Abstract
We demonstrate the possibility of coding parts, features that are higher level than boundaries, using a modified AT field after augmenting the interaction term of the AT energy with a non-local term and weakening the separation into boundary/not-boundary phases. 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. The work is a step in establishing the AT function as a bridge between low and high level visual processing.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Aslan, C., Tari, S.: An axis-based representation for recognition. In: ICCV, pp. 1339–1346 (2005)
Bar, L., Sochen, N., Kiryati, N.: Image deblurring in the presence of impulsive noise. Int. J. Comput. Vision 70(3), 279–298 (2006)
Braides, A.: Approximation of Free-discontinuity Problems. Lecture Notes in Mathematics, vol. 1694. Springer, Heidelberg (1998)
Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: CVPR, pp. 60–65 (2005)
Droske, M., Rumpf, M.: Multi scale joint segmentation and registration of image morphology. IEEE T-PAMI 29(12), 2181–2194 (2007)
Erdem, E., Sancar-Yilmaz, A., Tari, S.: Mumford-shah regularizer with spatial coherence. In: Sgallari, F., Murli, A., Paragios, N. (eds.) SSVM 2007. LNCS, vol. 4485, pp. 545–555. Springer, Heidelberg (2007)
Jung, M., Bresson, X., Chan, T.F., Vese, L.A.: Color image restoration using nonlocal mumford-shah regularizers. In: Cremers, D., Boykov, Y., Blake, A., Schmidt, F.R. (eds.) EMMCVPR 2009. LNCS, vol. 5681, pp. 373–387. Springer, Heidelberg (2009)
Jung, M., Vese, L.: Nonlocal variational image deblurring models in the presence of gaussian or impulse noise. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 401–412. Springer, Heidelberg (2009)
Kimia, B., Tannenbaum, A., Zucker, S.: Shapes, shocks, and deformations I: The components of two-dimensional shape and the reaction-diffusion space. Int. J. Comput. Vision 15(3), 189–224 (1995)
Kimmel, R., Kiryati, N., Bruckstein, A.: Sub-pixel distance maps and weighted distance transforms. J. of Math. Imag. and Vis. 6(2-3), 223–233 (1996)
Maragos, P., Butt, M.A.: Curve evolution, differential morphology and distance transforms as applied to multiscale and eikonal problems. Fundamentae Informatica 41, 91–129 (2000)
March, R., Dozio, M.: A variational method for the recovery of smooth boundaries. Image and Vision Computing 15(9), 705–712 (1997)
Meyer, F.: Topographic distance and watershed lines. Signal Processing 38, 113–125 (1994)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Com. Pure App. Math. 42, 577–685 (1989)
Osher, S., Sethian, J.: Fronts propagating with curvat. dependent speed: algs. based on Hamilton-Jacobi formulations. J. Comp. Phys. 79, 12–49 (1988)
Pätz, T., Preusser, T.: Ambrosio-tortorelli segmentation of stochastic images. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6315, pp. 254–267. Springer, Heidelberg (2010)
Pelillo, M., Siddiqi, K., Zucker, S.W.: Matching hierarchical structures using association graphs. IEEE Trans. Pattern Anal. Mach. Intell. 21, 1105–1120 (1999)
Pien, H.H., Desai, M., Shah, J.: Segmentation of mr images using curve evolution and prior information. IJPRAI 11(8), 1233–1245 (1997)
Preußer, T., Droske, M., Garbe, C., Rumpf, M., Telea, A.: A phase field method for joint denoising, edge detection and motion estimation. SIAM Journal on Applied Mathematics 68(3), 599–618 (2007)
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. Kluwer, Dordrecht (1994)
Shah, J.: Segmentation by nonlinear diffusion. In: CVPR, pp. 202–207 (1991)
Shah, J.: A common framework for curve evolution, segmentation and anisotropic diffusion. In: CVPR, pp. 136–142 (1996)
Shah, J.: Skeletons and segmentation of shapes. Technical report, Northeastern University (2005), http://www.math.neu.edu/shah/publications.html
Shah, J., Pien, H.H., Gauch, J.: Recovery of shapes of surfaces with discontinuities by fusion of shading and range data within a variational framework. IEEE Trans. on Image Processing 5(8), 1243–1251 (1996)
Tari, S.: Hierarchical shape decomposition via level sets. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 215–225. Springer, Heidelberg (2009)
Tari, S., Shah, J., Pien, H.: A computationally efficient shape analysis via level sets. In: MMBIA, pp. 234–243 (1996)
Tari, S., Shah, J., Pien, H.: Extraction of shape skeletons from grayscale images. CVIU 66(2), 133–146 (1997)
Teboul, S., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Variational approach for edge preserving regularization using coupled PDE’s. IEEE Trans. Imag. Pr. 7, 387–397 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tari, S., Genctav, M. (2012). From a Modified Ambrosio-Tortorelli to a Randomized Part Hierarchy Tree. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-24785-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24784-2
Online ISBN: 978-3-642-24785-9
eBook Packages: Computer ScienceComputer Science (R0)