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Issue No.03 - March (2008 vol.30)
pp: 412-423
In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the Euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor which is chosen depends only upon position and is in this sense isotropic. While directional information has been studied previously for other segmentation frameworks, here we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming based schemes. Finally we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery.
Directional segmentation, Finsler metric, dynamic programming, active contours, diffusion weighted imagery
John Melonakos, Eric Pichon, Sigurd Angenent, Allen Tannenbaum, "Finsler Active Contours", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 3, pp. 412-423, March 2008, doi:10.1109/TPAMI.2007.70713
[1] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Conformal Curvature Flows: From Phase Transitions to Active Vision,” Archive of Rational Mechanics and Analysis, vol. 134, no. 3, pp. 275-301, 1996.
[2] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic Active Contours,” Int'l J. Computer Vision, vol. 22, no. 11, pp. 61-79, 1997.
[3] D. Bao, S.-S. Chern, and Z. Shen, Introduction to Riemann-Finsler Geometry. Springer, 2000.
[4] M. Spivak, Introduction to Differential Geometry, vol. 2. Publish of Perish, 1974.
[5] S. Angenent and M. Gurtin, “Anisotropic Motion of a Phase Interface. Well-Posedness of the Initial Value Problem and Qualitative Properties of the Interface,” J. Reine und Angewandte Mathematik, vol. 446, pp. 1-47, 1994.
[6] M. Gage, “Evolving Plane Curves by Curvature in Relative Geometries,” Duke J. Math., vol. 72, pp. 441-466, 1993.
[7] E. Pichon, G. Sapiro, and A. Tannenbaum, Segmentation of Diffusion Tensor Imagery, “, Directions in Math. Systems Theory and Optimization,” Lecture Notes in Computer Science, no. 286, pp.239-247, 2003.
[8] E. Pichon, C. Westin, and A. Tannenbaum, “Hamilton-Jacobi-Bellman Approach to High-Angular Resolution Diffusion Tractography,” Proc. Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 180-187, 2005.
[9] E. Pichon and A. Tannenbaum, “Pattern Detection and Image Segmentation with Anisotropic Conformal Factor,” Proc. Int'l Conf. Image Processing, 2005.
[10] G. Bellettini, “Anisotropic and Crystalline Mean Curvature Flow,” A Sampler of Riemann-Finsler Geometry, pp.49-82, Cambridge Univ. Press, 2004.
[11] R. Kimmel and A. Bruckstein, “Regularized Zero-Crossings as Optimal Edge Detectors,” Int'l J. Computer Vision, vol. 53, no. 3, pp.225-243, 2003.
[12] A. Vasilevsky and K. Siddiqi, “Flux Maximizing Geometric Flows,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, no. 12, pp. 1565-1579, Dec. 2002.
[13] Y. Boykov, V. Kolmorgorov, D. Cremers, and A. Delong, “An Integral Solution to Surface Evolution Pdes via Geo-Cuts,” Proc. IEEE European Conf. Computer Vision, pp. 409-422, 2006.
[14] V. Kolmorgorov and Y. Boykov, “What Metrics Can Be Approximated by Geo-Cuts or Global Optimization of Length/Area and Flux,” Proc. IEEE Int'l Conf. Computer Vision, 2003.
[15] Y. Boykov and V. Kolmorgorov, “Computing Geodesics and Minimal Surfaces via Graph Cuts,” Proc. IEEE Int'l Conf. Computer Vision, pp. 26-33, 2003.
[16] J.-M. Morel and S. Solimini, Variational Methods for Image Segmentation. Birkhauser, 1994.
[17] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer, 2003.
[18] J. Sethian, Level Set Methods and Fast Marching Methods. Cambridge Univ. Press, 1999.
[19] G. Sapiro, Geometric Partial Differential Equations and Image Analysis. Cambridge Univ. Press, 2001.
[20] E. Mortensen, B. Morse, W. Barrett, and J. Udupa, “Adaptive Boundary Detection Using Live-Wire Two-Dimensional Dynamic Programming,” IEEE Proc. Computers in Cardiology, pp. 635-638, 1992.
[21] J.N. Tsitsiklis, “Efficient Algorithms for Globally Optimal Trajectories,” IEEE Trans. Automatic Control, vol. 50, no. 9, pp. 1528-1538, 1995.
[22] S. Angenent, “Parabolic Equations for Curves on Surfaces. II. Intersections, Blow-Up and Generalized Solutions,” Ann. Math. (2), vol. 133, no. 1, pp. 171-215, 1991.
[23] J.A. Oaks, “Singularities and Self-Intersections of Curves Evolving on Surfaces,” Indiana Univ. Math. J., vol. 43, no. 3, pp. 959-981, 1994.
[24] M. DoCarmo, Riemannian Geometry. Springer, 2003.
[25] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions. Springer, 2003.
[26] H. Soner, “Dynamic Programming and Viscosity Solutions,” Proc. Ann. AMS Meeting, 1999.
[27] J. Sethian and A. Vladimirsky, “Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Applications,” SIAM J. Numerical Analysis, vol. 41, no. 1, pp. 325-363, 2003.
[28] C. Kao, S. Osher, and Y. Tsai, “Fast Sweeping Methods for Static Hamilton-Jacobi Equations,” Technical Report 03-75, CAM, Univ. of California, Los Angeles, 2003.
[29] C. Kao, S. Osher, and J. Qian, “Lax-Friedrichs Sweeping Scheme for Static Hamilton-Jacobi Equations,” J. Computational Physics, vol. 196, no. 1, pp. 367-391, May 2004.
[30] M. Gage and R. Hamilton, “The Heat Equation Shrinking Convex Plane Curves,” J. Differential Geometry, vol. 23, pp. 69-96, 1986.
[31] M. Grayson, “The Heat Equation Shrinks Embedded Plane Curves to Round Points,” J. Differential Geometry, vol. 26, pp. 285-314, 1987.
[32] P. Basser, J. Mattiello, and D. LeBihan, “MR Diffusion Tensor Spectroscopy and Imaging,” Biophysical J., vol. 66, pp. 259-267, 1994.
[33] S. Mori, B. Crain, V. Chacko, and P. van Zijl, “Three-Dimensional Tracking of Axonal Projections in the Brain by Magnetic Resonance Imaging,” Annals of Neurology, vol. 45, no. 2, pp. 265-269, Feb. 1999.
[34] T. Conturo, N. Lori, T. Cull, E. Akbudak, A. Snyder, J. Shimony, R. McKinstry, H. Burton, and M. Raichle, “Tracking Neuronal Fiber Pathways in the Living Human Brain,” Proc. Nat'l Academy of Sciences, pp. 10422-10427, Aug. 1999.
[35] C.-F. Westin, S.E. Maier, B. Khidhir, P. Everett, F.A. Jolesz, and R. Kikinis, “Image Processing for Diffusion Tensor Magnetic Resonance Imaging,” Proc. Int'l Conf. Medical Image Computing and Computer-Assisted Intervention, pp. 441-452, 1999.
[36] P. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi, “In Vivo Fiber Tractography Using DT-MRI Data,” Magnetic Resonance in Medicine, vol. 44, pp. 625-632, 2000.
[37] P. Hagmann, T.G. Reese, W.-Y.I. Tseng, R. Meuli, J.-P. Thiran, and V.J. Wedeen, “Diffusion Spectrum Imaging Tractography in Complex Cerebral White Matter: An Investigation of the Centrum Semiovale,” Proc. Int'l Soc. Magnetic Resonance in Medicine, 2004.
[38] J.S. Campbell, “Diffusion Imaging of White Matter Fibre Tracts,” PhD dissertation, McGill Univ., 2004.
[39] L.C. Young, Calculus of Variations and Optimal Control Theory. W.B.Saunders, 1969.
[40] L. Ibanez, W. Schroeder, L. Ng, and J. Cates, “The ITK Software Guide. Kitware,” technical report, http://www.itk.orgItkSoft wareGuide.pdf, 2003.
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