Recovering convex edges of an image from noisy tomographic data

IEEE Transactions on Information Theory

Volumn 52, Issue 4, 2006, Pages 1322-1334

  Goldenshluger, Alexander ^a   Spokoiny, Vladimir ^b  

^a UNIVERSITY OF HAIFA   (Israel)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Edge detection; Minimax estimation; Optimal rates of convergence; Radon transform; Support function

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ESTIMATION; IMAGE PROCESSING; MATHEMATICAL TRANSFORMATIONS; NUMERICAL ANALYSIS;

CONVEX EDGES; MINIMAX ESTIMATION; OPTIMAL RATES OF CONVERGENCE; RADON TRANSFORM; SUPPORT FUNCTION;

EDGE DETECTION;

EID: 33645704779     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2006.871053     Document Type: Article

References (28)

1. P. Bickel and Y. Ritov, "Estimating a linear functional of a PET image," IEEE Trans. Med. Imag., vol. 14, no. 1, pp. 81-87, Mar. 1995.
2. E. J. Candés and D. Donoho, "Recovering edges in ill-posed inverse problems: Optimality of curvelet frames," Ann. Statist., vol. 30, pp. 784-842, 2002.
3. L. Cavalier and J.-Y. Koo, "Poisson intensity estimation for tomographic data using a wavelet shrinkage approach," IEEE Trans. Inf. Theory, vol. 48, no. 10, pp. 2794-2802, Oct. 2002.
4. A. Faridani, E. Ritman, and K. Smith, "Local tomography," SIAM J. Appl. Math., vol. 52, pp. 459-484, 1992.
5. R. Gardner, Geometric Tomography. Cambridge, U.K.: Cambridge Univ. Press, 1995.
6. I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions. Vol. 5. Integral Geometry and Representation Theory. New York/London, USA/U.K.: Academic, 1966.
7. A. Goldenshluger and V. Spokoiny, "On the shape-from-moments problem and recovering edges from noisy Radon data," Probab. Theory Rel. Fields, vol. 128, pp. 123-140, 2004.
8. A. Goldenshluger and A. Zeevi, "Recovering convex boundaries from blurred and noisy observations," unpublishesd manuscript, 2004.
9. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge, U.K.: Cambridge Univ. Press, 1996.
10. W. Härdle, B. U. Park, and A. Tsybakov, "Estimation of nonsharp support boundaries," J. Multivariate Anal., vol. 55, pp. 205-218, 1995.
11. A. O. Hero, R. Piramuthu, J. A. Fessler, and S. R. Titus, "Minimax emission computed tomography using high-resolution anatomical side information and B-spline models," IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 920-938, Mar. 1999.
12. I. Johnstone and B. Silverman, "Speed of estimation in the positron emission tomography and related inverse problems," Ann. Statist., vol. 18, pp. 251-280, 1990.
13. I. Johnstone and B. Silverman, "Discretization effects in statistical inverse problems," J. Complexity, vol. 7, pp. 1-34, 1991.
14. A. Korostelev and A. Tsybakov, Minimax Theory of Image Reconstruction (Lectures Notes in Statistics). New York: Springer-Verlag, 1993.
15. A. Korostelev and A. Tsybakov, "Asymptotic efficiency in estimation of a convex set," Probl. Inf. Transm., vol. 30, pp. 317-327, 1994.
16. A. Korostelev, L. Simar, and A. Tsybakov, "On estimation of monotone and convex boundaries," Publ. Inst. Statist. Univ. Paris., vol. 39, pp. 3-18, 1995.
17. E. Mammen and A. Tsybakov, "Asymptotical minimax recovery of sets with smooth boundaries," Ann. Statist., vol. 23, pp. 502-524, 1995.
18. J. Polzhel and V. Spokoiny, "Adaptive weights smoothing with applications to image restoration," J. R. Stat. Soc. Ser. B: Stat. Methodol., vol. 62, pp. 335-354, 2000.
19. 3," SIAM J. Math. Anal., vol. 24, pp. 1215-1225, 1993.
20. A. Ramm and A. Zaslavsky, "Reconstructing singularities of a function from its radon transform," Math. Comput. Modeling, vol. 18, pp. 109-138, 1993.
21. Y. Ritov, "Estimating mass and shape of domains in PET imaging," J. Nonparametric Statist., vol. 10, pp. 47-66, 1998.
22. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Cambridge, U.K.: Cambridge Univ. Press, 1993.
23. N. Srinivasa, K. R. Ramakrishnan, and K. Rajgopal, "Edge detection from projections," IEEE Trans. Med. Imag., vol. 11, no. 1, pp. 76-80, Mar. 1992.
24. M. Talagrand, "Sharper bounds for Gaussian and empirical processes," Ann. Probab., vol. 22, pp. 28-76, 1994.
25. A. van der Vaart and J. Wellner, Weak Convergence and Empirical Processes. New York: Springer-Verlag, 1996.
26. Y. Vardi, L. Shepp, and L. Kaufman, "A statistical model for positron emission tomography. With discussion," J. Amer. Statist. Assoc., vol. 80, pp. 8-37, 1985.
27. Y. Wang, "Change curve estimation via wavelets," J. Amer. Statst. Assoc., vol. 93, pp. 163-172, 1998.
28. J. C. Ye, Y. Bresler, and P. Moulin, "Asymptotic global confidence regions in parametric shape estimation problems," IEEE Trans. Inf. Theory, vol. 46, no. 5, pp. 1881-1895, Aug. 2000.