Advertisement

Compact Object Reconstruction

  • Ali Mohammad-Djafari
  • Charles Soussen
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this chapter we first present a review of the methods for the tomographie reconstruction of a compact homogeneous object that lies in a homogeneous background. Then we focus on contour estimation and polyhedral shape reconstructions. We give some sufficient conditions to obain exact reconstructions from a complete set of projections in the 2.1) case and present some extensions to the 3D case. Finally, due to the inherent diiculties of the exact reconstruction methods and their inappropriateness for practical situations,we propose an approximate reconstruction method that can handle the situation of very limited-angle projections.

Keywords

Active Contour Model Polygonal Shape Parallel Projection Geometric Moment Polyhedral Shape 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. T. Herman, H. K. Tuy, K. J. Langenberg, and P. C. Sabatier, Basic Methods of Tomography and Inverse Problems (Adam Hilger, Bristol, UK) 1987.Google Scholar
  2. [2]
    A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York), 1987.Google Scholar
  3. [3]
    K. D. Sauer and C. A. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Transactions on Nuclear Sciences 39, 1144–1152 (1992).CrossRefGoogle Scholar
  4. [4]
    K. D. Sauer and C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Transactions on Signal Processing SP-41, 534–548 (1993).CrossRefGoogle Scholar
  5. [5]
    R. Malladi, J. A. Sethian, and B. C. Vemuri, “Shape modeling with front propagation: A level-set approach,” IEEE Transactions on Pattern Analysis and Machine Intelligence PAMI-17, 158–175 (1995).CrossRefGoogle Scholar
  6. [6]
    D. J. Rossi and A. S. Willsky, “Reconstruction from projections based on detection and estimation of objects,” IEEE Transactions on Acoustics Speech and Signal Processing ASSP-32, 886–906 (1984).CrossRefGoogle Scholar
  7. [7]
    G. Demoment, “Image reconstruction and restoration: Overview of common estimation structure and problems,” IEEE Transactions on Acoustics Speech and Signal Processing ASSP-37, 2024–2036 (1989).CrossRefGoogle Scholar
  8. [8]
    S. Geman and D. McClure, “Statistical methods for tomographie image reconstruction,” In Proceedings of the 46th Session of the ISI, Bulletin of the ISI, volume 52, pp. 5–21, 1987.Google Scholar
  9. [9]
    L. Bedini, I. Gerace, and A. Tonazzini, “A deterministic algorithm for reconstructing images with interacting discontinuities,” Computer Vision and Graphics and Image Processing 56, 109–123 (1994).Google Scholar
  10. [10]
    C. A. Bouman and K. D. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Transactions on Image Processing IP-2, 296–310 (1993).CrossRefGoogle Scholar
  11. [11]
    A. Mohammad-Djafari and J. Idier, “A scale invariant Bayesian method to solve linear inverse problems,” In G. Heidbreder, Maximum Entropy and Bayesian Methods (Kluwer Academic Publishers, Dordrecht), pp. 121–134, 1996.CrossRefGoogle Scholar
  12. [12]
    M. Nikolova, J. Idier, and A. Mohammad-Djafari, “Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF,” IEEE Transactions on Image Processing 7, 571–585 (1998).CrossRefPubMedGoogle Scholar
  13. [13]
    S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,” Journal of Computational Physics 79, 12–49 (1988).CrossRefGoogle Scholar
  14. [14]
    F. Catte, P. Lions, J. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM Journal of Numerical Analysis 29, 182–193 (1992).CrossRefGoogle Scholar
  15. [15]
    R. T. Chin and K. F. Lai, “Deformable contours: Modeling and extraction,” IEEE Transactions on Pattern Analysis and Machine Intel-ligence PAMI-16, 601–608 (1994).Google Scholar
  16. [16]
    M. Kass, A. P. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” International Journal of Computer Vision 1, 321–331 (1988).CrossRefGoogle Scholar
  17. [17]
    L. H. Staib and J. S. Duncan, “Parametrically deformable contour models,” Computer Vision and Pattern Recognition, 98–103 (1989).Google Scholar
  18. [18]
    F. Santosa, “A level-set approach for inverse problems involving obstacles,” Control, Optimisation and Calculus of Variations 1, 17–33 (1996).CrossRefGoogle Scholar
  19. [19]
    Y. Amit, U. Grenander, and M. Piccioni, “Structural image restoration through deformable templates,” Journal of Acoustical Society America 86, 376–387 (1991).Google Scholar
  20. [20]
    K. M. Hanson, G. S. Cunningham, and R. J. McKee, “Uncertainty assessment for reconstructions based on deformable models,” International Journal of Imaging Systems and Technology 8, 506–512 (1997).CrossRefGoogle Scholar
  21. [21]
    D. Kolzow, A. Kuba, and A. Volcic, “An algorithm for reconstructing convex bodies from their projections,” Discrete and Computational Geometry 4, 205–237 (1989).CrossRefGoogle Scholar
  22. [22]
    J. L. Prince and A. S. Willsky, “Reconstructing convex sets from support line measurements,” IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 377–389 (1990).CrossRefGoogle Scholar
  23. [23]
    J. L. Prince and A. S. Willsky, “Convex set reconstruction using prior shape information,” Computer Vision and Graphics and Image Processing 53, 413–427 (1991).Google Scholar
  24. [24]
    P. Milanfar, W. C. Karl, and A. S. Willsky, “A moment-based variational approach to tomographie reconstruction,” IEEE Transactions on Image Processing 25, 772–781 (1994).Google Scholar
  25. [25]
    P. Milanfar, W. C. Karl, and A. S. Willsky, “Reconstructing binary polygonal objects from projections: A statistical view,” Computer Vision and Graphics and Image Processing 56, 371–391 (1994).Google Scholar
  26. [26]
    P. Milanfar, G. C. Verghese, W. C. Karl, and A. S. Willsky, “Reconstructing polygons from moments with connections to array processing,” IEEE Transactions on Signal Processing 43, 432–443 (1995).CrossRefGoogle Scholar
  27. [27]
    P. J. Davis, “Triangle formulas in the complex plane,” Mathematics of Computation 18, 569–577 (1964).CrossRefGoogle Scholar
  28. [28]
    P. J. Davis, “Plane regions determined by complex moments,” Journal of Approximation Theory 19, 148–153 (1977).CrossRefGoogle Scholar
  29. [29]
    C. Soussen and A. Mohammad-Djafari, “A 3D polyhedral shape recon-struction from tomographic projection data,” Technical Report, GPI-LSS, Gif-sur-Yvette, France (1998).Google Scholar
  30. [30]
    B. Mirtich, “Fast and accurate computation of polyhedral mass properties,” Journal of Graphic Tools 1, 31–50 (1996).CrossRefGoogle Scholar
  31. [31]
    A. Mohammad-Djafari, “Shape reconstruction in X-ray tomography,” In Proc. of SPIE 97, volume 3170, (SPIE, Bellingham, WA), pp. 240–251, 1997.CrossRefGoogle Scholar
  32. [32]
    A. Mohammad-Djafari, K. D. Sauer, Y. Khayi, and E. Cano, “Reconstruction of the shape of a compact object from a few number of projections,” In IEEE Int. Conf. on Image Processing (ICIP), volume 1,(IEEE, Piscataway, NJ), pp. 165–169, 1997.CrossRefGoogle Scholar
  33. [33]
    G. Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods (Springer Verlag, Berlin), 1995.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Ali Mohammad-Djafari
    • 1
  • Charles Soussen
    • 2
  1. 1.Laboratoire des Signaux et Systèmes(CNRS-ESE-UPS)Supélec, Plateau de MoulonGif-sur-YvetteFrance
  2. 2.Laboratoire des Signaux et Systèmes (CNRS-ESE-UPS)Supélec, Plateau de MoulonGif-sur-YvetteFrance

Personalised recommendations