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Minimal Surfaces Based Object Segmentation
April 1997 (vol. 19 no. 4)
pp. 394-398

Abstract—A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as minimal surfaces. The space where the surfaces are computed is induced from the 3D image (volumetric data) in which the objects are to be detected. The model links between classical deformable surfaces obtained via energy minimization, and intrinsic ones derived from curvature based flows. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation.

[1] D. Adalsteinsson and J.A. Sethian, "A Fast Level Set Method for Propagating Interfaces," J. Comparative Physics, vol. 118, pp. 269-277, 1995.
[2] L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, "Axioms and Fundamental Equations of Image Processing," Architecture Rational Mechanics, vol. 123, 1993.
[3] A. Blake and A. Zisserman, Visual Reconstruction.Cambridge, Mass.: MIT Press, 1987.
[4] V. Caselles, F. Catte, T. Coll, and F. Dibos, "A Geometric Model for Active Contours," Numerische Mathematik, vol. 66, pp. 1-31, 1993.
[5] V. Caselles, R. Kimmel, and G. Sapiro, "Geodesic Active Contours," Proc. IEEE ICCV-95, pp. 694-699, 1995.
[6] V. Caselles, R. Kimmel, G. Sapiro, and C. Sbert, "Minimal Surfaces: A Three-Dimensional Segmentation Approach," Technion Technical Report 973, June 1995,Israel.
[7] V. Caselles and C. Sbert, "What Is the Best Causal Scale-Space for 3D Images?," SIAM J. Applied Math, forthcoming.
[8] D. Chopp, "Computing Minimal Surfaces via Level Set Curvature Flows," J. Comparative Physics, vol. 106, no. 1, pp. 77-91, 1993.
[9] L.D. Cohen, "On Active Contour Models and Balloons," CVGIP: Image Understanding, vol. 53, pp. 211-218, 1991.
[10] I. Cohen, L.D. Cohen, and N. Ayache, "Using Deformable Surfaces to Segment 3D Images and Infer Differential Structure," CVGIP: Image Understanding, vol. 56, pp. 242-263, 1992.
[11] L.D. Cohen and I. Cohen, “Finite-Element Methods for Active Contour Models and Balloons for 2D and 3D Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 15, no. 11, pp. 1,131-1,147, Nov. 1993.
[12] L.D. Cohen and R. Kimmel, "Global Minimum for Active Contours Models: A Minimal Path Approach," Int'l J. Computer Vision, forthcoming. A short version appears in Proc. CVPR '96, San-Francisco, Calif., 1996.
[13] P. Fua and Y.G. Leclerc, "Model Driven Edge Detection," Machine Vision and Applications, vol. 3, pp. 45-56, 1990.
[14] M. Gage and R.S. Hamilton, "The Heat Equation Shrinking Convex Plane Curves," J. Differential Geometry, vol. 23, pp. 69-96, 1986.
[15] M. Grayson, "The Heat Equation Shrinks Embedded Plane Curves to Round Points," J. Differential Geometry, vol. 26, pp. 285-314, 1987.
[16] M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: Active Contour Models," Int'l J. Computer Vision, vol. 1, pp. 321-331, 1988.
[17] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, Gradient Flows and Geometric Active Contour Models Proc. IEEE Int'l Conf. Computer Vision, pp. 810-815, 1995.
[18] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, "Conformal Curvature Flows: From Phase Transitions to Active Vision," Archive for Rational Mechanics and Analysis, forthcoming.
[19] B.B. Kimia, A. Tannenbaum, and S.W. Zucker, "Shapes, Shocks, and Deformations, I," Int'l J. Computer Vision, vol. 15, pp. 189-224, 1995.
[20] R. Kimmel, A. Amir, A.M. Bruckstein, "Finding Shortest Paths on Surfaces Using Level Sets Propagation," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 635-640, June 1995.
[21] R. Malladi, R. Kimmel, D. Adalsteinsson, G. Sapiro, V. Caselles, and J.A. Sethian, "A Geometric Approach to Segmentation and Analysis of 3D Medical Images," Proc. Math. Methods Biomedical Image Analysis Workshop,San Francisco, June21-22, 1996.
[22] R. Malladi, J. Sethian, and B.C. Vemuri, "Shape Modeling with Front Propagation: A Level Set Approach," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, pp. 158-175, 1995.
[23] T. McInerney and D. Terzopoulos, "Topologically Adaptable Snakes," Proc. IEEE ICCV-95, pp. 840-845, 1995.
[24] P.J. Olver, G. Sapiro, and A. Tannenbaum, "Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing," SIAM J. Applied Math., forthcoming.
[25] S.J. Osher and J.A. Sethian, "Fronts Propagation with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations," J. Computational Physics, vol. 79, pp. 12-49, 1988.
[26] R. Osserman, Survey of Minimal Surfaces.Dover, 1986.
[27] G. Sapiro, R. Kimmel, and V. Caselles, "Object Detection and Measurements in Medical Images via Geodesic Active Contours," Proc. SPIE-Vision Geometry,San Diego, July 1995.
[28] G. Sapiro and A. Tannenbaum, "Affine Invariant Scale-Space," Int'l J. Computer Vision, vol. 11, no. 1, pp. 25-44, 1993.
[29] J. Shah, "Recovery of Shapes by Evolution of Zero-Crossings," technical report, Math. Dept., Northeastern Univ., Boston, Mass., 1995.
[30] R. Szeliski, D. Tonnesen, and D. Terzopoulos, "Modeling Surfaces of Arbitrary Topology with Dynamic Particles," Proc. IEEE Computer Vision and Pattern Recognition, pp. 82-85,New York, NY, June 1993.
[31] H. Tek and B.B. Kimia, Image Segmentation by Reaction-Diffusion Bubbles Proc. Int'l Conf. Computer Vision, pp. 156-162, 1995.
[32] D. Terzopoulos and D. Metaxas, “Dynamic 3D Models with Local and Global Deformations: Deformable Superquadrics,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 7, pp. 703-714, July 1991.
[33] D. Terzopoulos, A. Witkin, and M. Kass, "Constraints on Deformable Models: Recovering 3D Shape and Nonrigid Motions," Artificial Intelligence, vol. 36, pp. 91-123, 1988.
[34] R. Whitaker, “Algorithms for Implicit Deformable Models,” Proc. Fifth Int'l Conf. Computer Vision, pp. 822-827, June 1995.
[35] A. Yezzi, S. Kichenassamy, P. Olver, and A. Tannenbaum, A Geometric Snake Models for Segmentation of Medical Imagery IEEE Trans. Medical Imaging, vol. 16, pp. 199-209, 1997.
[36] S.C. Zhu, T.S. Lee, and A.L. Yuille, “Region Competition: Unifying Snakes, Region Growing, Energy/Bayes/MDL for Multiband Image Segmentation,” Proc. Fifth IEEE Int'l Conf. Computer Vision (ICCV), pp. 416-423, 1995.

Index Terms:
3D segmentation, minimal surfaces, deformable models, mean curvature motion, medical images.
Vincent Caselles, Ron Kimmel, Guillermo Sapiro, Catalina Sbert, "Minimal Surfaces Based Object Segmentation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 4, pp. 394-398, April 1997, doi:10.1109/34.588023
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