A justification of a fast surface tracking algorithm

In many three-dimensional (3D) imaging applications in science, medicine, and engineering, object information captured in a 3D digital image is visualized and analyzed based on object surfaces extracted from the image. Recently Gordon and Udupa (Comput. Vision Graphics Image Process.45, 1989, 196–214) reported a fast algorithm for tracking such surfaces, whose correct behavior has only been a conjecture but observed nonetheless in all uses of this algorithm in a variety of biomedical applications. In the present paper, we prove the correctness of their algorithm by showing that the surfaces generated by the algorithm correspond to connected components of boundary elements and that they possess Jordan boundary properties.

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

[2]  Hsun K. Liu,et al.  Two and three dimensional boundary detection , 1977 .

[3]  Jayaram K. Udupa,et al.  Interactive segmentation and boundary surface formation for 3-D digital images , 1982, Comput. Graph. Image Process..

[4]  Azriel Rosenfeld,et al.  Multidimensional Edge Detection by Hypersurface Fitting , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[6]  Jayaram K. Udupa,et al.  Computer aspects of 3D imaging in Medicine: a tutorial , 1991 .

[7]  Henry Fuchs,et al.  Optimal surface reconstruction from planar contours , 1977, CACM.

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

[9]  Donald P. Greenberg,et al.  Computer generated images for medical applications , 1978, SIGGRAPH.

[10]  Gabor T. Herman,et al.  The theory, design, implementation and evaluation of a three-dimensional surface detection algorit , 1981 .

[11]  Gabor T. Herman,et al.  A topological proof of a surface tracking algorithm , 1982, Comput. Vis. Graph. Image Process..

[12]  Jayaram K. Udupa,et al.  Fast surface tracking in three-dimensional binary images , 1989, Comput. Vis. Graph. Image Process..

[13]  Azriel Rosenfeld,et al.  Digital surfaces , 1991, CVGIP Graph. Model. Image Process..

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