Optimal Separable Algorithms to Compute the Reverse Euclidean Distance Transformation and Discrete Medial Axis in Arbitrary Dimension

In binary images, the distance transformation (DT) and the geometrical skeleton extraction are classic tools for shape analysis. In this paper, we present time optimal algorithms to solve the reverse Euclidean distance transformation and the reversible medial axis extraction problems for d-dimensional images. We also present a d-dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

[1]  Jun-ichiro Toriwaki,et al.  Reverse Distance Transformation and Skeletons Based upon the Euclidean Metric for n -Dimensional Digital Binary Pictures , 1994 .

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

[3]  Wim H. Hesselink,et al.  A General Algorithm for Computing Distance Transforms in Linear Time , 2000, ISMM.

[4]  Jen-Hui Chuang,et al.  Skeletonization of Three-Dimensional Object Using Generalized Potential Field , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  Tomio Hirata,et al.  A Unified Linear-Time Algorithm for Computing Distance Maps , 1996, Inf. Process. Lett..

[6]  Grégoire Malandain,et al.  3-D chamfer distances and norms in anisotropic grids , 2005, Image Vis. Comput..

[7]  Azriel Rosenfeld,et al.  Digital geometry - geometric methods for digital picture analysis , 2004 .

[8]  Yaorong Ge,et al.  On the Generation of Skeletons from Discrete Euclidean Distance Maps , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  Wim H. Hesselink,et al.  Euclidean Skeletons of 3D Data Sets in Linear Time by the Integer Medial Axis Transform , 2005, ISMM.

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

[11]  Ingela Nyström,et al.  Efficient shape representation by minimizing the set of centres of maximal discs/spheres , 1997, Pattern Recognit. Lett..

[12]  Sunghee Choi,et al.  The power crust, unions of balls, and the medial axis transform , 2001, Comput. Geom..

[13]  D. Lee,et al.  Skeletonization via Distance Maps and Level Sets , 1995 .

[14]  Michael Lindenbaum,et al.  Euclidean Voronoi labelling on the multidimensional grid , 1995, Pattern Recognit. Lett..

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

[16]  Franco P. Preparata,et al.  Sequencing-by-hybridization revisited: the analog-spectrum proposal , 2004, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

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

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

[19]  Kaleem Siddiqi,et al.  The Hamilton-Jacobi skeleton , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

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

[21]  Edouard Thiel,et al.  Optimizing 3D chamfer masks with norm constraints , 2000 .

[22]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[23]  Edouard Thiel,et al.  Exact medial axis with euclidean distance , 2005, Image Vis. Comput..

[24]  Weiguang Guan,et al.  A List-Processing Approach to Compute Voronoi Diagrams and the Euclidean Distance Transform , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[25]  Gunilla Borgefors,et al.  Distance transformations in digital images , 1986, Comput. Vis. Graph. Image Process..

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

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

[28]  Frederic Fol Leymarie,et al.  Simulating the Grassfire Transform Using an Active Contour Model , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  David Coeurjolly.  Algorithmique et géométrie discrète pour la caractérisation des courbes et des surfaces. (Algorithmic and digital geometry for curve and surface characterization) , 2002 .

[30]  Calvin R. Maurer,et al.  A Linear Time Algorithm for Computing Exact Euclidean Distance Transforms of Binary Images in Arbitrary Dimensions , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[31]  Gabriella Sanniti di Baja,et al.  Pruning Discrete and Semiocontinuous Skeletons , 1995, ICIAP.

[32]  Azriel Rosenfeld,et al.  Geometric Methods for Digital Picture Analysis , 2005 .

[33]  Benjamin B. Kimia,et al.  On the Local Form and Transitions of Symmetry Sets, Medial Axes, and Shocks , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[34]  Benjamin B. Kimia,et al.  A formal classification of 3D medial axis points and their local geometry , 2004, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[35]  Ugo Montanari,et al.  Continuous Skeletons from Digitized Images , 1969, JACM.

[36]  Dominique Attali,et al.  Computing and Simplifying 2D and 3D Continuous Skeletons , 1997, Comput. Vis. Image Underst..

[37]  Mariette Yvinec,et al.  An Algorithm for Constructing the Convex Hull of a Set of Spheres in Dimension D , 1996, Comput. Geom..

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

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

[40]  Ching-Shoei Chiang.  The Euclidean distance transform , 1992 .

[41]  Benedek Nagy.  A Comparison Among Distances Based on Neighborhood Sequences in Regular Grids , 2005, SCIA.

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

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