Modèles déformables contraints en reconstruction tomographique par temps de première arrivée

Modèles déformables contraints en reconstruction tomographique par temps de première arrivée

Gil Gaullier Pierre Charbonnier  Fabrice Heitz  Philippe Côte 

CEREMA/DTer Est/Laboratoire Régional de Strasbourg 11 Rue Jean Mentelin B.P. 9, 67035 Strasbourg

ICube, UMR 7357, Université de Strasbourg, CNRS 300 Bd Sébastien Brant CS 10413, 67412 Illkirch Cedex

Dept. GERS, IFSTTAR, Centre de Nantes Route de Bouaye CS 4, 44344 Bouguenais

Corresponding Author Email: 
gil.gaullier@gmail.com,pierre.charbonnier@cerema.fr
Page: 
215-243
|
DOI: 
https://doi.org/10.3166/TS.32.215-243
Received: 
5 December 2014
| |
Accepted: 
23 June 2015
| | Citation

OPEN ACCESS

Abstract: 

Reconstruction in traveltime tomography is a difficult, ill-posed, non-linear inverse problem, in which only few data are generally available. As a consequence pixel-based regularization and reconstruction methods often lead to poor solutions. In this paper we develop a new object-based / deformable model-based reconstruction approach which enables toset strong constraints on the shape of the object to reconstruct. The non-linearity is handled by a decoupling method which yields a sequence of linear shape optimization problems. The approach supports objects with complex topology, thanks to a level sets implementation. The reconstruction method is assessed on real data, obtained on a reduced-scale bench used in traveltime geoacoustic inversion.

RÉSUMÉ

La reconstruction en tomographie par temps de première arrivée est rendue difficile par son caractère mal posé, par la non-linéarité du problème direct et par le faible nombre de données généralement disponibles. Dans ce contexte, les méthodes de reconstruction et de régularisation orientées « pixel » ne fournissent généralement pas des reconstructions satisfaisantes. Afin de contraindre la solution du problème de reconstruction, nous proposons ici une nouvelle approche de reconstruction orientée « objet » permettant d’introduire un a priori géométrique de haut niveau sur la forme des objets à reconstruire, par le biais d’un modèle déformable. La méthode développée approche le problème direct non linéaire par une suite de problèmes linéaires, conduisant à un schéma de minimisation alternée d’une fonction d’énergie intégrant l’a priori de forme et gérant les changements de topologie par l’algorithme des ensembles de niveaux. L’efficacité de la méthode est illustrée sur un jeu de données réelles, acquises sur un banc dédié à la réalisation d’expérimentations géophysiques à échelle réduite, en particulier pour un faible nombre de données.

Keywords: 

inverse problems, traveltime tomography, deformable models, shape-constrained reconstruction. geoacoustic inversion

MOTS-CLÉS

problèmes inverses, tomographie par temps de première arrivée, modèles déformables, reconstruction orientée objet, contraintes de forme, inversion géo-acoustique

1. Introduction
2. Tomographie Par Temps De Première Arrivée
3. Une Méthode De Reconstruction Sous Contrainte De Forme
4. Résultats De Reconstruction De Forme
5. Conclusion
  References

Abraham O., Slimane K. B., Côte P. (1988). Factoring anisotropy into iterative geometric reconstruction algorithms for seismic tomography. International Journal of Rock Mechanics and Mining Sciences, vol. 35, no 1, p. 31-41.

Alvino C., Yezzi A. (2004). Tomographic reconstruction of piecewise smooth images. In Ieee computer society conference on computer vision and pattern recognition, p. 576-581. Washington D.C., USA, IEEE.

Aubert G., Barlaud M., Faugeras O., Jehan-Besson S. (2003). Image segmentation using active contours: calculus of variations or shape gradients? SIAM, Journal on Applied Mathematics, vol. 63, no 6, p. 2128-2154.

Battle X., Cunningham G., Hanson K. (1998). Tomographic reconstruction using 3D deformable models. Physics in Medicine and Biology, vol. 43, no 4, p. 983-990.

Bresler Y., Macovski A. (1987). Three-dimensional reconstruction from projections with incomplete and noisy data by object estimation. IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 35, no 8, p. 1139–1152.

Bretaudeau F., Leparoux D., Durand O., Abraham O. (2011). Small-scale modeling of onshore seismic experiment: A tool to validate numerical modeling and seismic imaging methods. Geophysics, vol. 76, no 5, p. T101–T112.

Bruandet J.-P., Peyrin F., Dinten J.-M., Barlaud M. (2002). 3D tomographic reconstruction of binary images from cone beam projections: a fast level set approach. In IEEE international symposium on biomedical imaging, p. 677-680. Washington D.C., USA, IEEE.

Burger M. (2003). A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces and Free Boundaries, vol. 5, p. 301-329.

Cartan H. (1967). Formes différentielles, éléments de calcul des variations. Paris, Hermann.

Caselles V., Kimmel R., Sapiro G. (1997, février). Geodesic active contours. International Journal of Computer Vision, vol. 22, no 1, p. 61-79.

Cerveny V. (2001). Seismic ray theory. Cambridge, R.U., Cambridge University Press.

Chan T., Vese L. (2001, février). Active contours without edges. IEEE Transactions on Image processing, vol. 10, no 2, p. 266-277.

Charbonnier P., Blanc-Féraud L., Aubert G., Barlaud M. (1997). Deterministic edge-preserving regularization in computed imaging. IEEE Transactions on Image Processing, vol. 6, no 2,p. 298-311.

Chiao P., Rogers W., Clinthorne N., Fessler J., Hero A. (1994). Model-based estimation for dynamic cardiac studies using ECT. IEEE Transactions on Medical Imaging, vol. 13, no 2, p. 217–226.

Cremers D., Osher S., Soatto S. (2006). Kernel density estimation and intrinsic alignment for shape priors in level set segmentation. International Journal of Computer Vision, vol. 69, no 3, p. 335-351.

Dorn O., Lesselier D. (2006). Level set methods for inverse scattering. Inverse Problems, vol. 22, no 4, p. R67-R131.

Feng H. (2002). Curve evolution object-based techniques for image reconstruction and seg mentation. Thèse de doctorat, Université de Boston, USA.

Feng H., Karl W., Castañon D. (2003, janvier). A curve evolution approach to object-based tomographic reconstruction. IEEE Transactions on Image Processing, vol. 12, no 1, p. 44-57.

Foulonneau A., Charbonnier P., Heitz F. (2006). Affine-invariant geometric shape priors for region-based active contours. IEEE Transactions On Pattern Analysis and Machine Intelli gence, vol. 28, no 8, p. 1352–1357.

Foulonneau A., Charbonnier P., Heitz F. (2008, mai). Multi-reference affine-invariant geo metric shape priors for region-based active contours. Rapport technique no RR-AF01-08. Strasbourg, LRS ERA 27 LCPC / LSIIT UMR 7005 CNRS. Consulté sur http://icube-publis.unistra.fr/papr/docs/files/2233/RR-AF01-08.pdf (Une version abrégée (sans les annexes) de ce rapport préliminaire a été publiée dans (Foulonneau et al., 2009).)

Foulonneau A., Charbonnier P., Heitz F. (2009). Multi-reference shape priors for active contours. International Journal on Computer Vision, vol. 81, no 1, p. 68-81.

Gaullier G. (2013). Modèles déformables contraints en reconstruction d’images de tomo graphie non linéaire par temps d’arrivée. Thèse de doctorat, Université de Strasbourg. (http://www.theses.fr/2013STRAD026)

Gaullier G., Charbonnier P., Heitz F. (2009). Introducing shape priors in object-based tomographic reconstruction. In ICIP 2009, 16th international conference on image processing, p. 1077-1080. Le Caire, Egypte, IEEE.

Geman D., Reynolds G. (1992). Constrained restoration and the recovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 14, no 3, p. 367-383.

Geman D., Yang C. (1995). Nonlinear image recovery with half-quadratic regularization and FFT’s. IEEE Transactions on Image Processing, vol. 4, no 7, p. 932-946.

Lecellier F., Jehan-Besson S., Fadili J. (2014). Statistical region-based active contours for segmentation: An overview. {IRBM}, vol. 35, no 1, p. 3 – 10.

Leventon M., Grimson W., Faugeras O. (2000, juin). Statistical shape influence in geodesic active contours. In Proc. of ieee conference on computer vision and pattern recognition (cvpr), p. 1316–1323. Hilton Head Island, USA, IEEE.

Li W., Leung S. (2013). A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness. Geophysical Journal International, vol. 195, no 1, p. 582–596.

Lin Y., Ortega A. (2013). Object-based high contrast travel time tomography. Submitted to SEG Geophysics, arXiv:1303.3052 [physics.data-an].

Mohammad-Djafari A. (1996). Image reconstruction of a compact object from a few number of projections. In International conference on signal and image processing (SIP’96), p. 325-329. Orlando, USA, IASTED.

Osher S., Sethian J. (1988). Fronts propagating with curvature-dependant speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics, vol. 79, no 1,p. 12-49.

Paragios N., Deriche R. (2002). Geodesic active regions and level set methods for supervised texture segmentation. International Journal of Computer Vision, vol. 46, no 3, p. 223–247.

Ramlau R., Ring W. (2007). A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data. Journal of Computational Physics, vol. 221, no 2, p. 539-557.

Rawlinson N., Sambridge M. (2003). Seismic traveltime tomography of the crust and lithosphere. Advances in Geophysics, vol. 46, p. 81-198.

Rossi D., Willsky A. (1984). Reconstruction from projections based on detection and estimation of objects-parts I and II: Performance analysis and robustness analysis. IEEE Transactions on Acoustic, Speech, and Signal Processing, vol. 32, no 4, p. 886–906.

Santosa F. (1996). A level-set approach for inverse problems involving obstacles. The European Series in Applied and Industrial Mathematics: Control, Optimization and Calculus of Variations, vol. 1, p. 17-33.

Sethian J. (1996). A fast marching level set method for monotonically advancing fronts. Proceedings of the National Academy of Sciences, vol. 93, no 4, p. 1591–1595.

Sokolowski J., Zolesio J. (1992). Introduction to shape optimization: shape sensitivity analysis (vol. 16). Berlin, Heidelberg, Springer.

Soussen C., Mohammad-Djafari A. (2004). Polygonal and polyhedral contour reconstruction in computed tomography. IEEE Transactions on Image Processing, vol. 13, no 11, p. 1507–1523.

Tsai A., Yezzi A., Wells W., Tempany C., Tucker D., Fan A. et al. (2003, février). A shapebased approach to the segmentation of medical imagery using level sets. IEEE Transactions on Medical Imaging, vol. 22, no 2, p. 137–154.

Tsai Y.-H., Cheng L., Osher S., Zhao H.-K. (2003). Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM Journal on Numerical Analysis, vol. 2, no 41, p. 673–694.

Zhu S., Yuille A. (1996, septembre). Region competition: unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no 9, p. 884-900.