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Issue No.02 - February (2010 vol.32)
pp: 321-334
Leo Grady , Siemens Corporate Research, East Princeton
ABSTRACT
Shortest paths have been used to segment object boundaries with both continuous and discrete image models. Although these techniques are well defined in 2D, the character of the path as an object boundary is not preserved in 3D. An object boundary in three dimensions is a 2D surface. However, many different extensions of the shortest path techniques to 3D have been previously proposed in which the 3D object is segmented via a collection of shortest paths rather than a minimal surface, leading to a solution which bears an uncertain relationship to the true minimal surface. Specifically, there is no guarantee that a minimal path between points on two closed contours will lie on the minimal surface joining these contours. We observe that an elegant solution to the computation of a minimal surface on a cellular complex (e.g., a 3D lattice) was given by Sullivan [47]. Sullivan showed that the discrete minimal surface connecting one or more closed contours may be found efficiently by solving a Minimum-cost Circulation Network Flow (MCNF) problem. In this work, we detail why a minimal surface properly extends a shortest path (in the context of a boundary) to three dimensions, present Sullivan's solution to this minimal surface problem via an MCNF calculation, and demonstrate the use of these minimal surfaces on the segmentation of image data.
INDEX TERMS
3D image segmentation, minimal surfaces, shortest paths, Dijkstra's algorithm, boundary operator, total unimodularity, linear programming, minimum-cost circulation network flow.
CITATION
Leo Grady, "Minimal Surfaces Extend Shortest Path Segmentation Methods to 3D", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 2, pp. 321-334, February 2010, doi:10.1109/TPAMI.2008.289
REFERENCES
 [1] K. Aoshima and M. Iri, “Comments on F. Hadlock's Paper: Finding a Maximum Cut of a Planar Graph in Polynomial Time,” SIAM J. Computing, vol. 6, pp. 86-87, 1977. [2] B. Appleton and H. Talbot, “Globally Optimal Surfaces by Continuous Maximal Flows,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 1, pp. 106-118, Jan. 2006. [3] R. Ardon and L.D. Cohen, “Fast Constrained Surface Extraction by Minimal Paths,” Int'l J. Computer Vision, vol. 69, no. 1, pp. 127-136, Aug. 2006. [4] R. Ardon, L.D. Cohen, and A. Yezzi, “A New Implicit Method for Surface Segmentation by Minimal Paths: Applications in 3D Medical Images,” Proc. Int'l Workshop Energy Minimization Methods in Computer Vision and Pattern Recognition, A.Rangarajan, ed., pp.520-535, 2005. [5] C.J. Armstrong, W.A. Barrett, and B. Price, “Live Surface,” Proc. Volume Graphics '06, vol. 22, pp. 661-670, Sept. 2006. [6] N. Biggs, Algebraic Graph Theory. Cambridge Univ. Press, 1974. [7] I. Bitter, A.E. Kaufman, and M. Sato, “Penalized-Distance Volumetric Skeleton Algorithm,” IEEE Trans. Visualization and Computer Graphics, vol. 7, no. 3, pp. 195-206, July-Sept. 2001. [8] M.J. Black, G. Sapiro, D.H. Marimont, and D. Heeger, “Robust Anisotropic Diffusion,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 421-432, Mar. 1998. [9] Y. Boykov and M.-P. Jolly, “Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images,” Proc. Int'l Conf. Computer Vision, pp. 105-112, 2001. [10] Y. Boykov and V. Kolmogorov, “Computing Geodesics and Minimal Surfaces via Graph Cuts,” Proc. Int'l Conf. Computer Vision, vol. 1, Oct. 2003. [11] A.J. Briggs, C. Detweiler, D. Scharstein, M. College, and A. Vandenberg-Rodes, “Expected Shortest Paths for Landmark-Based Robot Navigation,” Int'l J. Robotics Research, vol. 23, nos.7/8, pp. 717-728, 2004. [12] C. Buehler, S.J. Gortler, M.F. Cohen, and L. McMillan, “Minimal Surfaces for Stereo,” Proc. Seventh European Conf. Computer Vision, vol. III, pp. 885-899, May 2002. [13] L. Cohen and T. Deschamps, “Grouping Connected Components Using Minimal Path Techniques. Application to Reconstruction of Vessels in 2D and 3D Images,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 2, pp 102-109, 2001. [14] L.D. Cohen and R. Kimmel, “Global Minimum for Active Contour Models: A Minimal Path Approach,” Int'l J. Computer Vision, vol. 24, no. 1, pp. 57-78, 1997. [15] A.X. Falcão and J.K. Udupa, “A 3D Generalization of User-Steered Live-Wire Segmentation,” Medical Image Analysis, vol. 4, pp. 389-402, 2000. [16] A.X. Falcão, J.K. Udupa, S. Samarasekera, S. Sharma, B.H. Elliot, and R. de A. Lotufo, “User-Steered Image Segmentation Paradigms: Live Wire and Live Lane,” Graphical Models and Image Processing, vol. 60, no. 4, pp. 233-260, 1998. [17] L.R. Ford and D.R. Fulkerson, “A Primal-Dual Algorithm for the Capacitated Hitchcock Problem,” Naval Research Logistics Quarterly, vol. 4, pp. 47-54, 1957. [18] L.R. Ford and D.R. Fulkerson, Flows in Networks. Princeton Univ. Press, 1962. [19] J. Forrest, D. de la Nuez, and R. Lougee-Heimer, CLP User Guide. IBM Research, 2004. [20] A.V. Goldberg, E. Tardos, and R.E. Tarjan, “Network Flow Algorithms,” Paths, Flows and VLSI-Design, B. Korte, L. Lovasz, H. Proemel, and A. Schrijver, eds., pp. 101-164, Springer-Verlag, 1990. [21] L. Grady, “Computing Exact Discrete Minimal Surfaces: Extending and Solving the Shortest Path Problem in 3D with Application to Segmentation,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 69-78, June 2006. [22] F. Hadlock, “Finding a Maximum Cut of a Planar Graph in Polynomial Time,” SIAM J. Computing, vol. 4, no. 3, pp. 221-225, 1975. [23] G. Hamarneh, J. Yang, C. McIntosh, and M. Langille, “3D Live-Wire-Based Semi-Automatic Segmentation of Medical Images,” Proc. SPIE Medical Imaging '05: Image Processing, pp. 1597-1603, 2005. [24] P.J. Hilton and S. Wylie, Homology Theory. Cambridge Univ. Press, 1960. [25] D. Kirsanov, “Minimal Discrete Curves and Surfaces,” PhD thesis, Harvard Univ., 2004. [26] M. Knapp, A. Kanitsar, and M.E. Gröller, “Semi-Automatic Topology Independent Contour-Based 2${1\over 2}$ D Segmentation Using Live-Wire,” J. WSCG, vol. 12, no. 2, pp. 229-236, 2004. [27] V. Kolmogorov, “Primal-Dual Algorithm for Convex Markov Random Fields,” Technical Report MSR-TR-2005-117, Microsoft, Sept. 2005. [28] V. Kolmogorov and C. Rother, “Minimizing Nonsubmodular Functions with Graph Cuts—A Review,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 7, pp. 1274-1279, July 2007. [29] S. König and J. Hesser, “3D Live-Wires on Pre-Segmented Volume Data,” Proc. SPIE Medical Imaging '05: Image Processing, pp. 1674-1679, 2005. [30] S. Lefschetz, Algebraic Topology, vol. 27. Am. Math. Soc. Colloquium Publications, 1942. [31] K. Li, X. Wu, D.Z. Chen, and M. Sonka, “Optimal Surface Segmentation in Volumetric Images—A Graph-Theoretic Approach,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 1, pp. 119-134, Jan. 2006. [32] C. Mattiussi, “The Finite Volume, Finite Element and Finite Difference Methods as Numerical Methods for Physical Field Problems,” Advances in Imaging and Electron Physics, pp. 1-146, Academic Press, Inc., Apr. 2000. [33] F. Morgan, Geometric Measure Theory, third ed. Academic Press, 2000. [34] E. Mortensen and W. Barrett, “Interactive Segmentation with Intelligent Scissors,” Graphical Models in Image Processing, vol. 60, no. 5, pp. 349-384, 1998. [35] G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization. John Wiley & Sons, 1999. [36] S. Okada, “On Mesh and Node Determinants,” Proc. IRE, vol. 43, p. 1527, 1955. [37] C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization. Dover, 1998. [38] S.V. Porter, M. Mirmehdi, and B.T. Thomas, “A Shortest Path Representation for Video Summarisation,” Proc. 12th Int'l Conf. Image Analysis and Processing, pp. 460-465, Sept. 2003. [39] S. Roy and I. Cox, “A Maximum-Flow Formulation of the n-Camera Stereo Correspondence Problem,” Proc. Int'l Conf. Computer Vision, pp. 492-499, 1998. [40] Z. Salah and J.O.D. Bartz, “Live-Wire Revisited,” Proc. Workshop Bildverarbeitung in der Medizin, pp. 158-162, 2005. [41] A. Schenk, G. Prause, and H.-O. Peitgen, “Efficient Semiautomatic Segmentation of 3D Objects in Medical Images,” Proc. Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 186-195, 2000. [42] A. Schenk, G. Prause, and H.-O. Peitgen, “Local Cost Computation for Efficient Segmentation of 3D Objects with Live Wire,” Proc. SPIE Medical Imaging, M. Sonka and K.M. Hanson, eds., pp.1357-1364, 2001. [43] S. Seshu, “The Mesh Counterpart of Shekel's Theorem,” Proc. IRE, vol. 43, p. 342, 1955. [44] J.A. Sethian, “A Fast Marching Level Set Method for Monotonically Advancing Fronts,” Proc. Nat'l Academy of Sciences USA, vol. 93, no. 4, pp. 1591-1595, 1996. [45] J. Shi and J. Malik, “Normalized Cuts and Image Segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, Aug. 2000. [46] W.-K. Shih, S. Wu, and Y.S. Kuo, “Unifying Maximum Cut and Minimum Cut of a Planar Graph,” IEEE Trans. Computers, vol. 39, no. 5, pp. 694-697, May 1990. [47] J.M. Sullivan, “A Crystalline Approximation Theorem for Hypersurfaces,” PhD thesis, Princeton Univ., Oct. 1990. [48] C. Sun, “Fast Optical Flow Using 3D Shortest Path Techniques,” Image and Vision Computing, vol. 20, nos. 13/14, pp. 981-991, Dec. 2002. [49] E. Tonti, “On the Geometrical Structure of Electromagnetism,” Gravitation, Electromagnetism and Geometrical Structures, G.Ferraese, ed., pp. 281-308, Pitagora, 1996. [50] K. Truemper, “Algebraic Characterizations of Unimodular Matrices,” SIAM J. Applied Math., vol. 35, no. 2, pp. 328-332, Sept. 1978. [51] J.N. Tsitsiklis, “Efficient Algorithms for Globally Optimal Trajectories,” IEEE Trans. Automatic Control, vol. 40, no. 9, pp. 1528-1538, Sept. 1995. [52] A.J. Zomorodian, Topology for Computing. Cambridge Univ. Press, 2005.