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Anisotropic Curvature Motion for Structure Enhancing Smoothing of 3D MR Angiography Data

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Abstract

We propose a novel concept of shape prior for the processing of tubular structures in 3D images. It is based on the notion of an anisotropic area energy and the corresponding geometric gradient flow. The anisotropic area functional incorporates a locally adapted template as a shape prior for tubular vessel structures consisting of elongated, ellipsoidal shape models. The gradient flow for this functional leads to an anisotropic curvature motion model, where the evolution is driven locally in direction of the considered template. The problem is formulated in a level set framework, and a stable and robust method for the identification of the local prior is presented. The resulting algorithm is able to smooth the vessels, pushing solution toward elongated cylinders with round cross sections, while bridging gaps in the underlying raw data. The implementation includes a finite-element scheme for numerical accuracy and a narrow band strategy for computational efficiency.

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References

  1. D. Adalsteinsson and J.A. Sethian, “A fast level set method for propagating interfaces,” Journal of Computational Physics, Vol. 118, No. 2, pp. 269–277, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Ambrosio and H.M. Soner, “Level set approach to mean curvature flow in arbitrary codimension,” J. Diff. Geom., Vol. 43, pp. 693–737, 1996.

    MATH  MathSciNet  Google Scholar 

  3. B. Andrews, “Volume-preserving anisotropic mean curvature flow,” Indiana University Mathematics Journal, Vol. 50, pp. 783–827, 1991.

    Google Scholar 

  4. E. Bullit, A. Aylward, A. Liu, S. Mukherji, J. Stone, C. Coffey, G. Gerig, and S.M. Pizer, “3d graph description of the intracerebral vasculature from segmented mra and test of accuracy by comparison with X-ray angiograms,” in: Information Processing in Medical Imaging (IPMI), 1999 pp. 308–321.

  5. A. Chung and J. Noble, “Statistical 3d vessel segmentation using a rician distribution,” in Proc. Medical Image Conference and Computer Assisted Interventions (MICCAI), 1999, pp. 82–89.

  6. U. Clarenz, G. Dziuk, and M. Rumpf, “On generalized mean curvature flow in surface processing,” in Geometric Analysis and Nonlinear Partial Differential Equations, H. Karcher, S. Hildebrandt (Eds.), Springer, 2003, pp. 217–248.

  7. U. Clarenz, M. Rumpf, and A. Telea, “Robust feature detection and local classification for surfaces based on moment analysis,” IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 5, pp. 516–524, 2004.

    Article  Google Scholar 

  8. K. Deckelnick and G. Dziuk, “A fully discrete numerical scheme for weighted mean curvature flow,” Numerische Mathematik, Vol. 91, No. 3, pp. 423–452, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Droske and M. Rumpf, “A level set formulation for willmore flow,” Interfaces and Free Boundaries, Vol. 6, No. 3, pp. 361–378, 2004.

    MATH  MathSciNet  Google Scholar 

  10. L.C. Evans and J. Spruck, “Motion of level sets by mean curvature I,” J. Diff. Geom., Vol. 33, No. 3, pp. 635–681, 1991.

    MATH  MathSciNet  Google Scholar 

  11. M.T. Figueiredo and J.M.N. Leitao, “A nonsmoothing approach to the estimation of vessel contours in angiograms,” IEEE Trans. on Medical Imaging, Vol. 14, pp. 162 – 172, 1995.

    Article  Google Scholar 

  12. A. Frangi, W.J. Niessen, K.L. Vincken, and M.A. Viergever, Vessel enhancement filterin,” in Proc. Medical Image Conference and Computer Assisted Interventions (MICCAI), 1998, pp. 130–137.

  13. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” Int. J. Computer Vis., Vol. 1, No. 4, pp. 321–331, 1988.

    Article  Google Scholar 

  14. A.K. Klein, F. Lee, and A.A. Amini, “Quantitative coronary angiography with deformable spline models,” IEEE Trans. on Medical Imaging, Vol. 16, pp. 468–482, 1997.

    Article  Google Scholar 

  15. K. Krissian, “Flux-based anisotropic diffusion applied to enhancement of 3d angiograms,” IEEE Trans. on Medical Imaging, Vol. 21, pp. 1440–1442, 2002.

    Article  Google Scholar 

  16. K. Krissian, G. Malandain, and N. Ayache, “Directional anisotropic diffusion applied to segmentation of vessels in 3d images,” in Proc. Int’l Conf. Scale-Space, 1997, pp. 345–348.

  17. K. Krissian, G. Malandain, N. Ayache, R. Vaillant, and Y. Trousset, “Model based detection of tubular structures in 3d images,” Computer Vision and Image Understanding, Vol. 80, pp. 130–171, 2000.

    Article  MATH  Google Scholar 

  18. L. Lorigo, O. Faugeras, W. Grimson, R. Keriven, R. Kikinis, A. Nabavi, and C. Westin, “Codimension-two geodesic active contours for the segmentation of tubular structures,” in CVPR’2000, CVPR, 2000, pp 444–451.

  19. T. Preußer and M. Rumpf, “A level set method for anisotropic geometric diffusion in 3D image processing,” SIAM J. Appl. Math., Vol. 62, No. 5, pp. 1772–1793, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  20. V. Prinet, O. Monga, C. Ge, L. Sheng, and S. Ma, “Thin network extraction in 3d images: Application to medical angiograms,” in: Int. Conf. on Pattern Recognition, 1996 pp. 386–390.

  21. Y. Sato, S. Nakajima, N. Shiraga, H. Atsumi, S. Yoshida, T. Koller, G. Gerig, and R. Kikinis, “3d multi-scale line filter for segmentation and visualization of curvilinear structures in medical images,” IEEE Med. Image Anal., Vol. 2, pp. 143–168, 1998.

    Article  Google Scholar 

  22. J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.

  23. R. Whitaker, “Volumetric deformable models: active blobs,” in: Visualization in Biomedical Computing, 1994 pp 122–134.

  24. G. Wulff, “Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Kristallflächen,” Zeitschrift der Kristallographie, Vol. 34, pp. 449–530, 1901.

    Google Scholar 

  25. Y. Sun, R.J. Lucariello, and S.A. Chiaramida, “Directionsal low-pass filtering for improved accuracy and reproducibility of stenosis quantification in coronary arteriograms,” IEEE Trans. Med. Imaging, Vol. 14, pp. 242–248, 1995.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Nussallee 15, 53115, Bonn, Germany

    Oliver Nemitz & Martin Rumpf

  2. 3450 Merrill Engineering Building, School of Computing, University of Utah, 50 S. Central Campus Drive, Salt Lake City, UT, 84112

    Tolga Tasdizen & Ross Whitaker

Authors
  1. Oliver Nemitz
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  2. Martin Rumpf
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  3. Tolga Tasdizen
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  4. Ross Whitaker
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Correspondence to Oliver Nemitz.

Additional information

Oliver Nemitz received his Diploma in mathematics from the university of Duisburg, Germany in 2003. Then he started to work on his Ph.D. thesis in Duisburg. Since 2005 he is continuing the work on his Ph.D. project at the Institute for Numerical Simulation at Bonn University. His Ph.D. subject is fast algorithms for image manipulation in 3d, using PDE’s, variational methods, and level set methods.

Martin Rumpf received his Ph.D. in mathematics from Bonn University in 1992. He held a postdoctoral research position at Freiburg University. Between 1996 and 2001, he was an associate professor at Bonn University and from 2001 until 2004 full professor at Duisburg University. Since 2004 he is now full professor for numerical mathematics and scientific computing at Bonn University. His research interests are in numerical methods for nonlinear partial differential equations, geometric evolution problems, calculus of variations, adaptive finite element methods, image and surface processing.

Tolga Tasdizen received his B.S. degree in Electrical Engineering from Bogazici University, Istanbul in 1995. He received the M.S. and Ph.D. degrees in Engineering from Brown University in 1997 and 2001. From 2001 to 2004 he was a postdoctoral research associate with the Scientific Computing and Imaging Institute at the University of Utah. Since 2004 he has been with the School of Computing at the University of Utah as a research assistant professor. He also holds an adjunct assistant professor position with the Department of Neurology and the Center for Alzheimer’s Care, Imaging and Research, and a research scientist position with the Scientific Computing and Imaging Institute at the University of Utah.

Ross Whitaker received his B.S. degree in Electrical Engineering and Computer Science from Princeton University in 1986, earning Summa Cum Laude. From 1986 to 1988 he worked for the Boston Consulting Group, entering the University of North Carolina at Chapel Hill in 1989. At UNC he received the Alumni Scholarship Award, and completed his Ph.D. in Computer Science in 1994. From 1994–1996 he worked at the European Computer-Industry Research Centre in Munich Germany as a research scientist in the User Interaction and Visualization Group. From 1996–2000 he was an Assistant Professor in the Department of Electrical Engineering at the University of Tennessee. He is now an Associate Professor at the University of Utah in the College of Computing and the Scientific Computing and Imaging Institute.

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Nemitz, O., Rumpf, M., Tasdizen, T. et al. Anisotropic Curvature Motion for Structure Enhancing Smoothing of 3D MR Angiography Data. J Math Imaging Vis 27, 217–229 (2007). https://doi.org/10.1007/s10851-006-0645-2

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