International Journal of Computer Vision

, Volume 92, Issue 2, pp 146–161 | Cite as

Adaptive Continuous-Scale Morphology for Matrix Fields

  • Bernhard Burgeth
  • Luis Pizarro
  • Michael Breuß
  • Joachim Weickert
Article
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Accepted:

Abstract

In this article we consider adaptive, PDE-driven morphological operations for 3D matrix fields arising e.g. in diffusion tensor magnetic resonance imaging (DT-MRI). The anisotropic evolution is steered by a matrix constructed from a structure tensor for matrix valued data. An important novelty is an intrinsically one-dimensional directional variant of the matrix-valued upwind schemes such as the Rouy-Tourin scheme. It enables our method to complete or enhance anisotropic structures effectively. A special advantage of our approach is that upwind schemes are utilised only in their basic one-dimensional version, hence avoiding grid effects and leading to an accurate algorithm. No higher dimensional variants of the schemes themselves are required. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.

Mathematical morphology PDEs DT-MRI Tensor field Dilation Erosion 
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References

  1. Alvarez, L., & Mazorra, L. (1994). Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal on Numerical Analysis, 31, 590–605. MathSciNetMATHCrossRefGoogle Scholar
  2. Alvarez, L., Guichard, F., Lions, P.-L., & Morel, J.-M. (1993). Axioms and fundamental equations in image processing. Archive for Rational Mechanics and Analysis, 123, 199–257. MathSciNetMATHCrossRefGoogle Scholar
  3. Arehart, A. B., Vincent, L., & Kimia, B. B. (1993). Mathematical morphology: The Hamilton–Jacobi connection. In Proc. fourth international conference on computer vision (pp. 215–219). Berlin, May 1993. New York: IEEE Computer Society Press. Google Scholar
  4. Basser, P. J., Mattiello, J., & LeBihan, D. (1994). MR diffusion tensor spectroscopy and imaging. Biophysical Journal, 66, 259–267. CrossRefGoogle Scholar
  5. Bigün, J. (2006). Vision with direction. Berlin: Springer. Google Scholar
  6. Bigün, J., Granlund, G. H., & Wiklund, J. (1991). Multidimensional orientation estimation with applications to texture analysis and optical flow. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(8), 775–790. CrossRefGoogle Scholar
  7. Boris, J. P., & Book, D. L. (1973). Flux corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics, 11(1), 38–69. CrossRefGoogle Scholar
  8. Boris, J. P., & Book, D. L. (1976). Flux corrected transport. III. Minimal error FCT algorithms. Journal of Computational Physics, 20, 397–431. CrossRefGoogle Scholar
  9. Boris, J. P., Book, D. L., & Hain, K. (1975). Flux corrected transport. II. Generalizations of the method. Journal of Computational Physics, 18, 248–283. CrossRefGoogle Scholar
  10. Breuß, M., & Weickert, J. (2006). A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision, 25(2), 187–201. MathSciNetCrossRefGoogle Scholar
  11. Breuß, M., Burgeth, B., & Weickert, J. (2007). Anisotropic continuous-scale morphology. In J. Martí, J. M. Benedí, A. M. Mendonça, & J. Serrat (Eds.), Lecture notes in computer science : Vol. 4478. Pattern recognition and image analysis (pp. 515–522). Berlin: Springer. CrossRefGoogle Scholar
  12. Brockett, R. W., & Maragos, P. (1994). Evolution equations for continuous-scale morphological filtering. IEEE Transactions on Signal Processing, 42, 3377–3386. CrossRefGoogle Scholar
  13. Brox, T., Weickert, J., Burgeth, B., & Mrázek, P. (2006). Nonlinear structure tensors. Image and Vision Computing, 24(1), 41–55. CrossRefGoogle Scholar
  14. Burgeth, B., Bruhn, A., Didas, S., Weickert, J., & Welk, M. (2007a). Morphology for matrix-data: Ordering versus PDE-based approach. Image and Vision Computing, 25(4), 496–511. CrossRefGoogle Scholar
  15. Burgeth, B., Bruhn, A., Papenberg, N., Welk, M., & Weickert, J. (2007b). Mathematical morphology for tensor data induced by the Loewner ordering in higher dimensions. Signal Processing, 87(2), 277–290. MATHCrossRefGoogle Scholar
  16. Burgeth, B., Didas, S., Florack, L., & Weickert, J. (2007c). A generic approach to diffusion filtering of matrix-fields. Computing, 81, 179–197. MathSciNetMATHCrossRefGoogle Scholar
  17. Burgeth, B., Breuß, M., Didas, S., & Weickert, J. (2009a). PDE-based morphology for matrix fields: Numerical solution schemes. In S. Aja-Fernández, R. de Luis García, D. Tao, & X. Li (Eds.), Advances in pattern recognition. Tensors in image processing and computer vision (pp. 125–150). London: Springer. CrossRefGoogle Scholar
  18. Burgeth, B., Breuß, M., Pizarro, L., & Weickert, J. (2009b). PDE-driven adaptive morphology for matrix fields. In X.-C. Tai et al. (Eds.), Lecture notes in computer science : Vol. 5567. Proc. of the second international conference on scale space and variational methods in computer vision (pp. 247–258). Berlin: Springer. CrossRefGoogle Scholar
  19. Burgeth, B., Didas, S., & Weickert, J. (2009c). A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In D. Laidlaw & J. Weickert (Eds.), Mathematics and visualization. Visualization and processing of tensor fields (pp. 305–323). Berlin: Springer. CrossRefGoogle Scholar
  20. Chefd’Hotel, C., Tschumperlé, D., Deriche, R., & Faugeras, O. (2002). Constrained flows of matrix-valued functions: Application to diffusion tensor regularization. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science : Vol. 2350. Computer vision—ECCV 2002 (pp. 251–265). Berlin: Springer. CrossRefGoogle Scholar
  21. Di Zenzo, S. (1986). A note on the gradient of a multi-image. Computer Vision, Graphics and Image Processing, 33, 116–125. CrossRefGoogle Scholar
  22. Feddern, C., Weickert, J., Burgeth, B., & Welk, M. (2006). Curvature-driven PDE methods for matrix-valued images. International Journal of Computer Vision, 69(1), 91–103. CrossRefGoogle Scholar
  23. Förstner, W., & Gülch, E. (1987). A fast operator for detection and precise location of distinct points, corners and centres of circular features. In Proc. ISPRS intercommission conference on fast processing of photogrammetric data (pp. 281–305). Interlaken, Switzerland, June 1987. Google Scholar
  24. Gilboa, G., Sochen, N. A., & Zeevi, Y. Y. (2002). Regularized shock filters and complex diffusion. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science : Vol. 2350. Computer vision—ECCV 2002 (pp. 399–413). Berlin: Springer. CrossRefGoogle Scholar
  25. Guichard, F., & Morel, J.-M. (2003). A note on two classical enhancement filters and their associated PDE’s. International Journal of Computer Vision, 52(2/3), 153–160. CrossRefGoogle Scholar
  26. Horn, R. A., & Johnson, C. R. (1990). Matrix analysis. Cambridge: Cambridge University Press. MATHGoogle Scholar
  27. Kramer, H. P., & Bruckner, J. B. (1975). Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition, 7, 53–58. MathSciNetMATHCrossRefGoogle Scholar
  28. Laidlaw, D., & Weickert, J. (Eds.) (2009). Visualization and processing of tensor fields. Berlin: Springer. MATHGoogle Scholar
  29. Matheron, G. (1967). Eléments pour une théorie des milieux poreux. Paris: Masson. Google Scholar
  30. Matheron, G. (1975). Random sets and integral geometry. New York: Wiley. MATHGoogle Scholar
  31. Osher, S., & Fedkiw, R. P. (2002). Applied mathematical sciences : Vol. 153. Level set methods and dynamic implicit surfaces. New York: Springer. Google Scholar
  32. Osher, S., & Rudin, L. I. (1990). Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis, 27, 919–940. MATHCrossRefGoogle Scholar
  33. Osher, S., & Rudin, L. (1991). Shocks and other nonlinear filtering applied to image processing. In A. G. Tescher (Ed.), Proceedings of SPIE : Vol. 1567. Applications of digital image processing XIV (pp. 414–431). Bellingham: SPIE Press. Google Scholar
  34. Osher, S., & Sethian, J. A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79, 12–49. MathSciNetMATHCrossRefGoogle Scholar
  35. Pizarro, L., Burgeth, B., Breuß, M., & Weickert, J. (2009). A directional Rouy-Tourin scheme for adaptive matrix-valued morphology. In M. H. F. Wilkinson & J. B. T. M. Roerdink (Eds.), Lecture notes in computer science : Vol. 5720. Proc. of the ninth international symposium on mathematical morphology, ISMM (pp. 250–260). Berlin: Springer. Google Scholar
  36. Remaki, L., & Cheriet, M. (2003). Numerical schemes of shock filter models for image enhancement and restoration. Journal of Mathematical Imaging and Vision, 18(2), 153–160. MathSciNetCrossRefGoogle Scholar
  37. Rouy, E., & Tourin, A. (1992). A viscosity solutions approach to shape-from-shading. SIAM Journal on Numerical Analysis, 29, 867–884. MathSciNetMATHCrossRefGoogle Scholar
  38. Sapiro, G., Kimmel, R., Shaked, D., Kimia, B. B., & Bruckstein, A. M. (1993). Implementing continuous-scale morphology via curve evolution. Pattern Recognition, 26, 1363–1372. CrossRefGoogle Scholar
  39. Schavemaker, J. G. M., Reinders, M. J. T., & van den Boomgaard, R. (1997). Image sharpening by morphological filtering. In Proc. 1997 IEEE workshop on nonlinear signal and image processing, Mackinac Island, MI, September 1997. www.ecn.purdue.edu/NSIP/.
  40. Serra, J. (1967). Echantillonnage et estimation des phénomènes de transition minier. PhD thesis, University of Nancy, France. Google Scholar
  41. Serra, J. (1982). Image analysis and mathematical morphology (Vol. 1). London: Academic Press. MATHGoogle Scholar
  42. Serra, J. (1988). Image analysis and mathematical morphology (Vol. 2). London: Academic Press. Google Scholar
  43. Sethian, J. A. (1999). Level set methods and fast marching methods (2nd ed.). Cambridge: Cambridge University Press. Paperback edition. MATHGoogle Scholar
  44. Soille, P. (2003). Morphological image analysis (2nd ed.). Berlin: Springer. MATHGoogle Scholar
  45. van den Boomgaard, R. (1992). Mathematical morphology: Extensions towards computer vision. PhD thesis, University of Amsterdam, The Netherlands. Google Scholar
  46. van den Boomgaard, R. (1999). Numerical solution schemes for continuous-scale morphology. In M. Nielsen, P. Johansen, O. F. Olsen, & J. Weickert (Eds.), Lecture notes in computer science : Vol. 1682. Scale-space theories in computer vision (pp. 199–210). Berlin: Springer. CrossRefGoogle Scholar
  47. van Vliet, L. J., Young, I. T., & Beckers, A. L. D. (1989). A nonlinear Laplace operator as edge detector in noisy images. Computer Vision, Graphics and Image Processing, 45(2), 167–195. CrossRefGoogle Scholar
  48. Weickert, J. (2003). Coherence-enhancing shock filters. In B. Michaelis & G. Krell (Eds.), Lecture notes in computer science : Vol. 2781. Pattern recognition (pp. 1–8). Berlin: Springer. CrossRefGoogle Scholar
  49. Weickert, J., & Brox, T. (2002). Diffusion and regularization of vector- and matrix-valued images. In M. Z. Nashed & O. Scherzer (Eds.), Contemporary mathematics : Vol. 313. Inverse problems, image analysis, and medical imaging (pp. 251–268). Providence: AMS. Google Scholar
  50. Weickert, J., & Hagen, H. (Eds.) (2006). Visualization and processing of tensor fields. Berlin: Springer. MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Bernhard Burgeth
    • 1
  • Luis Pizarro
    • 1
    • 2
  • Michael Breuß
    • 1
  • Joachim Weickert
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.National Heart and Lung Institute, and Department of ComputingImperial College LondonLondonUK

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