The Use of Structure Tensors in the Analysis of Seismic Data

  • Maria Faraklioti
  • Maria Petrou
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 7)


The first and second order structure tensors, simply estimated by differencing the image, can be used to quantify the local structure of seismic data and their departure from laminar structure. They can be used to distinguish chaotic regions as well as regions of interest, like mounds and horizon terminations from stratified regions. They have been well established in the processing of 2D images, but their application to 3D volume data is still a largely unexplored field of research. This chapter reviews the properties of these tensors and their application to image processing in general, and demonstrates their usefulness in the analysis of 2D and 3D seismic data.


Hessian Matrix Corner Point Structure Tensor Order Tensor Diagonal Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    P. Bakker (2002) Image structure analysis for seismic interpretation. PhD thesis, Pattern Recognition Group, Faculty of Applied Physics, Delft University of Technology, Delft, The Netherlands, June 2002.Google Scholar
  2. 2.
    P. Bakker, L.J. van Vliet, and P.W. Verbeek (2001) Confidence and curvature of curvilinear structures in 3D. Proceedings of the Eighth IEEE International Conference on Computer Vision, Vancouver, Canada, July 2001.Google Scholar
  3. 3.
    E. Barth (2000) The minors of the structure tensor. Mustererkennung, 221–228.Google Scholar
  4. 4.
    E. Barth and M. Ferraro (2000) On the geometric structure of spatio-temporal patterns. Proc. Second International Workshop Algebraic Frames for the Perception-Action Cycle (AFPAC), Springer-Verlag, Heidelberg, 134–143.Google Scholar
  5. 5.
    E. Barth, I. Stuke, and C. Mota (2002) Analysis of motion and curvature in image sequences. Proc. Fifth IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI), Santa Fe, New Mexico, April 2002, 206–210.Google Scholar
  6. 6.
    P.J. Basser, J. Mattiello, and D. Le Bihan (1994) Estimation of the effective self-diffusion tensor from the NMR spin echo. Journal of Magnetic Resonance 103, 247–254.Google Scholar
  7. 7.
    P.J. Basser, J. Mattiello, and D. Le Bihan (1994) MR diffusion tensor spectroscopy and imaging. Biophys J. 66, 259–267.CrossRefGoogle Scholar
  8. 8.
    P.R. Baudet (1978) Rotationally invariant image operators. 4th International Conference on Pattern Recognition, 579–583.Google Scholar
  9. 9.
    A. Bhalerao and R. Wilson (2001) Estimating local and global structure using a gaussian intensity model. Proceedings of Medical Image Understanding and Analysis (MIUA).Google Scholar
  10. 10.
    A. Bhalerao and R. Wilson (2001) A Fourier approach to 3D local feature estimation from volume data. Proceedings of British Machine Vision Conference (BMVC), volume 2, 461–470.Google Scholar
  11. 11.
    K. Deguchi, T. Izumitani, and H. Hontani (2002) Detection and enhancement of line structures in an image by anisotropic diffusion. Pattern Recognition Letters 23, 1399–1405.CrossRefMATHGoogle Scholar
  12. 12.
    L. Dreschler and H.H. Nagel (1981) Volumetric model and 3D trajectory of a moving car derived from monocular TV frame sequences of a street scene. International Joint Conference on Artificial Intelligence (IJCAI), 692–697.Google Scholar
  13. 13.
    L. Florack (1997) Image Structure. Computational Imaging and Vision, volume 10, Kluwer Academic Publishers.Google Scholar
  14. 14.
    A.F. Frangi, W.J. Niessen, K.L. Vincken, and M.A. Viergever (1998) Multiscale vessel enhancement filtering. Medical Image Computing and Computer-Assisted Intervention (MICCAI), Lecture Notes in Computer Science, volume 1496, Springer-Verlag, 130–137.Google Scholar
  15. 15.
    G.H. Granlund and H. Knutsson (1995) Signal Processing for Computer Vision. Kluwer Academic Publisher.Google Scholar
  16. 16.
    C. Harris and M. Stephens (1988) A combined corner and edge detector. Proc. 4th Alvey Vision Conf., 189–192.Google Scholar
  17. 17.
    J. Hladuvka and E. Gröller (2001) Direction-driven shape-based interpolation of volume data. Proceedings of Vision, Modeling, and Visualization, Stuttgart, Germany, November 2001.Google Scholar
  18. 18.
    J. Hladuvka, A. Konig, and E. Gröller (2001) Exploiting eigenvalues of the Hessian matrix for volume decimation. Proceedings of the 9th International Conference in Central Europe on Computer Graphics, Visualisation and Computer Vision(WSCG).Google Scholar
  19. 19.
    B. Jähne (1993) Digital Image Processing. Springer-Verlag.Google Scholar
  20. 20.
    B. Jähne, H. Haussecker, H. Scharr, H. Spies, D. Schmundt, and U. Schurr (1998) Study of dynamical processes with tensor-based spatiotemporal image processing techniques. Proc. European Conference on Computer Vision (ECCV), 323–336.Google Scholar
  21. 21.
    M. Kass and A. Witkin (1987) Analyzing oriented patterns. Computer Vision, Graphics and Image Processing 37, 362–385.CrossRefGoogle Scholar
  22. 22.
    L. Kitchen and A. Rosenfeld (1988) Grey-level corner detection. Pattern Recognition letters 8, 311–318.CrossRefGoogle Scholar
  23. 23.
    H. Knutsson (1989) Representing local structure using tensors. Proceedings of the 6th Scandinavian Conference Image Analysis, Oulu, Finland, June 1989.Google Scholar
  24. 24.
    J.J. Koenderink (1984) The structure of images. Biological Cybernetics 50, 363–370.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    P. Kornprobst and G. Medioni (2000) Tracking segmented objects using tensor voting. Proc. of Conputer Vision and Pattern Recognition, 118–125.Google Scholar
  26. 26.
    Q. Lin (2001) Enhancement, Detection, and Visualization of 3D Volume Data. PhD thesis, Linköping University, Department of Electrical Engineering, Computer Vision Laboratory, Linköping, Sweden, October 2001.Google Scholar
  27. 27.
    T. Lindeberg (1994) Scale-Space Theory in Computer Vision. Kluwer Academic Publishers.Google Scholar
  28. 28.
    A.M. López, F. Lumbreras, J. Serrat, and J.J. Villanueva (1999) Evaluation of methods for ridge and valley detection. IEEE Pattern Analysis and Machine Intelligence 21(4), 327–335.CrossRefGoogle Scholar
  29. 29.
    H.P. Moravec (1977) Towards automatic visual obstacle avoidance. Proc. Int. joint. Conf. Artificial Intelligence, page 584.Google Scholar
  30. 30.
    M.S. Nixon and A.S. Aguado (2002) Feature Extraction and Image Processing. Butterworth-Heinemann.Google Scholar
  31. 31.
    J.A. Noble (1988) Finding corners. Image and Vision Computing Journal 6(2), 121–128.CrossRefGoogle Scholar
  32. 32.
    M. Petrou (1994) The differentiating filter approach to edge detection. Advances in Electronics and Electron Physics 88, 297–345.Google Scholar
  33. 33.
    T. Randen, E. Monsen, C. Signer, A. Abrahamsen, J. Hansen, T. Saeter, and J. Schlaf (2000) Three-dimensional texture attributes for seismic data analysis. Society of Exploration Geophysicists(SEG), Annual Meeting, Expanded Abstracts, 668–671.Google Scholar
  34. 34.
    A.R. Rao and B.G. Schunck (1989) Computing oriented texture fields. Conputer Vision and Pattern Recognition 32, 61–68.Google Scholar
  35. 35.
    B. Rieger, F.J. Timmermans, and L.J. van Vliet (2002) Estimation of curvature on surfaces in 3D grey-value images. Proc. 8th Annual Conference of the Advanced School for Computing and Imaging (ASCI), Delft, The Netherlands, June 2002, 170–177.Google Scholar
  36. 36.
    B. Rieger and L.J. van Vliet (2002) Curvature of n-dimensional space curves in grey-value images. IEEE Transactions on Image Processing 11(7), 738–745.MathSciNetCrossRefGoogle Scholar
  37. 37.
    Y. Sato, S. Nakajima, and N. Shiraga (1998) 3D multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical Image Analysis 2(2), 143–168.CrossRefGoogle Scholar
  38. 38.
    Y. Sato, C.F. Westin, A. Bhalerao, S. Nakajima, N. Shiraga, S. Tamura, and R. Kikinis (2000) Tissue classification based on 3D local intensity structures for volume rendering. IEEE Transactions on Visualization and Computer Graphics 6(2), 160–179.CrossRefGoogle Scholar
  39. 39.
    A.F. Solé, A. López, and G. Sapiro (2001) Crease enhancement diffusion. Computer Vision and Image Understanding 84, 241–248.CrossRefMATHGoogle Scholar
  40. 40.
    H. Spies and B. Jähne (2001) A general framework for image sequence analysis. Fachtagung Informationstechnik, Magdeburg, Germany, March 2001, 125–132.Google Scholar
  41. 41.
    J. Stoeckel, F.M. Vos, P.H. Vos, and A.M. Vossepoel (2000) An evaluation of ridge extraction methods for portal imaging. Proc. of International Conference on Pattern Recognition (ICPR), Barcelona, Spain, September 2000, 3433–3436.Google Scholar
  42. 42.
    C.K. Tang and G. Medioni (2002) Curvature-augmented tensor voting for shape inference from noisy 3D data. IEEE Transactions on Pattern Recognition and Machine Intelligence, 858–864.Google Scholar
  43. 43.
    C.K. Tang, G. Medioni, and M.S. Lee (2001) N-dimensional tensor voting, with application to epipolar geometry estimation. IEEE Transactions on Pattern Recognition and Machine Intelligence 23(8), 829–844.CrossRefGoogle Scholar
  44. 44.
    W.S. Tong, C.K. Tang, and G. Medioni (2001) First order tensor voting, and application to 3-D scale analysis. Proc. of Computer Vision and Pattern Recognition, 175–182.Google Scholar
  45. 45.
    L.J. van Vliet and P.W. Verbeek (1995) Estimators for orientation and anisotropy in digitized images. Proceedings of the first Conference of the Advanced School for Computing and Imaging (ASCI), 442–450.Google Scholar
  46. 46.
    P.W. Verbeek, L.J. van Vliet, and J. van de Weijer (1998) Improved curvature and anisotropy estimation for curved line bundles. Proc. 14th Int. Conf. on Pattern Recognition, Brisbane, Australia, August 1998.Google Scholar
  47. 47.
    J. Weickert (1997) A review of nonlinear diffusion filtering. Scale-Space Theory in Computer Vision, Lecture Notes in Comp. Science, volume 1252, Springer-Verlag, Berlin, 3–28.Google Scholar
  48. 48.
    J. Weickert (1998) Anisotropic Diffusion in Image Processing. ECMI Series, Teubner-Verlag, Stuttgart.MATHGoogle Scholar
  49. 49.
    J. Weickert (1999) Coherence-enhancing diffusion filtering. Int. J. Computer Vision 31, 111–127.CrossRefGoogle Scholar
  50. 50.
    C.F. Westin, S.E. Maier, B. Khidhir, P. Everett, F.A. Jolesz, and R. Kikinis (1999) Image processing for diffusion tensor magnetic resonance imaging. Second International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), Springer-Verlag, 441–452.Google Scholar
  51. 51.
    C.F. Westin, A. Bhalerao, H. Knutsson, and R. Kikinis (1997) Using local 3D structure for segmentation of bone from computer tomography images. Proceedings of IEEE Conf on Conputer Vision and Pattern Recognition (CVPR), 794–800.Google Scholar
  52. 52.
    C.F. Westin, L. Wigstrom, T. Loock, L. Sjokvist, R. Kikinis, and H. Knutsson (2001) Three-dimensional adaptive filtering in magnetic resonance angiography. Journal of Magnetic Resonance Imaging 14, 63–71.CrossRefGoogle Scholar
  53. 53.
    Z. Zheng, H. Wang, and E.K. Teoh (1999) Analysis of gray level corner detection. Pattern Recognition Letters 20, 149–162.CrossRefMATHGoogle Scholar
  54. 54.
    O.A. Zuniga and R.M. Haralick (1983) Corner detection using the facet model. Proc. Conf. Pattern Recognition Image Processing, 30–37.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Maria Faraklioti
    • 1
  • Maria Petrou
    • 1
  1. 1.School of Electronics and Physical SciencesUniversity of SurreyGuildfordUK

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