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Fast Evolution of Image Manifolds and Application to Filtering and Segmentation in 3D Medical Images
September/October 2004 (vol. 10 no. 5)
pp. 525-535
In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS) [1], [2] for Beltrami and the subjective surface computation. The Beltrami flow [3], [4], [5] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [6] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition” the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.

[1] J. Weickert, B.M. ter Haar Romeny, and M. Viergener, Efficient and Reliable Scheme for Non-Linear Diffusion and Filtering IEEE Trans. Image Processing, vol. 7, pp. 398-410, 1998.
[2] R. Goldenberg, R. Kimmel, E. Rivlin, and M. Rudzsky, Fast Geodesic Active Contours IEEE Trans. Image Processing, vol. 10, pp. 1467-1475, 2001.
[3] R. Kimmel, N. Sochen, and R. Malladi, From High Energy Physics to Low Level Vision Proc. Scale Space Theories in Computer Vision, July 1997.
[4] N. Sochen, R. Kimmel, and R. Malladi, “A Geometrical Framework for Low Level Vision,” IEEE Trans. Image Processing, vol. 7, no. 3, pp. 310-318, 1998.
[5] R. Kimmel, R. Malladi, and N. Sochen, Images as Embedded Maps and Minimal Surfaces: Movies, Color, Texture, and Volumetric Medical Images Int'l J. Computer Vision, 1999.
[6] A. Sarti, R. Malladi, and J.A. Sethian, Subjective Surfaces: A Method for Completing Missing Boundaries Int'l J. Computer Vision, vol. 46, no. 3, pp. 201-221, 2002.
[7] P. Perona and J. Malik, "Scale-Space and Edge Detection Using Anisotropic Diffusion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629639, July 1990.
[8] S. Osher and J.A. Sethian, Fronts Propagating with Curvature Dependent Speed: Algorithms Based on the Hamilton-Jacobi Formulation J. Computational Physics, vol. 79, pp. 12-49, 1988.
[9] L. Alvarez, P.L. Lions, and J.M. Morel, Image Selective Smoothing and Edge Detection by Non-Linear Diffusion II SIAM J. Numerical Analysis, vol. 29, no. 3, pp. 845-866, 1992.
[10] L. Rudin, S. Osher, and E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms Physica D, vol. 60, pp. 259-268, 1992.
[11] R. Malladi and J.A. Sethian, Image Processing via Level Set Curvature Flow Proc. Nat'l Academy of Sciences of the USA, vol. 92, pp. 7046-7050, July 1995.
[12] R. Malladi and J.A. Sethian, Image Processing: Flows under Min/Max Curvature and Mean Curvature Graphical Models and Image Processing, vol. 58, no. 2, pp. 127-141, 1996.
[13] Geometry-Driven Diffusion in Computer Vision, B.M. ter Haar Romeny, ed. Kluwer Academic, 1994.
[14] A. Yezzi, Modified Curvature Motion for Image Smoothing and Enhancement IEEE Trans. Image Processing, vol. 7, no. 3, 1998.
[15] R. Kimmel and N. Sochen, Orientation Diffusion or How to Comb a Porcupine? J. Visual Comm. and Image Representation, 2002.
[16] B. Tang, G. Sapiro, and V. Caselles, Diffusion of General Data on Non-Flat Manifolds via Harmonic Map Theory Int'l J. Computer Vision, pp. 149-161, 2000.
[17] T. Chan and J. Shen, Variational Restoration of Non-Flat Image Features: Models and Algorithms SIAM J. Applied Math., vol. 61, no. 4, pp. 1338-1361, 2001.
[18] J. Weickert, Anisotropic Diffusion in Image Processing PhD thesis, Kaiserslautern Univ., Germany, 1996.
[19] V. Caselles, R. Kimmel, and G. Sapiro, Geodesic Active Contours Int'l J. Computer Vision, vol. 22, no. 1, pp. 61-79, 1997.
[20] G.I. Barenblatt, Self-Similar Intermediate Asymptotics for NonLinear Degenerate Parabolic Free-Boundary Problems which Occur in Image Processing Proc. Nat'l Academy of Sciences of the USA, vol. 98, no. 23, pp. 12878-12881, Nov. 2001.
[21] R. Malladi and I. Ravve, Fast Difference Schemes for Edge-Enhancing Beltrami Flow Proc. Seventh European Conf. Computer Vision, May 2002.
[22] A.M. Polyakov, Quantum Geometry of Bosonic Strings Physics Letters B, vol. 130B, no. 3, pp. 207-210, 1981.
[23] C.R. Vogel and M.E. Oxman, Fast, Robust Total Variation-Based Reconstruction of Noisy, Blurred Images IEEE Trans. Image Processing, vol. 7, pp. 813-824, 1998.
[24] R. Malladi and I. Ravve, Fast Difference Scheme for Anisotropic Beltrami Smoothing and Edge Contrast Enhancement of Gray Level and Color Images LBNL Report 48796, Lawrence Berkeley Nat'l Laboratory, Univ. of California, Berkeley, Aug. 2001.
[25] E.S.S. Nussbaum, Brain Aneurysms and Vascular Malformations. Xlibris Corp., 2000.
[26] J.M. Kniss, G.L. Kindlmann, and C. Hansen, Multidimensional Transfer Functions for Interactive Volume Rendering IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 3, pp. 270-285, July-Sept. 2002.
[27] R.J. LeVeque, Numerical Methods for Conservation Laws. Birkhauser, 1992.
[28] I. Ravve and R. Malladi, Fast Methods for Beltrami Flow and Subjective Surfaces Proc. Int'l Workshop Visualization and Math. (VisMath '02), May 2002.
[29] R. Kimmel, 3D Shape Reconstruction from Autostereograms and Stereo J. Visual Comm. and Image Representation, vol. 13, pp. 324-333, Mar. 2002.
[30] J.A. Sethian, A Fast Marching Level Set Method for Monotonically Advancing Fronts Proc. Natural Academy of Sciences, vol. 93, no. 4, pp. 1591-1595, Feb. 1996.
[31] R. Malladi and J.A. Sethian, A Real-Time Algorithm for Medical Shape Recovery Proc. IEEE Int'l Conf. Computer Vision, pp. 304-310, Jan. 1998.
[32] J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, second ed. Cambridge Univ. Press, Univ. of California, Berkeley, 1999.
[33] T. Deschamps, Curve and Shape Extraction with Minimal Path and Level-Sets Technique Applications to 3D Medical Imaging PhD thesis, UniversitéParis-IX Dauphine, Dec. 2001.

Index Terms:
Beltrami flow, subjective surfaces, unconditionally stable scheme, segmentation, Eikonal equation, fast-marching, volume visualization.
Thomas Deschamps, Ravi Malladi, Igor Ravve, "Fast Evolution of Image Manifolds and Application to Filtering and Segmentation in 3D Medical Images," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 5, pp. 525-535, Sept.-Oct. 2004, doi:10.1109/TVCG.2004.26
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