A Scalable Fluctuating Distance Field: An Application to Tumor Shape Analysis

Part of the Association for Women in Mathematics Series book series (AWMS, volume 1)

Abstract

Tumor growth involves highly complicated processes and complex dynamics, which typically lead to deviation of tumor shape from a compact structure. In order to quantify the tumor shape variations in a follow-up scenario, a shape registration based on a scalable fluctuating shape field is described. In the earlier work of fluctuating distance fields (Tari and Genctav, J Math Imaging Vis 1–18, 2013; Tari, Fluctuating distance fields, parts, three-partite skeletons. In: Innovations for shape analysis. Springer, Berlin/New York, pp 439–466, 2013), the shape field consists of positive and negative values whose zero crossing separates the central and the peripheral volumes of a silhouette. We add a non-linear constraint upon the original fluctuating field idea in order to introduce a “fluctuation scale”, which indicates an assumption about peripherality. This provides the induction of an hierarchy hypothesis onto the field. When fixed, the field becomes robust for scale changes for analysis of correspondence. We utilize the scalable fluctuating field first in segmentation of the protruded regions in a tumor, which are significant for the radiotherapy planning and assessment procedures. Furthermore, the unique information encoded in the shape field is utilized as an underlying shape representation for follow-up registration applications. The representation performance of the scalable field for a fixed ‘fluctuation scale’ is demonstrated in comparison to the conventional distance transform approach for the registration problem.

References

  1. 1.
    Attene, M., Katz, S., Mortara, M., Patané, G., Spagnuolo, M., Tal, A.: Mesh segmentation-a comparative study. In: IEEE International Conference on Shape Modeling and Applications, SMI 2006, Matsushima, p. 7. IEEE (2006)Google Scholar
  2. 2.
    August, J., Siddiqi, K., Zucker, S.W.: Ligature instabilities in the perceptual organization of shape. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Fort Collins, vol. 2. IEEE (1999)Google Scholar
  3. 3.
    Bai, X., Latecki, L.J.: Path similarity skeleton graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1282–1292 (2008)CrossRefGoogle Scholar
  4. 4.
    Biederman, I.: Recognition-by-components: a theory of human image understanding. Psychol. Rev. 94, 115 (1987)CrossRefGoogle Scholar
  5. 5.
    Blum, H., Nagel, R.N.: Shape description using weighted symmetric axis features. Pattern Recognit. 10(3), 167–180 (1978)CrossRefMATHGoogle Scholar
  6. 6.
    Blum, H., et al.: A transformation for extracting new descriptors of shape. Models Percept. Speech Vis. Form 19(5), 362–380 (1967)Google Scholar
  7. 7.
    Brady, M., Asada, H.: Smoothed local symmetries and their implementation. Int. J. Robot. Res. 3(3), 36–61 (1984)CrossRefGoogle Scholar
  8. 8.
    Brox, T., Bruhn, A., Papenberg, N., Weickert, J.: High accuracy optical flow estimation based on a theory for warping. In: Computer Vision-ECCV 2004, Prague, pp. 25–36 (2004)Google Scholar
  9. 9.
    Burbeck, C.A., Pizer, S.M.: Object representation by cores: identifying and representing primitive spatial regions. Vis. Res. 35(13), 1917–1930 (1995)CrossRefGoogle Scholar
  10. 10.
    Chen, X., Golovinskiy, A., Funkhouser, T.: A benchmark for 3d mesh segmentation. ACM Trans. Graph. (TOG) 28, 73 (2009). ACMGoogle Scholar
  11. 11.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Crane, K., Weischedel, C., Wardetzky, M.: Geodesics in heat: a new approach to computing distance based on heat flow. ACM Trans. Graph. (TOG) 32(5), 152 (2013)Google Scholar
  13. 13.
    De Goes, F., Goldenstein, S., Velho, L.: A hierarchical segmentation of articulated bodies. Comput. Graph. Forum 27, 1349–1356 (2008). Wiley Online LibraryGoogle Scholar
  14. 14.
    Do Carmo, M.P., Do Carmo, M.P.: Differential Geometry of Curves and Surfaces, vol. 2. Prentice-Hall, Englewood Cliffs (1976)Google Scholar
  15. 15.
    Feldman, J., Singh, M.: Bayesian estimation of the shape skeleton. Proc. Natl. Acad. Sci. 103(47), 18,014–18,019 (2006)Google Scholar
  16. 16.
    Golovinskiy, A., Funkhouser, T.: Consistent segmentation of 3d models. Comput. Graph. 33(3), 262–269 (2009)CrossRefGoogle Scholar
  17. 17.
    Hamamci, A., Kucuk, N., Karaman, K., Engin, K., Unal, G.: Tumor-cut: segmentation of brain tumors on contrast enhanced mr images for radiosurgery applications. IEEE Trans. Med. Imaging 31(3), 790–804 (2012)CrossRefGoogle Scholar
  18. 18.
    Hoffman, D.D., Richards, W.A.: Parts of recognition. Cognition 18, 65–96 (1984)CrossRefGoogle Scholar
  19. 19.
    Horn, B., Schunck, B.: Determining optical flow. Artif. Intell. 17, 185–203 (1981)CrossRefGoogle Scholar
  20. 20.
    Joshi, S., Pizer, S., Fletcher, P.T., Yushkevich, P., Thall, A., Marron, J.: Multiscale deformable model segmentation and statistical shape analysis using medial descriptions. IEEE Trans. Med. Imaging 21(5), 538–550 (2002)CrossRefGoogle Scholar
  21. 21.
    Kalogerakis, E., Hertzmann, A., Singh, K.: Learning 3d mesh segmentation and labeling. ACM Trans. Graph. (TOG) 29(4), 102 (2010)Google Scholar
  22. 22.
    Khan, F.M.: The Physics of Radiation Therapy. Lippincott Williams & Wilkins, Philadelphia/London (2009)Google Scholar
  23. 23.
    Konukoglu, E., Pennec, X., Clatz, O., Ayache, N.: Tumor growth modeling in oncological image analysis. In: Bankman, I. (ed.) Handbook of Medical Image Processing and Analysis – New edition, chap. 18, pp. 297–307. Burlington, San Diego, London (2008)Google Scholar
  24. 24.
    Lai, Y.K., Zhou, Q.Y., Hu, S.M., Wallner, J., Pottmann, D., et al.: Robust feature classification and editing. IEEE Trans. Vis. Comput. Graph. 13(1), 34–45 (2007)Google Scholar
  25. 25.
    Liu, R., Zhang, H., Shamir, A., Cohen-Or, D.: A part-aware surface metric for shape analysis. Comput. Graph. Forum 28, 397–406 (2009). Wiley Online LibraryGoogle Scholar
  26. 26.
    Macrini, D., Dickinson, S., Fleet, D., Siddiqi, K.: Bone graphs: medial shape parsing and abstraction. Comput. Vis. Image Underst. 115(7), 1044–1061 (2011)CrossRefGoogle Scholar
  27. 27.
    Meyer, F.: Topographic distance and watershed lines. Signal Process. 38(1), 113–125 (1994)CrossRefMATHGoogle Scholar
  28. 28.
    Mi, X., DeCarlo, D.: Separating parts from 2d shapes using relatability. In: IEEE 11th International Conference on Computer Vision, ICCV 2007, Rio de Janeiro, pp. 1–8. IEEE (2007)Google Scholar
  29. 29.
    Paragios, N., Rousson, M., Ramesh, V.: Non-rigid registration using distance functions. Comput. Vis. Image Underst. 89(2), 142–165 (2003)CrossRefMATHGoogle Scholar
  30. 30.
    Pizer, S.M., Fritsch, D.S., Yushkevich, P.A., Johnson, V.E., Chaney, E.L.: Segmentation, registration, and measurement of shape variation via image object shape. IEEE Trans. Med. Imaging 18(10), 851–865 (1999)CrossRefGoogle Scholar
  31. 31.
    Pottmann, H., Steiner, T., Hofer, M., Haider, C., Hanbury, A.: The isophotic metric and its application to feature sensitive morphology on surfaces. In: Computer Vision-ECCV 2004, Prague, pp. 18–23 (2004)Google Scholar
  32. 32.
    Shaked, D., Bruckstein, A.M.: Pruning medial axes. Comput. Vis. Image Underst. 69(2), 156–169 (1998)CrossRefGoogle Scholar
  33. 33.
    Shamir, A.: A survey on mesh segmentation techniques. Comput. Graph. Forum 27, 1539–1556 (2008). Wiley Online LibraryGoogle Scholar
  34. 34.
    Styner, M., Gerig, G., Lieberman, J., Jones, D., Weinberger, D.: Statistical shape analysis of neuroanatomical structures based on medial models. Med. Image Anal. 7(3), 207–220 (2003)CrossRefGoogle Scholar
  35. 35.
    Styner, M., Lieberman, J.A., Pantazis, D., Gerig, G.: Boundary and medial shape analysis of the hippocampus in schizophrenia. Med. Image Anal. 8(3), 197–203 (2004)CrossRefGoogle Scholar
  36. 36.
    Tari, S.: Fluctuating distance fields, parts, three-partite skeletons. In: Innovations for Shape Analysis, pp. 439–466. Springer, Berlin/Heidelberg (2013)Google Scholar
  37. 37.
    Tari, S., Genctav, M.: From a non-local ambrosio-tortorelli phase field to a randomized part hierarchy tree. J. Math. Imaging Vis. 49(1), 69–86. Springer (2014)Google Scholar
  38. 38.
    Tari, Z., Shah, J., Pien, H.: Extraction of shape skeletons from grayscale images. Comput. Vis. Image Underst. 66(2), 133–146 (1997)CrossRefGoogle Scholar
  39. 39.
    Tombropoulos, R., Schweikard, A., Latombe, J.C., Adler, J.: Treatment planning for image-guided robotic radiosurgery. In: Computer Vision, Virtual Reality and Robotics in Medicine, pp. 131–137. Springer, Berlin/Heidelberg (1995)Google Scholar

Copyright information

© Springer International Publishing Switzerland & The Association for Women in Mathematics 2015

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesSabanci UniversityIstanbulTurkey

Personalised recommendations