The Community for Technology Leaders
RSS Icon
Issue No.01 - January/February (2010 vol.16)
pp: 109-119
Thomas Schultz , MPI Informatik, Saarbruecken
Holger Theisel , University of Magdeburg, Magdeburg
Hans-Peter Seidel , MPI Informatik, Saarbruecken
Crease surfaces are two-dimensional manifolds along which a scalar field assumes a local maximum (ridge) or a local minimum (valley) in a constrained space. Unlike isosurfaces, they are able to capture extremal structures in the data. Creases have a long tradition in image processing and computer vision, and have recently become a popular tool for visualization. When extracting crease surfaces, degeneracies of the Hessian (i.e., lines along which two eigenvalues are equal) have so far been ignored. We show that these loci, however, have two important consequences for the topology of crease surfaces: First, creases are bounded not only by a side constraint on eigenvalue sign, but also by Hessian degeneracies. Second, crease surfaces are not, in general, orientable. We describe an efficient algorithm for the extraction of crease surfaces which takes these insights into account and demonstrate that it produces more accurate results than previous approaches. Finally, we show that diffusion tensor magnetic resonance imaging (DT-MRI) stream surfaces, which were previously used for the analysis of planar regions in diffusion tensor MRI data, are mathematically ill-defined. As an example application of our method, creases in a measure of planarity are presented as a viable substitute.
Height crease, ridge surface, valley surface, tensor topology, DT-MRI stream surface.
Thomas Schultz, Holger Theisel, Hans-Peter Seidel, "Crease Surfaces: From Theory to Extraction and Application to Diffusion Tensor MRI", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 1, pp. 109-119, January/February 2010, doi:10.1109/TVCG.2009.44
[1] L. Armijo, “Minimization of Functions Having Lipschitz Continuous First Partial Derivatives,” Pacific J. Math., vol. 16, no. 1, pp. 1-3, 1966.
[2] P.J. Basser, J. Mattiello, and D.L. Bihan, “Estimation of the Effective Self-Diffusion Tensor from the NMR Spin Echo.” J.Magnetic Resonance, vol. B, no. 103, pp. 247-254, 1994.
[3] A.G. Belyaev, E.V. Anoshkina, and T.L. Kunii, “Ridges, Ravines and Singularities,” Topological Modeling for Visualization, A.T.Fomenko and T.L. Kunii, eds., chapter 18, Springer, 1997.
[4] H. Blum and R.N. Nagel, “Shape Description Using Weighted Symmetric Axis Features,” Pattern Recognition, vol. 10, pp. 167-180, 1978.
[5] J.Y. Chang, K.M. Lee, and S.U. Lee, “Multiview Normal Field Integration Using Level Set Methods,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), pp. 1-8, 2007.
[6] J. Damon, “Generic Structure of Two-Dimensional Images under Gaussian Blurring,” SIAM J. Applied Math., vol. 59, no. 1, pp. 97-138, 1998.
[7] T. Delmarcelle and L. Hesselink, “The Topology of Symmetric, Second-Order Tensor Fields,” Proc. IEEE Conf. Visualization, R.D.Bergeron and A.E. Kaufman, eds., pp. 140-147, 1994.
[8] D. Eberly, Ridges in Image and Data Analysis, Computational Imaging and Vision, vol. 7. Kluwer Academic Publishers. 1996.
[9] D. Eberly, R. Gardner, B. Morse, S. Pizer, and C. Scharlach, “Ridges for Image Analysis,” J. Math. Imaging and Vision, vol. 4, pp. 353-373, 1994.
[10] R.T. Frankot and R. Chellappa, “A Method for Enforcing Integrability in Shape from Shading Algorithms,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 439-451, July 1988.
[11] J.D. Furst and S.M. Pizer, “Marching Ridges,” Proc. Int'l Assoc. Science and Technology for Development (IASTED) Int'l Conf. Signal and Image Processing, pp. 22-26, 2001.
[12] J.D. Furst, S.M. Pizer, and D.H. Eberly, “Marching Cores: A Method for Extracting Cores from 3D Medical Images,” Proc. Workshop Math. Methods in Biomedical Image Analysis, pp. 124-130, 1996.
[13] R.M. Haralick, “Ridges and Valleys on Digital Images,” Computer Vision, Graphics, and Image Processing, vol. 22, pp. 28-38, 1983.
[14] B. Jeremić, G. Scheuermann, J. Frey, Z. Yang, B. Hamann, K.I. Joy, and H. Hagen, “Tensor Visualizations in Computational Geomechanics,” Int'l J. Numerical and Analytical Methods in Geomechanics, vol. 26, pp. 925-944, 2002.
[15] G. Kindlmann, “Superquadric Tensor Glyphs,” Proc. Eurographics/IEEE Symp. Visualization (SymVis), pp. 147-154, 2004.
[16] G. Kindlmann, X. Tricoche, and C.-F. Westin, “Anisotropy Creases Delineate White Matter Structure in Diffusion Tensor MRI,” R.Larsen, M. Nielsen, and J. Sporring, eds., Proc. Ninth Int'l Conf. Medical Image Computing and Computer-Assisted Intervention (MICCAI), pp. 126-133, 2006.
[17] G. Kindlmann, X. Tricoche, and C.-F. Westin, “Delineating White Matter Structure in Diffusion Tensor MRI with Anisotropy Creases,” Medical Image Analysis, vol. 11, no. 5, pp. 492-502, 2007.
[18] G. Kindlmann and C.-F. Westin, “Diffusion Tensor Visualization with Glyph Packing,” IEEE Trans. Visualization and Computer Graphics, vol. 12, no. 5, pp. 1329-1335, Sept./Oct. 2006.
[19] J.J. Koenderink and A.J. van Doorn, “Local Features of Smooth Shapes: Ridges and Courses,” Geometric Methods in Computer Vision II, B.C. Vemuri, ed., pp. 2-13, 1993.
[20] T. Lindeberg, “Edge Detection and Ridge Detection with Automatic Scale Selection,” Int'l J. Computer Vision, vol. 30, no. 2, pp.117-154, 1998.
[21] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Proc. ACM SIGGRAPH '87, pp. 163-169, 1987.
[22] J.R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, rev. ed. Wiley, 1998.
[23] H. Miura and S. Kida, “Identification of Tubular Vortices in Turbulence,” J. Physical Soc. Japan, vol. 66, no. 5, pp. 1331-1334, 1997.
[24] S. Mori, B.J. Crain, V.P. Chacko, and P.C.M. van Zijl, “Three-Dimensional Tracking of Axonal Projections in the Brain by Magnetic Resonance Imaging,” Ann. Neurology, vol. 45, no. 2, pp.265-269, 1999.
[25] B.S. Morse, Computation of Object Cores from Grey-Level Images, PhD thesis, Univ. of North Carolina at Chapel Hill, 1994.
[26] R. Peikert and M. Roth, “The ‘Parallel Vectors’ Operator—A Vector Field Visualization Primitive,” Proc. IEEE Visualization Conf. '99, D.S. Ebert, M. Gross, and B. Hamann, eds., pp. 263-270, 1999.
[27] R. Peikert and F. Sadlo, “Height Ridge Computation and Filtering for Visualization,” Proc. IEEE Pacific Visualization 2008, I. Fujishiro, H. Li, and K.-L. Ma, eds., pp. 119-126, 2008.
[28] B.T. Phong, “Illumination for Computer Generated Pictures,” Comm. ACM, vol. 18, no. 6, pp. 311-317, 1975.
[29] S.M. Pizer, C.A. Burbeck, J.M. Coggins, D.S. Fritsch, and B.S. Morse, “Object Shape Before Boundary Shape: Scale-Space Medial Axes,” J. Math. Imaging and Vision vol. 4, pp. 303-313, 1994.
[30] B. Preim and D. Bartz, Visualization in Medicine. Theory, Algorithms, and Applications. Morgan Kaufmann, 2007.
[31] R.J. Rost, OpenGL Shading Manual, second ed. Addison-Wesley, 2006.
[32] F. Sadlo and R. Peikert, “Efficient Visualization of Lagrangian Coherent Structures by Filtered AMR Ridge Extraction,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 6, pp. 1456-1463, Nov. 2007.
[33] J. Sahner, T. Weinkauf, and H.-C. Hege, “Galilean Invariant Extraction and Iconic Representation of Vortex Core Lines,” Proc. Eurographics/IEEE Visualization and Graphics Technical Committee (VGTC) Symp. Visualization (EuroVis), K. Brodlie, D. Duke, and K.Joy, eds., pp. 151-160, 2005.
[34] J. Sahner, T. Weinkauf, N. Teuber, and H.-C. Hege, “Vortex and Strain Skeletons in Eulerian and Lagrangian Frames,” IEEE Trans. Visualization and Computer Graphics, vol. 13, no. 5, pp. 980-990, Sept./Oct. 2007.
[35] T. Schultz and H.-P. Seidel, “Using Eigenvalue Derivatives for Edge Detection in DT-MRI Data,” Pattern Recognition, G. Rigoll, ed., pp. 193-202, 2008.
[36] R. Sondershaus and S. Gumhold, “Meshing of Diffusion Surfaces for Point-Based Tensor Field Visualization,” Proc. 12th Int'l Meshing Roundtable (IMR), pp. 177-188, 2003.
[37] J. Süßmuth and G. Greiner, “Ridge Based Curve and Surface Reconstruction,” Proc. Eurographics Symp. Geometry Processing, A. Belyaev and M. Garland, eds., pp. 243-251, 2007.
[38] G. Taubin, “Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation,” Proc. Fifth Int'l Conf. Computer Vision (ICCV), pp. 902-907, 1995.
[39] H. Theisel and H.-P. Seidel, “Feature Flow Fields,” Proc. Eurographics/IEEE-TCVG Symp. Visualization (VisSym), G.-P. Bonneau, S. Hahmann, and C.D. Hansen, eds., pp. 141-148, 2003.
[40] X. Tricoche, G. Kindlmann, and C.-F. Westin, “Invariant Crease Lines for Topological and Structural Analysis of Tensor Fields,” IEEE Trans. Visualization and Computer Graphics, vol. 14, no. 6, pp.1627-1634, Nov./Dec. 2008.
[41] A. Vilanova, G. Berenschot, and C. van Pul, “DTI Visualization with Stream Surfaces and Evenly-Spaced Volume Seeding,” Proc. Joint Eurographics-IEEE Technical Committee on Visualization and Graphics (TCVG) Symp. Visualization (VisSym), O. Deussen, C.Hansen, D.A. Keim, and D. Saupe, eds., pp. 173-182, 2004.
[42] A. Vilanova, S. Zhang, G. Kindlmann, and D.H. Laidlaw, “An Introduction to Visualization of Diffusion Tensor Imaging and Its Applications,” Visualization and Processing of Tensor Fields, J.Weickert and H. Hagen, eds., pp. 121-153, Springer, 2006.
[43] C.-F. Westin, S. Peled, H. Gudbjartsson, R. Kikinis, and F.A. Jolesz, “Geometrical Diffusion Measures for MRI from Tensor Basis Analysis,” Proc. Int'l Soc. Magnetic Resonance in Medicine, p. 1742, 1997.
[44] S. Zhang, C. Demiralp, and D.H. Laidlaw, “Visualizing Diffusion Tensor MR Images Using Streamtubes and Streamsurfaces,” IEEE Trans. Visualization and Computer Graphics, vol. 9, no. 4, pp. 454-462, Oct.-Dec. 2003.
[45] S. Zhang, D.H. Laidlaw, and G. Kindlmann, “Diffusion Tensor MRI Visualization,” The Visualization Handbook, C.D. Hansen and C.R. Johnson, eds., pp. 327-340, Elsevier, 2005.
[46] X. Zheng and A. Pang, “Topological Lines in 3D Tensor Fields,” Proc. IEEE Visualization, pp. 313-320, 2004.
[47] X. Zheng, B. Parlett, and A. Pang, “Topological Structures of 3D Tensor Fields,” Proc. IEEE Visualization Conf., pp. 551-558, 2005.
23 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool