Localisation of topological features using 3D object representations

Holes, tunnels and cavities of two-dimensional (2D) and 3D objects are concise topological features used for object representation and recognition. In this study, the authors are representing any cubical tessellation (regular or not) of 2D and 3D objects and dealing with the extraction and the localisation of these features by using homology-based approach. The cubical tessellation (regular or not) of objects is translated into algebraic language suitable for building a reduced cell complex structure. The extraction of the homology information is equivalent to the estimation of the rank of the homology groups of the reduced complex. The localisation means the reconstruction of the object cycles from the generators of the homology groups. The reduction operation of the cell complex leads to an efficient algorithm. Note that, several objects can be analysed simultaneously by the algorithm conceived in our approach. This algorithm is validated by using 2D and 3D binary images.

[1]  Guillaume Damiand,et al.  Computing Homology Generators for Volumes Using Minimal Generalized Maps , 2008, IWCIA.

[2]  Mohamad Jamil Seaidoun,et al.  A fast exact euclidean distance transform with application to computer vision and digital image processing , 1993 .

[3]  Leila De Floriani,et al.  An iterative algorithm for homology computation on simplicial shapes , 2011, Comput. Aided Des..

[4]  Konstantin Mischaikow,et al.  Cubical homology and the topological classification of 2D and 3D imagery , 2001, Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205).

[5]  Neil F. Stewart,et al.  A framework for system specification using chains on cell complexes , 1999, Comput. Aided Des..

[6]  Ozgur Ege,et al.  Some Results on Simplical Homology Groups of 2D Digital Images. , 2012 .

[7]  Rocío González-Díaz,et al.  Integral Operators for Computing Homology Generators at Any Dimension , 2008, CIARP.

[8]  Rocío González-Díaz,et al.  Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images , 2016, DGCI.

[9]  Jacques-Olivier Lachaud,et al.  Computation of homology groups and generators , 2006, Comput. Graph..

[10]  Madjid Allili,et al.  Geometric Construction of a Coboundary of a Cycle , 2001, Discret. Comput. Geom..

[11]  B. S. Manjunath,et al.  Unsupervised Segmentation of Color-Texture Regions in Images and Video , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[12]  Guillaume Damiand,et al.  Computing Homology for Surfaces with Generalized Maps: Application to 3D Images , 2006, ISVC.

[13]  Djemel Ziou,et al.  Topological feature extraction in binary images , 2001, Proceedings of the Sixth International Symposium on Signal Processing and its Applications (Cat.No.01EX467).

[14]  Adrian Ion,et al.  Controlling Geometry of Homology Generators , 2007 .

[15]  Andrew Stein,et al.  Analysis of blood vessel topology by cubical homology , 2002, Proceedings. International Conference on Image Processing.

[16]  Djemel Ziou,et al.  Generating cubical complexes from image data and computation of the Euler number , 2002, Pattern Recognit..

[17]  David Eppstein,et al.  Graph-Theoretic Solutions to Computational Geometry Problems , 2009, WG.

[18]  Djemel Ziou,et al.  A global CAT approach for graylevel diffusion , 2003, Seventh International Symposium on Signal Processing and Its Applications, 2003. Proceedings..

[19]  Guillaume Damiand,et al.  Removal Operations in nD Generalized Maps for Efficient Homology Computation , 2012, CTIC.

[20]  Rocío González-Díaz,et al.  Algebraic Topological Analysis of Time-Sequence of Digital Images , 2005, CASC.

[21]  Pedro Real Jurado,et al.  Advanced Homology Computation of Digital Volumes Via Cell Complexes , 2008, SSPR/SPR.

[22]  Djemel Ziou,et al.  Topological feature extraction using algebraic topology , 2007, Electronic Imaging.

[23]  Theodosios Pavlidis,et al.  Picture Segmentation by a Tree Traversal Algorithm , 1976, JACM.

[24]  Pedro Real Jurado,et al.  Cell AT-Models for Digital Volumes , 2009, GbRPR.

[25]  Rocío González-Díaz,et al.  3D well-composed polyhedral complexes , 2015, Discret. Appl. Math..

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

[27]  Pedro Real Jurado,et al.  A Homological-Based Description of Subdivided nD Objects , 2011, CAIP.

[28]  Kenneth Steiglitz,et al.  Operations on Images Using Quad Trees , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[29]  Djemel Ziou,et al.  A Computational Algebraic Topology Model for the Deformation of Curves , 2002, AMDO.

[30]  Djemel Ziou,et al.  Image matching using algebraic topology , 2006, Electronic Imaging.

[31]  Gauthier Lafruit,et al.  Adaptive 3D Content for Multi-Platform On-Line Games , 2007, CW 2007.

[32]  Yll Haxhimusa,et al.  Directly computing the generators of image homology using graph pyramids , 2009, Image Vis. Comput..

[33]  Rocío González-Díaz,et al.  Chain homotopies for object topological representations , 2009, Discret. Appl. Math..

[34]  Basil G. Mertzios,et al.  Real-time computation of two-dimensional moments on binary images using image block representation , 1998, IEEE Trans. Image Process..