graph-theoretical model for periodic discrete structures☆ , 1968 .

[161]  G. Matheron Random Sets and Integral Geometry , 1976 .

[162]  Arnold W. M. Smeulders,et al.  Vector code probability and metrication error in the representation of straight lines of finite length , 1982, Comput. Graph. Image Process..

[163]  Seymour Shlien,et al.  Segmentation of digital curves using linguistic techniques , 1983, Comput. Vis. Graph. Image Process..

[164]  F. Sánchez-Marín,et al.  The curvature function evolved in scale-space as a representation of biological shapes , 1997, Comput. Biol. Medicine.

[165]  Keiichi Abe,et al.  Topological structural analysis of digitized binary images by border following , 1985, Comput. Vis. Graph. Image Process..

[166]  Keiichi Abe,et al.  Binary picture thinning by an iterative parallel two-subcycle operation , 1987, Pattern Recognit..

[167]  Hans-Peter Seidel,et al.  Linear One-Sided Stability of MAT for Weakly Injective Domain , 2004, Journal of Mathematical Imaging and Vision.

[168]  Chul E. Kim On cellular straight line segments , 1982, Comput. Graph. Image Process..

[169]  Godfried T. Toussaint,et al.  A simple O(n log n) algorithm for finding the maximum distance between two finite planar sets , 1982, Pattern Recognit. Lett..

[170]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[171]  P. Seymour,et al.  A new proof of the four-colour theorem , 1996 .

[172]  Stefano Levialdi,et al.  On shrinking binary picture patterns , 1972, CACM.

[173]  Longin Jan Latecki 3D well-composed pictures , 1995, Optics & Photonics.

[174]  Azriel Rosenfeld Geodesics in Digital Pictures , 1978, Inf. Control..

[175]  Mandyam D. Srinath,et al.  Corner detection from chain-code , 1990, Pattern Recognit..

[176]  Richard C. Staunton,et al.  An analysis of hexagonal thinning algorithms and skeletal shape representation , 1996, Pattern Recognit..

[177]  Reinhard Klette,et al.  Digital Planar Segment Based Polyhedrization for Surface Area Estimation , 2001, IWVF.

[178]  Roberto Scopigno,et al.  Discretized Marching Cubes , 1994, Proceedings Visualization '94.

[179]  Nicholas Ayache,et al.  Topological segmentation of discrete surfaces , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[180]  Tony Lindeberg,et al.  Scale-Space Theory in Computer Vision , 1993, Lecture Notes in Computer Science.

[181]  Azriel Rosenfeld,et al.  Digital topology: Introduction and survey , 1989, Comput. Vis. Graph. Image Process..

[182]  A. Smeulders,et al.  Discrete straight line segments: parameters, primitives and properties , 1991 .

[183]  Azriel Rosenfeld,et al.  Some Parallel Thinning Algorithms for Digital Pictures , 1971, JACM.

[184]  P. P. Das,et al.  Hyperspheres in digital geometry , 1990, Inf. Sci..

[185]  Dominique Gouyou-Beauchamps,et al.  Sur les Facteurs des Suites de Sturm , 1990, Theor. Comput. Sci..

[186]  Milan Sonka,et al.  A Fully Parallel 3D Thinning Algorithm and Its Applications , 1996, Comput. Vis. Image Underst..

[187]  Wan-Chi Siu,et al.  New Single-Pass Algorithm for Parallel Thinning , 1995, Comput. Vis. Image Underst..

[188]  Ulrich Rösler,et al.  Maskenverfahren zur Manipulation von binären Rasterbildern linienförmiger Objekte , 1980, J. Inf. Process. Cybern..

[189]  George M. Reed On the characterization of simple closed surfaces in three-dimensional digital images , 1984, Comput. Vis. Graph. Image Process..

[190]  D. F. Lawden Elliptic Functions and Applications , 1989 .

[191]  Luc Vincent,et al.  Exact Euclidean distance function by chain propagations , 1991, Proceedings. 1991 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[192]  Giorgio Levi,et al.  A Grey-Weighted Skeleton , 1970, Inf. Control..

[193]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[194]  H. S. M. Coxeter,et al.  Geometry and Algebra in Ancient Civilizations , 1985 .

[195]  W. T. Tutte Graph Theory , 1984 .

[196]  Partha Pratim Das,et al.  Use of medial axis transforms for computing normals at boundary points , 2002, Pattern Recognit. Lett..

[197]  C. Sturm,et al.  Cours d'analyse de l'école polytechnique , 1884 .

[198]  Deborah Silver,et al.  Parameter-Controlled Volume Thinning , 1999, Graph. Model. Image Process..

[199]  Wolfgang Tutschke Grundlagen der reellen Analysis , 1971 .

[200]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[201]  Reinhard Klette Parallel operations on binary images , 1980 .

[202]  Kensaku Mori,et al.  Distance Transformation and Skeletonization of 3D Pictures and Their Applications to Medical Images , 2000, Digital and Image Geometry.

[203]  Hideo Ogawa Corner detection on digital curves based on local symmetry of the shape , 1989, Pattern Recognit..

[204]  Jack Sklansky,et al.  Measuring Concavity on a Rectangular Mosaic , 1972, IEEE Transactions on Computers.

[205]  Harold Davenport,et al.  On a Principle of Lipschitz , 1951 .

[206]  Anne Vialard Geometrical parameters extraction from discrete paths , 1996, DGCI.

[207]  Robert F. Sproull,et al.  Principles of interactive computer graphics (2nd ed.) , 1979 .

[208]  Partha Pratim Das,et al.  The t-Cost distance in digital geometry , 1992, Inf. Sci..

[209]  Stina Svensson,et al.  Digital Distance Transforms in 3D Images Using Information from Neighbourhoods up to 5×5×5 , 2002, Comput. Vis. Image Underst..

[210]  G. L. Dirichlet Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. , 1850 .

[211]  Jim Piper,et al.  Computing distance transformations in convex and non-convex domains , 1987, Pattern Recognit..

[212]  Longin Jan Latecki,et al.  Digital Topology , 1994 .

[213]  Li-De Wu,et al.  On the Chain Code of a Line , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[214]  George Tourlakis,et al.  Homological Methods for the Classification of Discrete Euclidean Structures , 1977 .

[215]  Nabil Jean Naccache,et al.  In response to "a comment on an investigation into the skeletonization approach of hilditch" , 1986, Pattern Recognit..

[216]  Rémy Malgouyres Computing the Fundamental Group in Digital Spaces , 2001, Int. J. Pattern Recognit. Artif. Intell..

[217]  Patrick Shen-Pei Wang,et al.  A comment on “a fast parallel algorithm for thinning digital patterns” , 1986, CACM.

[218]  Adam Huang,et al.  Approximating normals for marching cubes applied to locally supported isosurfaces , 2002, IEEE Visualization, 2002. VIS 2002..

[219]  John N. Tsitsiklis,et al.  Local Versus Nonlocal Computation of Length of Digitized Curves , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[220]  P. Danielsson Euclidean distance mapping , 1980 .

[221]  D. Rosen On the areas and boundaries of quantized objects , 1980 .

[222]  Ching Y. Suen,et al.  Thinning Methodologies - A Comprehensive Survey , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[223]  Christian Lantuéjoul,et al.  Geodesic methods in quantitative image analysis , 1984, Pattern Recognit..

[224]  Bidyut Baran Chaudhuri,et al.  A new shape preserving parallel thinning algorithm for 3D digital images , 1997, Pattern Recognit..

[225]  James C. Mullikin,et al.  The vector distance transform in two and three dimensions , 1992, CVGIP Graph. Model. Image Process..

[226]  Reinhard Klette,et al.  Computer vision - three-dimensional data from images , 1998 .

[227]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[228]  Azriel Rosenfeld,et al.  A Min-Max Medial Axis Transformation , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[229]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[230]  T. Pavlidis Algorithms for Graphics and Image Processing , 1981, Springer Berlin Heidelberg.

[231]  Azriel Rosenfeld,et al.  How a Digital Computer can Tell whether a Line is Straight , 1982 .

[232]  L. Shapiro Connected Component Labeling and Adjacency Graph Construction , 1996 .

[233]  Isabelle Debled-Rennesson,et al.  Etude et reconnaissance des droites et plans discrets , 1995 .

[234]  Larry S. Davis,et al.  Understanding Shape: Angles and Sides , 1977, IEEE Transactions on Computers.

[235]  Reinhard Klette,et al.  Angle counts for isothetic polygons and polyhedra , 2003, Pattern Recognit. Lett..

[236]  Partha Pratim Das More on path generated digital metrics , 1989, Pattern Recognit. Lett..

[237]  Azriel Rosenfeld,et al.  Local and global topology preservation in locally finite sets of tiles , 2001, Inf. Sci..

[238]  Atsushi Imiya,et al.  Marching cubes method with connectivity , 1999, Proceedings 1999 International Conference on Image Processing (Cat. 99CH36348).

[239]  Robert A. Melter Convexity is necessary - A correction , 1988, Pattern Recognit. Lett..

[240]  Rolf. Klötzler,et al.  Beiträge zur Analysis , 1981 .

[241]  Partha Pratim Das,et al.  Generalized distances in digital geometry , 1987, Inf. Sci..

[242]  Peter Thanisch,et al.  Linear time algorithm for finding a picture's connected components , 1984, Image Vis. Comput..

[243]  H. G. Forder The foundations of Euclidean geometry , 1927 .

[244]  Azriel Rosenfeld,et al.  A method of curve partitioning using arc-chord distance , 1987, Pattern Recognit. Lett..

[245]  Ching Y. Suen,et al.  A fast parallel algorithm for thinning digital patterns , 1984, CACM.

[246]  W. Sierpinski Elementary Theory of Numbers , 1964 .

[247]  Yazid M. Sharaiha,et al.  Binary digital image processing - a discrete approach , 1999 .

[248]  Olivier D. Faugeras,et al.  Polyhedral approximation of 3-D objects without holes , 1984, Comput. Vis. Graph. Image Process..

[249]  Sartaj Sahni,et al.  Finding Connected Components and Connected Ones on a Mesh-Connected Parallel Computer , 1980, SIAM J. Comput..

[250]  Peter Gritzmann,et al.  Successive Determination and Verification of Polytopes by their X-Rays , 1992, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[251]  AZRIEL ROSENFELD,et al.  Digital Straight Line Segments , 1974, IEEE Transactions on Computers.

[252]  Nabil Jean Naccache,et al.  SPTA: A proposed algorithm for thinning binary patterns , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[253]  Azriel Rosenfeld,et al.  A fuzzy medial axis transformation based on fuzzy disks , 1991, Pattern Recognit. Lett..

[254]  Ugo Montanari,et al.  A Method for Obtaining Skeletons Using a Quasi-Euclidean Distance , 1968, J. ACM.

[255]  King-Sun Fu,et al.  A parallel thinning algorithm for 3-D pictures , 1981 .

[256]  D. Gale A theorem on flows in networks , 1957 .

[257]  Toshihide Ibaraki,et al.  Distances defined by neighborhood sequences , 1986, Pattern Recognit..

[258]  H. Minkowski,et al.  Geometrie der Zahlen , 1896 .

[259]  Gabriella Sanniti di Baja,et al.  Using distance transforms to decompose 3D discrete objects , 2002, Image Vis. Comput..

[260]  Theodosios Pavlidis,et al.  Structural pattern recognition , 1977 .

[261]  Christian Ronse A simple proof of Rosenfeld's characterization of digital straight line segments , 1985, Pattern Recognit. Lett..

[262]  E. Creutzburg,et al.  Geometric methods for on-line recognition of digital straight-line segments , 1988 .

[263]  Arnold W. M. Smeulders,et al.  Discrete Representation of Straight Lines , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[264]  Azriel Rosenfeld,et al.  Connectivity in Digital Pictures , 1970, JACM.

[265]  Reinhard Klette,et al.  A comparative evaluation of length estimators , 2002, Object recognition supported by user interaction for service robots.

[266]  Azriel Rosenfeld,et al.  Digital straightness , 2001, Electron. Notes Theor. Comput. Sci..

[267]  Hans-Peter Seidel,et al.  Hyperbolic Hausdorff Distance for Medial Axis Transform , 2001, Graph. Model..

[268]  Jovisa D. Zunic,et al.  On the Number of Digital Discs , 2004, Journal of Mathematical Imaging and Vision.

[269]  Dan Pedoe Notes on the History of Geometrical Ideas I. Homogeneous Coordinates , 1975 .

[270]  Azriel Rosenfeld A Note on Perimeter and Diameter in Digital Pictures , 1974, Inf. Control..

[271]  Hlawka Topologie der Polyeder und kombinatorische Topologie der Komplexe , 1941 .

[272]  Frank Y. Shih,et al.  Size-invariant four-scan Euclidean distance transformation , 1998, Pattern Recognit..

[273]  Arie E. Kaufman,et al.  Normal estimation in 3 D discrete space , 1992, The Visual Computer.

[274]  W. Schmidt Volume, surface area and the number of integer points covered by a convex set , 1972 .

[275]  Albert M. Vossepoel,et al.  A note on "distance transformations in digital images" , 1988, Comput. Vis. Graph. Image Process..

[276]  Jun-ichiro Toriwaki,et al.  New algorithms for euclidean distance transformation of an n-dimensional digitized picture with applications , 1994, Pattern Recognit..

[277]  Yun Xia,et al.  Skeletonization via the Realization of the Fire Front's Propagation and Extinction in Digital Binary Shapes , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[278]  Maurice Nivat,et al.  The number of convex polyominoes reconstructible from their orthogonal projections , 1996, Discret. Math..

[279]  Theodosios Pavlidis,et al.  An asynchronous thinning algorithm , 1982, Comput. Graph. Image Process..

[280]  Bidyut Baran Chaudhuri,et al.  3D Digital Topology under Binary Transformation with Applications , 1996, Comput. Vis. Image Underst..

[281]  Bei Li,et al.  A knowledge-based thinning algorithm , 1991, Pattern Recognit..

[282]  A. Rosenfeld The fuzzy geometry of image subsets , 1984, Pattern Recognit. Lett..

[283]  Michael Lindenbaum,et al.  A New Parameterization of Digital Straight Lines , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[284]  Peter E. Hart,et al.  GRAPHICAL-DATA-PROCESSING RESEARCH STUDY AND EXPERIMENTAL INVESTIGATION , 1964 .

[285]  Paolo Giulio Franciosa,et al.  On the Optimal Binary Plane Partition for Sets of Isothetic Rectangles , 1992, Inf. Process. Lett..

[286]  Chul E. Kim,et al.  Digital Parallelism, Perpendicularity, and Rectangles , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[287]  Reinhard Klette,et al.  Surface area estimation for digitized regular solids , 2000, SPIE Optics + Photonics.

[288]  D. P. Young,et al.  A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics , 1990 .

[289]  Daniel Nogly,et al.  Digital Topology on Graphs , 1996, Comput. Vis. Image Underst..

[290]  Satoru Kawai Topology quasi-preservation by local parallel operations , 1983, Comput. Vis. Graph. Image Process..

[291]  Vladimir Kovalevsky,et al.  Algorithms and Data Structures for Computer Topology , 2000, Digital and Image Geometry.

[292]  Gilles Schaeffer Conjugaison d'arbres et cartes combinatoires aléatoires , 1998 .

[293]  Ugo Montanari,et al.  A note on minimal length polygonal approximation to a digitized contour , 1970, CACM.

[294]  Azriel Rosenfeld,et al.  Multiresolution image processing and analysis , 1984 .

[295]  Ugo Montanari On Limit Properties in Digitization Schemes , 1970, JACM.

[296]  O. Cuisenaire Distance transformations: fast algorithms and applications to medical image processing , 1999 .

[297]  Frank Y. Shih,et al.  Fully parallel thinning with tolerance to boundary noise , 1994, Pattern Recognit..

[298]  Paul L. Rosin Techniques for Assessing Polygonal Approximations of Curves , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[299]  R. Osserman The isoperimetric inequality , 1978 .

[300]  Azriel Rosenfeld,et al.  'Continuous' functions on digital pictures , 1986, Pattern Recognit. Lett..

[301]  Marcel Worring,et al.  Digital curvature estimation , 1993 .

[302]  Bernd Hamann,et al.  The asymptotic decider: resolving the ambiguity in marching cubes , 1991, Proceeding Visualization '91.

[303]  Solomon Lefschetz,et al.  Introduction to Topology , 1950 .

[304]  Petr Dokladal,et al.  Grey-scale image segmentation : a topological approach , 2000 .

[305]  Fernand Meyer,et al.  Topographic distance and watershed lines , 1994, Signal Process..

[306]  Frank Rhodes Discrete euclidean metrics , 1992, Pattern Recognit. Lett..

[307]  Pekka J. Toivanen,et al.  Erratum to "New geodesic distance transforms for gray-scale images" [Pattern Recognition Letters 17 (1996) 437-450] , 1996, Pattern Recognit. Lett..

[308]  Jean-François Dufourd,et al.  Functional specification and prototyping with oriented combinatorial maps , 2000, Comput. Geom..

[309]  Attila Kuba,et al.  A Parallel 3D 12-Subiteration Thinning Algorithm , 1999, Graph. Model. Image Process..

[310]  Azriel Rosenfeld,et al.  Distances between fuzzy sets , 1985, Pattern Recognit. Lett..

[311]  Ioan Tomescu,et al.  Path generated digital metrics , 1983, Pattern Recognit. Lett..

[312]  Georges Voronoi Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites. , 1908 .

[313]  Jayaram K. Udupa,et al.  A search algorithm for skeletonization of thick patterns , 1974, Comput. Graph. Image Process..

[314]  William F. Osgood On the existence of the Green's function for the most general simply connected plane region , 1900 .

[315]  Ruowei Zhou,et al.  A novel single-pass thinning algorithm and an effective set of performance criteria , 1995, Pattern Recognit. Lett..

[316]  E. Landau,et al.  Ausgewählte Abhandlungen zur Gitterpunktlehre , 1962 .

[317]  Marek Doros On some properties of the generation of discrete circular arcs on a square grid , 1984, Comput. Vis. Graph. Image Process..

[318]  Isabelle Debled-Rennesson,et al.  Segmentation of Discrete Curves into Fuzzy Segments , 2003, Electron. Notes Discret. Math..

[319]  T. Saito,et al.  A sequential thinning algorithm for three dimensional digital pictures using the Euclidean distance transformation , 1995 .

[320]  K. Voss,et al.  Images, Objects, and Surfaces in Zn , 1991, Int. J. Pattern Recognit. Artif. Intell..

[321]  Oswald Veblen Decomposition of an $n$-space by a polyhedron , 1913 .

[322]  E. Preteux Watershed and skeleton by influence zones: A distance-based approach , 2004, Journal of Mathematical Imaging and Vision.

[323]  Murray H. Loew,et al.  Analysis and modeling of digitized straight-line segments , 1988, [1988 Proceedings] 9th International Conference on Pattern Recognition.

[324]  Carl Friedrich Gauss Disquisitiones generales circa superficies curvas , 1981 .

[325]  D. J. Langridge ON THE COMPUTATION OF SHAPE , 1972 .

[326]  Antoine Manzanera,et al.  MB: a coherent collection of 2D parallel thinning algorithms , 2002 .

[327]  Klaus Voss Coding of digital straight lines by continued fractions , 1991 .

[328]  Bill Roscoe,et al.  Characterisations of simply−connected finite polyhedra in 3−space , 1985 .

[329]  Akira Nakamura,et al.  Some results concerning connected fuzzy digital pictures , 1991, Pattern Recognit. Lett..

[330]  Azriel Rosenfeld,et al.  Supportedness and tameness differentialless geometry of plane curves , 1998, Pattern Recognit..

[331]  Gisela Klette,et al.  A comparative discussion of distance transforms and simple deformations in digital image processing , 2003 .

[332]  William Kingdon Clifford,et al.  THE PHILOSOPHY OF THE PURE SCIENCES , 1886 .

[333]  Pieter P. Jonker,et al.  Skeletons in N dimensions using shape primitives , 2002, Pattern Recognit. Lett..

[334]  Peter Veelaert,et al.  On the flatness of digital hyperplanes , 1993, Journal of Mathematical Imaging and Vision.

[335]  Ivan E. Sutherland,et al.  Sketchpad: a man-machine graphical communication system , 1899, AFIPS '63 (Spring).

[336]  Azriel Rosenfeld,et al.  Simple connectivity is not locally computable for connected 3D images , 1990, Comput. Vis. Graph. Image Process..

[337]  Bhabatosh Chanda,et al.  A single scan boundary removal thinning algorithm for 2-D binary object , 1993, Pattern Recognit. Lett..

[338]  Vladimir Kovalevsky Curvature in Digital 2D Images , 2001, Int. J. Pattern Recognit. Artif. Intell..

[339]  Azriel Rosenfeld,et al.  Interchangeable pairs of pixels in two-valued digital images , 2001, Pattern Recognit..

[340]  Longin Jan Latecki,et al.  Topological connectedness and 8-connectedness in digital pictures , 1993 .

[341]  Z. Kulpa On the properties of discrete circles, rings, and disks , 1979 .

[342]  Ching Y. Suen,et al.  A Systematic Evaluation of Skeletonization Algorithms , 1993, Int. J. Pattern Recognit. Artif. Intell..

[343]  Godfried T. Toussaint,et al.  PATTERN RECOGNITION AND GEOMETRICAL COMPLEXITY. , 1980 .

[344]  Frank Rhodes Some characterizations of the chessboard metric and the city block metric , 1990, Pattern Recognit. Lett..

[345]  John Mylopoulos,et al.  Some Results in Computational Topology , 1973, JACM.

[346]  Rémy Malgouyres Graphs Generalizing Closed Curves with Linear Construction of the Hamiltonian Cycle - Parametrization of Discretized Curves , 1995, Theor. Comput. Sci..

[347]  Jayant Shah,et al.  Extraction of Shape Skeletons from Grayscale Images , 1997, Comput. Vis. Image Underst..

[348]  Mohamed Tajine,et al.  Recognizing arithmetic straight lines and planes , 1996, DGCI.

[349]  Azriel Rosenfeld,et al.  Well-Composed Sets , 1995, Comput. Vis. Image Underst..

[350]  Azriel Rosenfeld,et al.  Digital knots , 2000, Pattern Recognit..

[351]  Naonori Ueda,et al.  Graph-Based Thinning for Binary Images , 1993, Int. J. Pattern Recognit. Artif. Intell..

[352]  A. Cormack,et al.  Early Tomography and Related Topics , 1981 .

[353]  Dennis Rosen,et al.  Errors in digitized area measurements: circles and rectangles , 1992, Pattern Recognit. Lett..

[354]  W. Sierpinski,et al.  Sur le probléme des courbes gauches en Topologie , 2022 .

[355]  Lennart Thurfjell,et al.  A new three-dimensional connected components labeling algorithm with simultaneous object feature extraction capability , 1992, CVGIP Graph. Model. Image Process..

[356]  Deborah Walters,et al.  General Ribbons: A Model for Stylus-Generated Images , 1999, Comput. Vis. Image Underst..

[357]  Wayne Niblack,et al.  Generating skeletons and centerlines from the distance transform , 1992, CVGIP Graph. Model. Image Process..

[358]  A. Rosenfeld,et al.  Fuzzy rectangles , 1990, Pattern Recognit. Lett..

[359]  Atsushi Imiya,et al.  Nonlinear Optimization for Polygonalization , 2003, DGCI.

[360]  Azriel Rosenfeld Fuzzy plane geometry: triangles , 1994, Pattern Recognit. Lett..

[361]  Partha Pratim Das,et al.  Thinning of 3-D images using the Safe Point Thinning Algorithm (SPTA) , 1989, Pattern Recognit. Lett..

[362]  Otis Philbrick,et al.  A STUDY OF SHAPE RECOGNITION USING THE MEDIAL AXIS TRANSFORMATION. , 1966 .

[363]  Paul Urysohn Mémoire sur les multiplicités Cantoriennes , 1925 .

[364]  Larry S. Davis,et al.  A Corner-Finding Algorithm for Chain-Coded Curves , 1977, IEEE Transactions on Computers.

[365]  Stéphane Marchand-Maillet,et al.  Skeleton location and evaluation based on local digital width in ribbon-like images , 1997, Pattern Recognit..

[366]  Silvana G. Dellepiane,et al.  Extraction of intensity connectedness for image processing , 1995, Pattern Recognit. Lett..

[367]  Jon G. Rokne,et al.  On Properties of Discretized Convex Curves , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[368]  Azriel Rosenfeld,et al.  Determining simplicity and computing topological change in strongly normal partial tilings of R2 or R3 , 2000, Pattern Recognit..

[369]  Bhabatosh Chanda,et al.  Topology preservation in 3D digital space , 1994, Pattern Recognit..

[370]  Alexandre Lenoir,et al.  Topology Preservation Within Digital Surfaces , 2000, Graph. Model..

[371]  Marek Doros,et al.  Algorithms for generation of discrete circles, rings, and disks , 1979 .

[372]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[373]  T. Fukumura,et al.  An Analysis of Topological Properties of Digitized Binary Pictures Using Local Features , 1975 .

[374]  Longin Jan Latecki Multicolor well-composed pictures , 1995, Other Conferences.

[375]  Alexandre Lenoir,et al.  Des outils pour les surfaces discretes, estimation d'invariants geometriques, preservation de la topologie, trace de geodesiques, visualisation , 1999 .

[376]  Alfred M. Bruckstein,et al.  Pruning Medial Axes , 1998, Comput. Vis. Image Underst..

[377]  Frank Y. Shih,et al.  Optimization on euclidean distance transformation using grayscale morphology , 1992, J. Vis. Commun. Image Represent..

[378]  Bidyut Baran Chaudhuri,et al.  A new approach to computing the Euler characteristic , 1995, Pattern Recognit..

[379]  Laurent Deniau Proposition d'un operateur geometrique pour l'analyse et l'identification de signaux et d'images , 1997 .

[380]  Mohamed Tajine,et al.  On Local Definitions of Length of Digital Curves , 2003, DGCI.

[381]  Reinhard Klette,et al.  An Approximative Calculation of Relative Convex Hulls for Surface Area Estimation of 3D Digital Objects , 2002, ICPR.

[382]  Azriel Rosenfeld,et al.  Digital Calculus , 1997, Information Sciences.

[383]  Maurice Nivat,et al.  The Medians of Discrete Sets , 1998, Inf. Process. Lett..

[384]  Jack Koplowitz,et al.  Corner detection for chain coded curves , 1995, Pattern Recognit..

[385]  Vladimir Kovalevsky Applications of Digital Straight Segments to Economical Image Encoding , 1997, DGCI.

[386]  Ioan Tomescu,et al.  Metric bases in digital geometry , 1984, Comput. Vis. Graph. Image Process..

[387]  Herbert Freeman,et al.  Determining the minimum-area encasing rectangle for an arbitrary closed curve , 1975, CACM.

[388]  P. R. Meyer,et al.  Computer graphics and connected topologies on finite ordered sets , 1990 .

[389]  M. Spivak A comprehensive introduction to differential geometry , 1979 .

[390]  Vladimir A. Kovalevsky,et al.  Finite topology as applied to image analysis , 1989, Comput. Vis. Graph. Image Process..

[391]  Jovisa D. Zunic Limit shape of convex lattice polygons having the minimal Linfinity diameter w.r.t. the number of their vertices , 1998, Discret. Math..

[392]  T. Y. Kong,et al.  Topological Algorithms for Digital Image Processing , 1996 .

[393]  Sergei Petrovich Novikov,et al.  Basic Elements of Differential Geometry and Topology , 1990 .

[394]  Arnold W. M. Smeulders,et al.  Decomposition of discrete curves into piecewise straight segments in linear time , 1991 .

[395]  Piet W. Verbeek,et al.  An Efficient Uniform Cost Algorithm Applied to Distance Transforms , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[396]  Frank Y. Shih,et al.  A new safe-point thinning algorithm based on the mid-crack code tracing , 1995, IEEE Trans. Syst. Man Cybern..

[397]  Tran-Thong A symmetric linear algorithm for line segment generation , 1982, Comput. Graph..

[398]  Wei Shen,et al.  On Geometric and Orthogonal Moments , 2000, Int. J. Pattern Recognit. Artif. Intell..

[399]  C. V. Kameswara Rao,et al.  Checking connectivity preservation properties of some types of picture processing operations , 1978 .

[400]  J. Françon,et al.  Evaluation de la normale au bord d'un objet discret 3D , 1998 .

[401]  Partha Pratim Das,et al.  Metricity of super-knight's distance in digital geometry , 1990, Pattern Recognit. Lett..

[402]  Nathan Sivin Chinese Mathematics. A Concise History . By Li Yan and Du Shiran. Translated by John N. Crossley and Anthony W.-C. Lun.[Oxford: Clarendon Press, 1987. 290 pp.] - Histoire des Mathématiques Chinoises . By Jean-Claude Martzloff. [Paris: Masson, 1988. 375 pp.] , 1989 .

[403]  Reinhard Klette,et al.  Topology of grid continua , 1998, Optics & Photonics.

[404]  S. D. Poisson Mémoire sur la courbure des surfaces. , 1832 .

[405]  Maurice Nivat,et al.  Reconstruction of Connected Sets from Two Projections , 1999 .

[406]  Laurent Perroton A New 26-Connected Objects Surface Tracking Algorithm and Its Related Pram Version , 1995, Int. J. Pattern Recognit. Artif. Intell..

[407]  Rachad M. Shoucri,et al.  Note on the determination of a digital straight line from chain codes , 1985, Comput. Vis. Graph. Image Process..

[408]  Jiann-Der Lee,et al.  Three-Dimensional Topology Preserving Reduction on the 4-Subfields , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[409]  P. P. Das,et al.  Octagonal distances for digital pictures , 1990, Inf. Sci..

[410]  Chul E. Kim,et al.  Three-Dimensional Digital Line Segments , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[411]  Azriel Rosenfeld,et al.  Surfaces in Three-Dimensional Digital Images , 1981, Inf. Control..

[412]  Benoit M. Macq,et al.  Fast Euclidean Distance Transformation by Propagation Using Multiple Neighborhoods , 1999, Comput. Vis. Image Underst..

[413]  M. Shamos Problems in computational geometry , 1975 .

[414]  Rémy Malgouyres A Definition of Surfaces of Z: A new 3D Discrete Jordan Theorem , 1997, Theor. Comput. Sci..

[415]  Zenon Kulpa More about areas and perimeters of quantized objects , 1983, Comput. Vis. Graph. Image Process..

[416]  Hans-Ullrich Döhler,et al.  Compact contour codes for convex binary patterns , 1985 .

[417]  Reinhard Klette,et al.  Length Estimation for Curves with Different Samplings , 2000, Digital and Image Geometry.

[418]  Prabir Bhattacharya,et al.  Gray Image Skeletonization with Hollow Preprocessing Using Distance Transformation , 1999, Int. J. Pattern Recognit. Artif. Intell..

[419]  Alan M. McIvor,et al.  A comparison of local surface geometry estimation methods , 1997, Machine Vision and Applications.

[420]  Joëlle Vittone Caractérisation et reconnaissance de droites et de plans en géométrie discrète , 1999 .

[421]  Rémy Malgouyres Local characterization of strong surfaces within strongly separating objects , 1998, Pattern Recognit. Lett..

[422]  Arie E. Kaufman,et al.  Fundamentals of Surface Voxelization , 1995, CVGIP Graph. Model. Image Process..

[423]  Patrick Shen-Pei Wang,et al.  A New Parallel Thinning Methodology , 1994, Int. J. Pattern Recognit. Artif. Intell..

[424]  Azriel Rosenfeld,et al.  Three-Dimensional Digital Topology , 1981, Inf. Control..

[425]  Jun-ichiro Toriwaki,et al.  Local Patterns and Connectivity Indexes in a Three Dimensional Digital Picture , 2002 .

[426]  You Yi Zhang Redundancy of parallel thinning , 1997, Pattern Recognit. Lett..

[427]  Azriel Rosenfeld,et al.  A Note on the Use of Local MIN and MAX Operations in Digital Picture Processing , 1977 .

[428]  René Descartes,et al.  Discours de la méthode : pour bien conduire sa raison et chercher la vérité dans les sciences , 1950 .

[429]  Lucas J. van Vliet,et al.  Binary and Grey-Value Skeletons: Metrics and Algorithms , 1993, Int. J. Pattern Recognit. Artif. Intell..

[430]  Azriel Rosenfeld,et al.  Centers of sets of pixels , 2000, Discret. Appl. Math..

[431]  G. P. Dinneen Programming pattern recognition , 1955, AFIPS '55 (Western).

[432]  Jean Françon Discrete Combinatorial Surfaces , 1995, CVGIP Graph. Model. Image Process..

[433]  Patrick Shen-Pei Wang,et al.  Analytical Comparison of Thinning Algorithms , 1993, Int. J. Pattern Recognit. Artif. Intell..

[434]  Steven Skiena,et al.  Implementing discrete mathematics - combinatorics and graph theory with Mathematica , 1990 .

[435]  Jorge Urrutia,et al.  Illuminating Rectangles and Triangles in the Plane , 1993, J. Comb. Theory, Ser. B.

[436]  Wladyslaw Skarbek,et al.  Generalized Hilbert scan in image printing , 1992, Theoretical Foundations of Computer Vision.

[437]  Ivan Stojmenovic,et al.  A Parametrization of Digital Planes by Least-Squares Fits and Generalizations , 1996, CVGIP Graph. Model. Image Process..

[438]  Vladimir A Kovalevsky Discrete topology and contour definition , 1984, Pattern Recognit. Lett..

[439]  Azriel Rosenfeld,et al.  Appendix: Digital topology — A brief introduction and bibliography , 1996 .

[440]  Stephen Wolfram,et al.  The Mathematica book (4th edition) , 1999 .

[441]  Jonathan C. Roberts An Overview of Rendering from Volume Data --- including Surface and Volume Rendering , 1993 .

[442]  N. Kritikos Über konvexe Flächen und einschließende Kugeln , 1927 .

[443]  Patrick Shen-Pei Wang,et al.  Analysis and Design of Parallel Thinning Algorithms - A Generic Approach , 1995, Int. J. Pattern Recognit. Artif. Intell..

[444]  Michael B. Smyth Region-based Discrete Geometry , 2000, J. Univers. Comput. Sci..

[445]  Zenon Kulpa,et al.  Area and perimeter measurement of blobs in discrete binary pictures , 1977 .

[446]  Daniel Wagner,et al.  Hausdorff Discretization and Its Comparison to Other Discretization Schemes , 1999, DGCI.

[447]  Gilles Bertrand,et al.  Complete Local Characterization of Strong 26-Surfaces: Continuous Analogs for Strong 26-Surfaces , 1999, Int. J. Pattern Recognit. Artif. Intell..

[448]  Ioannis Pitas,et al.  Performance Analysis of a Morphological Voronoi Tessellation Algorithm , 1994, ISMM.

[449]  Jose L. Navalón,et al.  A thinning algorithm based on contours , 1987 .

[450]  Renato Stefanelli A comment on an investigation into the skeletonization approach of Hilditch , 1986, Pattern Recognit..

[451]  Philippe Bolon,et al.  Discrete distance operator on rectangular grids , 1995, Pattern Recognit. Lett..

[452]  King-Sun Fu,et al.  A general scheme for constructing skeleton models , 1982, Inf. Sci..

[453]  Sankar K. Pal Fuzzy skeletonization of an image , 1989, Pattern Recognit. Lett..

[454]  Azriel Rosenfeld,et al.  A Characterization of Parallel Thinning Algorithms , 1975, Inf. Control..

[455]  K. Voss Discrete Images, Objects, and Functions in Zn , 1993 .

[456]  Stéphane Ubéda,et al.  A Parallel Thinning Algorithm Using K x K Masks , 1993, Int. J. Pattern Recognit. Artif. Intell..

[457]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[458]  Paul C. K. Kwok,et al.  A thinning algorithm by contour generation , 1988, CACM.

[459]  V. Faddeeva,et al.  Computational methods of linear algebra: : chapter 1, basic material from linear algebra , 1952 .

[460]  Azriel Rosenfeld,et al.  Measuring the sizes of concavities , 1985, Pattern Recognit. Lett..

[461]  Ethan M. Coven,et al.  Sequences with minimal block growth , 2005, Mathematical systems theory.

[462]  Jayaram K. Udupa,et al.  Connected, oriented, closed boundaries in digital spaces: Theory and algorithms , 1996 .

[463]  Nahum Kiryati,et al.  Digital representation schemes for 3d curves , 1997, Pattern Recognit..

[464]  Vito Di Gesù,et al.  Detection of regions of interest via the Pyramid Discrete Symmetry Transform , 1996, TFCV.

[465]  Akira Nakamura,et al.  Digital images of geometric pictures , 1985, Comput. Vis. Graph. Image Process..

[466]  Azriel Rosenfeld,et al.  Fuzzy Digital Topology , 1979, Inf. Control..

[467]  K. Wagner,et al.  Über eine Verallgemeinerung des Jordanschen Kurvensatzes auf zweifach geordnete Mengen , 1959 .

[468]  Azriel Rosenfeld,et al.  A Note on Thinning , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[469]  Alan Stewart A one-pass thinning algorithm with interference guards , 1994, Pattern Recognit. Lett..

[470]  Peter Veelaert,et al.  Geometric Constructions in the Digital Plane , 1999, Journal of Mathematical Imaging and Vision.

[471]  Azriel Rosenfeld,et al.  Recognition of Surfaces in Three-Dimensional Digital Images , 1982, Inf. Control..

[472]  Azriel Rosenfeld,et al.  Computer representation of planar regions by their skeletons , 1967, CACM.

[473]  Azriel Rosenfeld,et al.  Picture Processing by Computer , 1969, CSUR.

[474]  Gilles Bertrand,et al.  Tessellations by connection , 2002, Pattern Recognit. Lett..

[475]  R. Lowen Convex fuzzy sets , 1980 .

[476]  Jun-ichiro Toriwaki,et al.  Parallel Local Operations for a New Distance Transformation of a Line Pattern and Their Applications , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[477]  Azriel Rosenfeld Some notes on digital triangles , 1983, Pattern Recognit. Lett..

[478]  Son Pham,et al.  Digital straight segments , 1986, Comput. Vis. Graph. Image Process..

[479]  佐藤 孝紀,et al.  A Hierarchical Data Structure for Picture Processing , 1976 .

[480]  P. McMullen,et al.  On Hammer's X-Ray Problem , 1980 .

[481]  Partha Pratim Das,et al.  Estimation of errors between Euclidean and m-neighbor distance , 1989, Inf. Sci..

[482]  W.G Kropatsch,et al.  Detecting the straightness of digital curves in O(N) steps , 1989, Comput. Vis. Graph. Image Process..

[483]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[484]  Patrick Henry Winston,et al.  Learning structural descriptions from examples , 1970 .

[485]  Kálmán Palágyi,et al.  A 3-subiteration 3D thinning algorithm for extracting medial surfaces , 2002, Pattern Recognit. Lett..

[486]  Rémy Malgouyres,et al.  A concise characterization of 3D simple points , 2003, Discret. Appl. Math..

[487]  Azriel Rosenfeld,et al.  Holes and Genus of 2D and 3D Digital Images , 1993, CVGIP Graph. Model. Image Process..

[488]  Jean-Marc Chassery,et al.  Recognition of Digital Naive Planes and Polyhedrization , 2000, DGCI.

[489]  Gabriella Sanniti di Baja Well-Shaped, Stable, and Reversible Skeletons from the (3, 4)-Distance Transform , 1994, J. Vis. Commun. Image Represent..

[490]  Marinette Revenu,et al.  Fast computation of the normal vector field of the surface of a 3-D discrete object , 1996, DGCI.

[491]  A. Favre,et al.  Parallel syntactic thinning by recoding of binary pictures , 1983, Comput. Vis. Graph. Image Process..

[492]  H. Minkowski Volumen und Oberfläche , 1903 .

[493]  Amelia Fong Lochovsky Algorithms for realtime component labelling of images , 1988 .

[494]  Peter F. M. Nacken,et al.  Chamfer metrics in mathematical morphology , 1994, Journal of Mathematical Imaging and Vision.

[495]  Luciano da Fontoura Costa,et al.  On Voronoi Diagrams and Medial Axes , 2002, Journal of Mathematical Imaging and Vision.

[496]  C. V Kameswara Rao,et al.  A parallel shrinking algorithm for binary patterns , 1976 .

[497]  Robert A. Melter Some characterizations of city block distance , 1987, Pattern Recognit. Lett..

[498]  Azriel Rosenfeld A note on 'geometric transforms' of digital sets , 1983, Pattern Recognit. Lett..

[499]  Farzin Mokhtarian,et al.  Scale-Based Description and Recognition of Planar Curves and Two-Dimensional Shapes , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[500]  Amitava Datta,et al.  A parallel algorithm for decomposition of binary objects through skeletonization , 1991, Pattern Recognit. Lett..

[501]  Zenon Kulpa,et al.  Algorithms for circular propagation in discrete images , 1983, Comput. Vis. Graph. Image Process..

[502]  Gabriella Sanniti di Baja,et al.  Geometric Properties of the Union of Maximal Neighborhoods , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[503]  Mohamed Tajine,et al.  An approach to discretization based on the Hausdorff metric , 1998 .

[504]  Bimal Kumar Ray,et al.  An algorithm for detection of dominant points and polygonal approximation of digitized curves , 1992, Pattern Recognit. Lett..

[505]  Gilles Bertrand,et al.  Some Sequential Algorithms for a Generalized Distance Transformation Based on Minkowski Operations , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[506]  Nils Anders Danielsson Axiomatic Discrete Geometry , 2002 .

[507]  Partha Pratim Das,et al.  Distance functions in digital geometry , 1987, Inf. Sci..

[508]  Gabriella Sanniti di Baja,et al.  Curve skeletonization of surface-like objects in 3D images guided by voxel classification , 2002, Pattern Recognit. Lett..

[509]  T. Y. Kong,et al.  On the soundness of surface voxelizations , 1996 .

[510]  Azriel Rosenfeld,et al.  Winding and Euler numbers for 2D and 3D digital images , 1991, CVGIP Graph. Model. Image Process..

[511]  Wen-Hsiang Tsai,et al.  A new one-pass parallel thinning algorithm for binary images , 1992, Pattern Recognit. Lett..

[512]  Azriel Rosenfeld,et al.  Angle Detection on Digital Curves , 1973, IEEE Transactions on Computers.

[513]  Moncef Gabbouj,et al.  Parallel Image Component Labeling With Watershed Transformation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[514]  A. ROSENFELD,et al.  Distance functions on digital pictures , 1968, Pattern Recognit..

[515]  Jean Chorowicz,et al.  Parametrisable skeletonization of binary and multi-level images , 1990, Pattern Recognit. Lett..

[516]  Daniel Cohen-Or,et al.  3D Line Voxelization and Connectivity Control , 1997, IEEE Computer Graphics and Applications.

[517]  R. Mahesh K. Sinha,et al.  Comments on 'fast thinning algorithm for binary images' , 1986, Image Vis. Comput..

[518]  R. Yagel,et al.  - 1-eNormal Estimation in 3 D Discrete Spac , 1992 .

[519]  Jun-ichiro Toriwaki,et al.  A generalized distance transformation of a line pattern with gray values and its applications , 1982, Comput. Graph. Image Process..

[520]  Jayaram K. Udupa,et al.  Boundary and object labelling in three-dimensional images , 1990, Comput. Vis. Graph. Image Process..

[521]  Azriel Rosenfeld,et al.  If we use 4- or 8-connectedness for both the objects and the background, the Euler characteristics is not locally computable , 1990, Pattern Recognition Letters.

[522]  L. Santaló Integral geometry and geometric probability , 1976 .

[523]  Azriel Rosenfeld,et al.  Distance on a Hexagonal Grid , 1976, IEEE Transactions on Computers.

[524]  M. Lothaire Algebraic Combinatorics on Words , 2002 .

[525]  Jacques-Olivier Lachaud,et al.  Continuous Analogs of Digital Boundaries: A Topological Approach to Iso-Surfaces , 2000, Graph. Model..

[526]  Ivan Stojmenovic,et al.  A new characterization of digital lines by least square fits , 1993, Pattern Recognit. Lett..

[527]  Punam K. Saha,et al.  Fuzzy Distance Transform: Theory, Algorithms, and Applications , 2002, Comput. Vis. Image Underst..

[528]  Azriel Rosenfeld,et al.  Geodesic Visibility in Graphs , 1998, Inf. Sci..

[529]  Hanan Samet,et al.  Applications of spatial data structures , 1989 .

[530]  Pierre Soille,et al.  Morphological Image Analysis: Principles and Applications , 2003 .

[531]  Sargur N. Srihari,et al.  Boundary Detection in Multidimensions , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[532]  Wen Xu,et al.  CGT: A fast thinning algorithm implemented on a sequential computer , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[533]  J. Steiner Sur le maximum et le minimum des figures dans le plan, sur la sphère et dans l'espace en général. Second mémoire. , 1842 .

[534]  Azriel Rosenfeld,et al.  Thinning Algorithms for Gray-Scale Pictures , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[535]  Chul E. Kim Three-Dimensional Digital Planes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[536]  Laure Tougne,et al.  Multigrid Convergence and Surface Area Estimation , 2002, Theoretical Foundations of Computer Vision.

[537]  S. R. Keller,et al.  On the surface area of the ellipsoid , 1979 .

[538]  Ching Y. Suen,et al.  An optimal O(n) algorithm for identifying line segments from a sequence of chain codes , 1995, Pattern Recognit..

[539]  Frans C. A. Groen,et al.  Three-Dimensional Skeletonization: Principle and Algorithm , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[540]  Oswald Veblen,et al.  Theory on plane curves in non-metrical analysis situs , 1905 .

[541]  Mohamed Tajine,et al.  Hausdorff Discretization for Cellular Distances and Its Relation to Coverand Supercover Discretizations , 2001, J. Vis. Commun. Image Represent..

[542]  Alexandre Lenoir,et al.  Fast Estimation of Mean Curvature on the Surface of a 3D Discrete Object , 1997, DGCI.

[543]  Atsushi Imiya,et al.  Discretization of three-dimensional objects: approximation and convergence , 1998, Optics & Photonics.

[544]  Azriel Rosenfeld,et al.  Adjacency in Digital Pictures , 1974, Inf. Control..

[545]  Rémy Malgouyres,et al.  A Digital Linking Number for Discrete Curves , 2001, Int. J. Pattern Recognit. Artif. Intell..

[546]  William E. Lorensen,et al.  Decimation of triangle meshes , 1992, SIGGRAPH.

[547]  Leonid Kantorovich,et al.  Funktionalanalysis in normierten Räumen , 1978 .

[548]  Jun-ichiro Toriwaki,et al.  Paths and distance functions on three-dimensional digitized pictures , 1983, Pattern Recognit. Lett..

[549]  Attila Kuba,et al.  A 3D 6-subiteration thinning algorithm for extracting medial lines , 1998, Pattern Recognit. Lett..

[550]  Laurent Papier Polyedrisation et visualisation d'objets discrets tridimensionnels , 1999 .

[551]  M. Lothaire,et al.  Applied Combinatorics on Words , 2005 .

[552]  Azriel Rosenfeld,et al.  Arcs and Curves in Digital Pictures , 1973, JACM.

[553]  Gilles Bertrand,et al.  Gray-scale image processing using topological operators , 1999, Optics & Photonics.

[554]  Azriel Rosenfeld,et al.  The Digital Topology of Sets of Convex Voxels , 2000, Graph. Model..

[555]  Xiaobo Li,et al.  A 3D Surface Tracking Algorithm , 1996, Comput. Vis. Image Underst..

[556]  Christian Ronse Criteria for approximation of linear and affine functions , 1986 .

[557]  Per-Erik Danielsson An Improved Segmentation and Coding Algorithm for Binary and Nonbinary Images , 1982, IBM J. Res. Dev..

[558]  Reinhard Klette,et al.  A Comparative Evaluation of Length Estimators of Digital Curves , 2004, IEEE Trans. Pattern Anal. Mach. Intell..

[559]  Peter Veelaert Digital Planarity of Rectangular Surface Segments , 1994, IEEE Trans. Pattern Anal. Mach. Intell..

[560]  Alfred M. Bruckstein,et al.  The number of digital straight lines on an N×N grid , 1990, IEEE Trans. Inf. Theory.

[561]  Shigeki Yokoi,et al.  On Generalized Distance Transformation of Digitized Pictures , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[562]  Michael Werman,et al.  Recognition and characterization of digitized curves , 1987, Pattern Recognit. Lett..

[563]  J. Sklansky,et al.  Skeleton generation fromx, y boundary sequences , 1981 .

[564]  John Mylopoulos,et al.  On the Topological Properties of Quantized Spaces, I. The Notion of Dimension , 1971, JACM.

[565]  Chee-Keng Yap,et al.  Approximate Euclidean shortest path in 3-space , 1994, SCG '94.

[566]  Uwe Schnell Lattice inequalities for convex bodies and arbitrary lattices , 1993 .

[567]  H. von Mangoldt,et al.  Einfuhrung in die hohere Mathematik. III , 1932, Mathematical Gazette.

[568]  Willi Rinow Lehrbuch der Topologie , 1975 .

[569]  Chul E. Kim,et al.  Digital squares , 1988, [1988 Proceedings] 9th International Conference on Pattern Recognition.

[570]  J. Reeve On the Volume of Lattice Polyhedra , 1957 .

[571]  Herbert Süße,et al.  Adaptive Modelle und Invarianten für zweidimensionale Bilder , 1995, Berichte aus der Informatik.

[572]  J. Koplowitz,et al.  Design Of Perimeter Estimators For Digitized Planar Shapes , 1988, Other Conferences.

[573]  H. Coxeter,et al.  Introduction to Geometry , 1964, The Mathematical Gazette.

[574]  Christopher Conrad,et al.  Preserving Topology by a Digitization Process , 1998, Journal of Mathematical Imaging and Vision.

[575]  Lawrence N. Stout Two discrete forms of the Jordan curve theorem , 1988 .

[576]  Pierre Soille Spatial distributions from contour lines: An efficient methodology based on distance transformations , 1991, J. Vis. Commun. Image Represent..

[577]  Yazid M. Sharaiha,et al.  A graph-theoretic approach to distance transformations , 1994, Pattern Recognit. Lett..

[578]  G. Ringel,et al.  Solution of the heawood map-coloring problem. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[579]  Vladimir Kovalevsky,et al.  Multidimensional cell lists for investigating 3-manifolds , 2003, Discret. Appl. Math..

[580]  Christian Ronse,et al.  Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images , 1988, Discret. Appl. Math..

[581]  Azriel Rosenfeld,et al.  New views of linearity and connectedness in digital geometry , 1989, Pattern Recognit. Lett..

[582]  Jerome Rothstein,et al.  Parallel and sequential specification of a context sensitive language for straight lines on grids , 1976 .

[583]  P. A. B. Pleasants,et al.  Characterization of two-distance sequences , 1992 .

[584]  P. P. Das Lattice of octagonal distances in digital geometry , 1990, Pattern Recognit. Lett..

[585]  Christian Ronse A note on the approximation of linear and affine functions: the case of bounded slope , 1990 .

[586]  Magdy Gaafar Convexity Verification, Block-Chords, and Digital Straight Lines , 1977 .

[587]  Reinhard Klette Multigrid Convergence of Geometric Features , 2000, Digital and Image Geometry.

[588]  Pierre Soille,et al.  Generalized geodesy via geodesic time , 1994, Pattern Recognit. Lett..

[589]  Joost van de Weijer,et al.  Improved curvature and anisotropy estimation for curved line bundles , 1998, Proceedings. Fourteenth International Conference on Pattern Recognition (Cat. No.98EX170).

[590]  T. Yung Kong,et al.  Generic axiomatized digital surface-structures , 2001, Electron. Notes Theor. Comput. Sci..

[591]  Chul E. Kim,et al.  Digital Convexity, Straightness, and Convex Polygons , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[592]  Derick Wood,et al.  An Isothetic View of Computational Geometry† †Work carried out under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692. , 1985 .

[593]  Her-Kun Chang,et al.  Extracting medial curves on 3D images , 2002, Pattern Recognit. Lett..

[594]  G. Dirac Some Theorems on Abstract Graphs , 1952 .

[595]  R. Oppermann Elements of the topology of plane sets of points: by M. H. A. Newman. 221 pages, illustrations, 15 × 23 cms. New York, The Macmillan Company, 1939. Price $3.50 , 1939 .

[596]  Cherng-Min Ma Connectivity Preservation of 3D 6-Subiteration Thinning Algorithms , 1996, CVGIP Graph. Model. Image Process..

[597]  L S Juan Diaz-de-Leon,et al.  On the computation of the Euler number of a binary object , 1996 .

[598]  A. Vershik,et al.  The limit shape of convex lattice polygons and related topics , 1994 .

[599]  Shu-Yen Wan,et al.  Parallel Thinning Algorithms on 3D (18, 6) Binary Images , 2000, Comput. Vis. Image Underst..

[600]  Partha Pratim Das,et al.  Metricity preserving transforms , 1989, Pattern Recognit. Lett..

[601]  Mohamed Tajine,et al.  Topological properties of Hausdorff discretization, and comparison to other discretization schemes , 2002, Theor. Comput. Sci..

[602]  Willy Scherrer Ein Satz über Gitter und Volumen , 1922 .

[603]  Roberto Cipolla,et al.  Skeletonization using an extended Euclidean distance transform , 1995, Image Vis. Comput..

[604]  H. Carter Fuzzy Sets and Systems — Theory and Applications , 1982 .

[605]  Son Pham Digital circles with non-lattice point centers , 2005, The Visual Computer.

[606]  Christian Ronse,et al.  Set-Theoretical Algebraic Approaches to Connectivity in Continuous or Digital Spaces , 2004, Journal of Mathematical Imaging and Vision.

[607]  Grégoire Malandain,et al.  Euclidean skeletons , 1998, Image Vis. Comput..

[608]  Jayaram K. Udupa,et al.  A justification of a fast surface tracking algorithm , 1992, CVGIP Graph. Model. Image Process..

[609]  Jane Wilhelms,et al.  Topological considerations in isosurface generation , 1994, TOGS.

[610]  Stina Svensson,et al.  On Reversible Skeletonization Using Anchor-Points from Distance Transforms , 1999, J. Vis. Commun. Image Represent..

[611]  A T Young,et al.  Mariner 6 television pictures: first report. , 1969, Science.

[612]  O. Ore Theory of Graphs , 1962 .

[613]  Stéphane Marchand-Maillet,et al.  Discrete Convexity, Straightness, and the 16-Neighborhood , 1997, Comput. Vis. Image Underst..

[614]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[615]  Hanan Samet,et al.  A general approach to connected-component labeling for arbitrary image representations , 1992, JACM.

[616]  Satoru Kawai On the topology preservation property of local parallel operations , 1982, Comput. Graph. Image Process..

[617]  Jayaram K. Udupa Multidimensional Digital Boundaries , 1994, CVGIP Graph. Model. Image Process..

[618]  Azriel Rosenfeld A note on average distances in digital sets , 1987, Pattern Recognit. Lett..

[619]  Chul E. Kim On the Cellular Convexity of Complexes , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[620]  NABIL JEAN NACCACHE,et al.  An investigation into the skeletonization approach of hilditch , 1984, Pattern Recognit..

[621]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology : II. Openings and closings , 1991, CVGIP Image Underst..

[622]  D. Bleyl Automatische Bildverarbeitung in Medizin und Biologie. Herausgegeben von H. Simon, D. Kunze, K. Voss und W. R. Herrmann. Verlag Theodor Steinkopff, Dresden 1975. Preis: 25,80 M , 1976 .

[623]  Azriel Rosenfeld,et al.  Sequential Operations in Digital Picture Processing , 1966, JACM.

[624]  Frederic Fol Leymarie,et al.  Fast raster scan distance propagation on the discrete rectangular lattice , 1992, CVGIP Image Underst..

[625]  Jim Piper Efficient implementation of skeletonisation using interval coding , 1985, Pattern Recognit. Lett..

[626]  Mohamed Tajine,et al.  Discretization in Hausdorff Space , 2004, Journal of Mathematical Imaging and Vision.

[627]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[628]  P. Libby The Scientific American , 1881, Nature.

[629]  J. Wilhelms,et al.  Octrees for faster isosurface generation , 1992, TOGS.

[630]  Christophe Lohou Contribution à l'analyse topologique des images : étude d'algorithmes de squelettisation pour images 2D et 3D, selon une approche topologie digitale ou topologie discrète. , 2001 .

[631]  R. A. Melter A survey of digital metrics , 1991 .

[632]  Geoff A. W. West,et al.  Salience Distance Transforms , 1995, CVGIP Graph. Model. Image Process..

[633]  F. Frances Yao,et al.  Optimal binary space partitions for orthogonal objects , 1990, SODA '90.

[634]  Christian Ronse,et al.  A Topological Characterization of Thinning , 1986, Theor. Comput. Sci..

[635]  Patrick Shen-Pei Wang,et al.  A Simple and Robust Thinning Algorithm , 1999, Int. J. Pattern Recognit. Artif. Intell..

[636]  M. Douglas McIlroy Best approximate circles on integer grids , 1983, TOGS.

[637]  Azriel Rosenfeld,et al.  Digital Picture Processing , 1976 .

[638]  Chul E. Kim,et al.  Digital Disks , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[639]  D. Klarner Cell Growth Problems , 1967, Canadian Journal of Mathematics.

[640]  Linda G. Shapiro,et al.  A new connected components algorithm for virtual memory computers , 1983, Comput. Vis. Graph. Image Process..

[641]  Andrew Mehnert,et al.  On Computing the Exact Euclidean Distance Transform on Rectangular and Hexagonal Grids , 1999, Journal of Mathematical Imaging and Vision.

[642]  Isabelle Debled-Rennesson,et al.  3D Discrete Normal Vectors , 1999, DGCI.

[643]  D. Hugh Redelmeier,et al.  Counting polyominoes: Yet another attack , 1981, Discret. Math..

[644]  Rémy Malgouyres Homotopy in Two-Dimensional Digital Images , 2000, Theor. Comput. Sci..

[645]  J. Steiner,et al.  Einfache Beweise der isoperimetrischen Hauptsätze. , 1838 .

[646]  P. P. Das,et al.  Knight's distance in digital geometry , 1988, Pattern Recognit. Lett..

[647]  Ding-Zhu Du,et al.  Diameter and radius in the Manhattan metric , 1990, Discret. Comput. Geom..

[648]  Shi-Jinn Horng,et al.  Optimal Computing the Chessboard Distance Transform on Parallel Processing Systems, , 1999, Comput. Vis. Image Underst..

[649]  A. W. Roscoe,et al.  A theory of binary digital pictures , 1985, Comput. Vis. Graph. Image Process..

[650]  King-Sun Fu,et al.  Using the FFT to determine digital straight line chain codes , 1982, Comput. Graph. Image Process..

[651]  Rangasami L. Kashyap,et al.  Building Skeleton Models via 3-D Medial Surface/Axis Thinning Algorithms , 1994, CVGIP Graph. Model. Image Process..

[652]  Johann Benedict Listing der Census räumlicher Complexe, oder Verallgemeinerung des Euler´schen Satzes von den Polyädern , 1862 .

[653]  J. A. Hillier,et al.  A Method for Finding the Shortest Route Through a Road Network , 1960 .

[654]  Ben J. H. Verwer,et al.  Local distances for distance transformations in two and three dimensions , 1991, Pattern Recognit. Lett..

[655]  Thomas Bülow,et al.  Minimum-Length Polygons in Simple Cube-Curves , 2000, DGCI.

[656]  Joseph N. Wilson,et al.  Handbook of computer vision algorithms in image algebra , 1996 .

[657]  Nahum Kiryati,et al.  Length Estimation in 3-D Using Cube Quantization , 2004, Journal of Mathematical Imaging and Vision.

[658]  Ralph Kopperman,et al.  A Jordan surface theorem for three-dimensional digital spaces , 1991, Discret. Comput. Geom..

[659]  Gabriella Sanniti di Baja,et al.  Skeletonization algorithm running on path-based distance maps , 1996, Image Vis. Comput..

[660]  Francoise J. Preteux,et al.  Unified mathematical framework for a compact and fully parallel n-D skeletonization procedure , 1999, Optics & Photonics.

[661]  Azriel Rosenfeld,et al.  An Improved Method of Angle Detection on Digital Curves , 1975, IEEE Transactions on Computers.