From a Non-Local Ambrosio-Tortorelli Phase Field to a Randomized Part Hierarchy Tree

Article

Abstract

In its most widespread imaging and vision applications, Ambrosio and Tortorelli (AT) phase field is a technical device for applying gradient descent to Mumford and Shah simultaneous segmentation and restoration functional or its extensions. As such, it forms a diffuse alternative to sharp interfaces or level sets and parametric techniques. The functionality of the AT field, however, is not limited to segmentation and restoration applications. We demonstrate the possibility of coding parts—features that are higher level than edges and boundaries—after incorporating higher level influences via distances and averages. The iteratively extracted parts using the level curves with double point singularities are organized as a proper binary tree. Inconsistencies due to non-generic configurations for level curves as well as due to visual changes such as occlusion are successfully handled once the tree is endowed with a probabilistic structure. As a proof of concept, we present (1) the most probable configurations from our randomized trees; and (2) correspondence matching results between illustrative shape pairs.

The work is a significant step towards establishing exponentially decaying diffuse distance fields as bridges between low level visual processing and shape computations.

Keywords

Bridging low level and high level vision Shape computation Screened Poisson PDE Implicit representations Linear model for reaction-diffusion 

References

  1. 1.
    Ambrosio, L., Tortorelli, V.: On the approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Aslan, C., Tari, S.: An axis-based representation for recognition. In: ICCV, pp. 1339–1346 (2005) Google Scholar
  3. 3.
    Aslan, C., Erdem, A., Erdem, E., Tari, S.: Disconnected skeleton: Shape at its absolute scale. IEEE Trans. Pattern Anal. 30(12), 2188–2203 (2008) CrossRefGoogle Scholar
  4. 4.
    Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: A quantum mechanical approach to shape analysis. In: ICCV—Workshop on Dynamic Shape Capture and Analysis (2011) Google Scholar
  5. 5.
    Bai, X., Wang, B., Yao, C., Liu, W., Tu, Z.: Co-transduction for shape retrieval. IEEE Trans. Image Process. 21(5), 2747–2757 (2012) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bajaj, C.L., Pascucci, V., Schikore, D.R.: The contour spectrum. In: Proceedings of the 8th conference on Visualization (1997) Google Scholar
  7. 7.
    Ballester, C., Caselles, V., Igual, L., Garrido, L.: Level lines selection with variational models for segmentation and encoding. J. Math. Imaging Vis. 27(1), 5–27 (2007) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Bar, L., Sochen, N., Kiryati, N.: Image deblurring in the presence of impulsive noise. Int. J. Comput. Vis. 70(3), 279–298 (2006) CrossRefGoogle Scholar
  9. 9.
    Biasotti, S., Cerri, A., Frosini, P., Giorgi, D., Landi, C.: Multidimensional size functions for shape comparison. J. Math. Imaging Vis. 32(2), 161–179 (2008) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Braides, A.: Approximation of Free-discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Berlin (1998) MATHGoogle Scholar
  11. 11.
    Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: CVPR, pp. 60–65. Springer, Berlin (2005) Google Scholar
  12. 12.
    Burgeth, B., Weickert, J., Tari, S.: Minimally stochastic schemes for singular diffusion equations. In: Tai, X.C., Lie, K.A., Chan, T.F., Osher, S. (eds.) Image Processing Based on Partial Differential Equations, Mathematics and Visualization, pp. 325–339. Springer, Berlin (2006) Google Scholar
  13. 13.
    Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001) CrossRefMATHGoogle Scholar
  14. 14.
    Cremers, D., Tischhäuser, F., Weickert, J., Schnörr, C.: Diffusion snakes: Introducing statistical shape knowledge into the Mumford-Shah functional. Int. J. Comput. Vis. 50(3), 295–313 (2002) CrossRefMATHGoogle Scholar
  15. 15.
    Dimitrov, P., Lawlor, M., Zucker, S.: Distance images and intermediate-level vision. In: SSVM, pp. 653–664. Springer, Berlin (2011) Google Scholar
  16. 16.
    Droske, M., Rumpf, M.: Multi scale joint segmentation and registration of image morphology. IEEE Trans. Pattern Anal. 29(12), 2181–2194 (2007) CrossRefGoogle Scholar
  17. 17.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Erdem, E., Tari, S.: Mumford-Shah regularizer with contextual feedback. J. Math. Imaging Vis. 33(1), 67–84 (2009) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Erdem, E., Sancar-Yilmaz, A., Tari, S.: Mumford-Shah regularizer with spatial coherence. In: SSVM, pp. 545–555. Springer, Berlin (2007) Google Scholar
  20. 20.
    Gebal, K., Bærentzen, J.A., Aanæs, H., Larsen, R.: Shape analysis using the Auto Dinfusion Function. Comput. Graph. Forum 28, 1405–1413 (2009) CrossRefGoogle Scholar
  21. 21.
    Gilboa, G., Darbon, J., Osher, S., Chan, T.: Nonlocal convex functionals for image regularization. UCLA CAM-report 06-57, (2006) Google Scholar
  22. 22.
    Gorelick, L., Galun, M., Sharon, E., Basri, R., Brandt, A.: Shape representation and classification using the Poisson equation. IEEE Trans. Pattern Anal. 28(12), 1991–2005 (2006) CrossRefGoogle Scholar
  23. 23.
    Jin, Y., Jost, J., Wang, G.: A nonlocal version of the Osher-Sol-Vese model. J. Math. Imaging Vis. 44, 99–113 (2012) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Jung, M., Vese, L.: Nonlocal variational image deblurring models in the presence of Gaussian or impulse noise. In: SSVM, pp. 401–412. Springer, Berlin (2009) Google Scholar
  25. 25.
    Jung, M., Bresson, X., Chan, T., Vese, L.: Color image restoration using nonlocal Mumford-Shah regularizers. In: EMMCVPR, pp. 373–387. Springer, Berlin (2009) Google Scholar
  26. 26.
    Kontschieder, P., Donoser, M., Bischof, H.: Beyond pairwise shape similarity analysis. In: ACCV 2009. Lecture Notes in Computer Science, vol. 5996, pp. 655–666. Springer, Berlin (2010) CrossRefGoogle Scholar
  27. 27.
    Lee, T.S., Yuille, A.: Efficient coding of visual scenes by grouping and segmentation. In: Doya, K., Ishii, S., Pouget, A., Rao, R. (eds.) Bayesian Brain: Probabilistic Approaches to Neural Coding, pp. 141–185. MIT Press, New York (2007) Google Scholar
  28. 28.
    Lee, T.S., Mumford, D., Romero, R., Lamme, V.A.: The role of the primary visual cortex in higher level vision. Vis. Res. 38(15–16), 2429–2454 (1998) CrossRefGoogle Scholar
  29. 29.
    March, R., Dozio, M.: A variational method for the recovery of smooth boundaries. Image Vis. Comput. 15(9), 705–712 (1997) CrossRefGoogle Scholar
  30. 30.
    Meyer, F.: Topographic distance and watershed lines. Signal Process. 38, 113–125 (1994) CrossRefMATHGoogle Scholar
  31. 31.
    Morse, S.P.: Concepts of use in contour map processing. Commun. ACM 12(3), 147–152 (1969) CrossRefMATHGoogle Scholar
  32. 32.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Patz, T., Preusser, T.: Ambrosio-Tortorelli segmentation of stochastic images. In: ECCV, pp. 254–267. Springer, Berlin (2010) Google Scholar
  34. 34.
    Patz, T., Kirby, R., Preusser, T.: Ambrosio-Tortorelli segmentation of stochastic images: model extensions, theoretical investigations and numerical methods. Int. J. Comput. Vis. (2012). doi:10.1007/s11263-012-0578-8, 23 pp. Google Scholar
  35. 35.
    Pelillo, M., Siddiqi, K., Zucker, S.: Matching hierarchical structures using association graphs. IEEE Trans. Pattern Anal. 21(11), 1105–1120 (1999) CrossRefGoogle Scholar
  36. 36.
    Peng, T., Jermyn, I., Prinet, V., Zerubia, J.: Extended phase field higher-order active contour models for networks. Int. J. Comput. Vis. 88(1), 111–128 (2010) CrossRefGoogle Scholar
  37. 37.
    Pien, H., Desai, M., Shah, J.: Segmentation of MR images using curve evolution and prior information. Int. J. Pattern Recognit. 11(8), 1233–1245 (1997) CrossRefGoogle Scholar
  38. 38.
    Preußer, T., Droske, M., Garbe, C., Rumpf, M., Telea, A.: A phase field method for joint denoising, edge detection and motion estimation. SIAM J. Appl. Math. 68(3), 599–618 (2007) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Proesman, M., Pauwels, E., van Gool, L.: Coupled geometry-driven diffusion equations for low-level vision. In: Romeny, B. (ed.) Geometry Driven Diffusion in Computer Vision. Lecture Notes in Computer Science. Kluwer, Amsterdam (1994) Google Scholar
  40. 40.
    Reuter, M.: Hierarchical shape segmentation and registration via topological features of Laplace-Beltrami eigenfunctions. Int. J. Comput. Vis. 89(2), 287–308 (2010) CrossRefGoogle Scholar
  41. 41.
    Rosin, P.L., West, G.: Salience distance transforms. Graph. Models Image Process. 57(6), 483–521 (1995) CrossRefMATHGoogle Scholar
  42. 42.
    Rosman, G., Bronstein, M.M., Bronstein, A.M., Kimmel, R.: Nonlinear dimensionality reduction by topologically constrained isometric embedding. Int. J. Comput. Vis. 89(1), 56–68 (2010) CrossRefGoogle Scholar
  43. 43.
    Shah, J.: Segmentation by nonlinear diffusion. In: CVPR, pp. 202–207. Springer, Berlin (1991) Google Scholar
  44. 44.
    Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: CVPR, pp. 136–142. Springer, Berlin (1996) Google Scholar
  45. 45.
    Shah, J.: Skeletons and segmentation of shapes. Tech. rep, Northeastern University (2005). See http://www.math.neu.edu/~shah/publications.html
  46. 46.
    Shah, J., Pien, H., Gauch, J.: Recovery of shapes of surfaces with discontinuities by fusion of shading and range data within a variational framework. IEEE Trans. Image Process. 5(8), 1243–1251 (1996) CrossRefGoogle Scholar
  47. 47.
    Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signaturebased on heat diffusion. In: Comput. Graph. Forum (2009) Google Scholar
  48. 48.
    Tari, S.: Hierarchical shape decomposition via level sets. In: ISMM, pp. 215–225. Springer, Berlin (2009) Google Scholar
  49. 49.
    Tari, S.: Fluctuating distance fields. In: Breuss, M., Bruckestein, A., Maragos, P. (eds.) Innovations in Shape Analysis—Proceedings of Dagstuhl Workshop, Mathematics and Visualization. Springer, Berlin (2013) Google Scholar
  50. 50.
    Tari, S., Genctav, M.: From a modified Ambrosio-Tortorelli to a randomized part hierarchy tree. In: SSVM, pp. 267–278. Springer, Berlin (2011) Google Scholar
  51. 51.
    Tari, S., Shah, J.: Local symmetries of shapes in arbitrary dimension. In: ICCV, pp. 1123–1128 (1998) Google Scholar
  52. 52.
    Tari, S., Shah, J., Pien, H.: Extraction of shape skeletons from grayscale images. Comput. Vis. Image Underst. 66(2), 133–146 (1997) CrossRefGoogle Scholar
  53. 53.
    Teboul, S., Blanc-Fraud, L., Aubert, G., Barlaud, M.: Variational approach for edge preserving regularization using coupled PDE’s. IEEE Trans. Image Process. 7, 387–397 (1998) CrossRefGoogle Scholar
  54. 54.
    Yang, X., Bai, X., Koknar-Tezel, S., Latecki, J.: Densifying distance spaces for shape and image retrieval. J. Math. Imaging Vis. (2012). doi:10.1007/s10851-012-0363-x Google Scholar
  55. 55.
    Zhu, S.C., Yuille, A.L.: FORMS: a flexible object recognition and modeling system. Int. J. Comput. Vis. 20(3), 187–212 (1996) CrossRefGoogle Scholar
  56. 56.
    Zucker, S.: Distance images and the enclosure field: applications in intermediate-level computer and biological vision. In: Breuss, M., Bruckestein, A., Maragos, P. (eds.) Innovations in Shape Analysis—Proceedings of Dagstuhl Workshop, Mathematics and Visualization. Springer, Berlin (2013) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Middle East Technical UniversityAnkaraTurkey

Personalised recommendations