Convolution-backprojection method for the k-plane transform, and calderón's identity for ridgelet transforms

Applied and Computational Harmonic Analysis

Volumn 16, Issue 3, 2004, Pages 231-242

  Rubin, Boris ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

Convolution backprojection method; K plane Radon transform; Ridgelet transforms; Wavelet transforms

Indexed keywords

APPROXIMATION THEORY; FOURIER TRANSFORMS; FUNCTION EVALUATION; INTEGRATION; PARTIAL DIFFERENTIAL EQUATIONS; WAVELET TRANSFORMS;

CONVOLUTION-BACKPROJECTION METHOD; INTERTWINING OPERATORS; K-PLANE RADON TRANSFORM; RIDGELET TRANSFORMS;

HARMONIC ANALYSIS;

EID: 2442507859     PISSN: 10635203     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.acha.2004.03.003     Document Type: Letter

References (35)

1. C.A. Berenstein, D.F. Walnut, Local inversion of the Radon transform in even dimensions using wavelets, in: S.G. Gindikin, P. Michor (Eds.), Proc. Conf. 75 Years of Radon Transform, Vienna, Austria, 1992, International Press, Hong Kong, 1994, pp. 45-69.
2. p-functions on real hyperbolic space, Preprint, 2001.
3. A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113-190.
4. E.J. Candès, Ridgelets: theory and applications, Ph.D. thesis, Technical report, Department of Statistics, Stanford University, 1998.
5. E.J. Candès, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal. 6 (2) (1999) 197-218.
6. E.J. Candès, New ties between computational harmonic analysis and approximation theory, in: C.K. Chui, L.L. Schumaker, J. Stoeckler (Eds.), Approximation Theory X: Wavelets, Splin, Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 87-153.
7. I. Daubechies, Ten lectures on wavelets, in: CBMS-NSF Series in Appl. Math., SIAM, Philadelphia, 1992.
8. n, Proc. Natl. Acad. Sci. USA 96 (5) (1999) 1828-1833.
9. M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley theory and the study of function spaces, in: CBMS Conf. Lecture Notes, vol. 79, American Mathematical Society, Providence, RI, 1991.
10. S. Helgason, The Radon Transform, second ed., Birkhäuser, Boston, 1999.
11. M. Holschneider, Inverse Radon transforms through inverse wavelet transforms, Inverse Problems 7 (1991) 853-861.
12. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955.
13. F. Keinert, Inversion of k-plane transforms and applications in computer tomography, SIAM Rev. 31 (1989) 273-289.
14. N. Murata, Function approximation by three-layered networks and its error bounds - an integral representation theorem, Technical report METR 94-19, University of Tokyo, October 1994.
15. N. Murata, An integral representation with ridge functions and approximation bounds of three-layered network, Neural Net. 9 (6) (1996) 947-956.
16. F. Natterer, The Mathematics of Computerized Tomography, Wiley, New York, 1986.
17. J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. Kl. 69 (1917) 262-277 (Russian translation in the Russian edition of S. Helgason The Radon Transform, Moscow, Mir, 1983, pp. 134-148).
18. J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. Kl. 69 (1917) 262-277 (Russian translation in the Russian edition of S. Helgason The Radon Transform, Moscow, Mir, 1983, pp. 134-148).
19. B. Rubin, Inversion and characterization of Radon transforms via continuous wavelet transforms, Preprint No. 13. The Hebrew University of Jerusalem, 1995/1996.
20. B. Rubin, Fractional integrals and potentials, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82, Longman, Harlow, 1996.
21. B. Rubin, Spherical Radon transform and related wavelet transforms, Appl. Comput. Harmon. Anal. 5 (1998) 202-215.
22. B. Rubin, Fractional calculus and wavelet transforms in integral geometry, Fractional Calculus Appl. Anal. 1 (2) (1998) 193-219.
23. p-spaces, J. Fourier Anal. Appl. 4 (1998) 175-197.
24. B. Rubin, Inversion of k-plane transforms via continuous wavelet transforms, J. Math. Anal. Appl. 220 (1998) 187-203.
25. B. Rubin, Inversion of Radon transforms using wavelet transforms generated by wavelet measures, Math. Scand. 85 (1999) 285-300.
26. B. Rubin, Calderón-type reproducing formula, in: Encyclopaedia of Mathematics, Kluwer Academic, Dordrecht, 2000, pp. 104-105 (10 volumes, 1988-1994), Supplement II; Reprinted in: Fractional Calculus Appl. Anal. 3 (1) (2000) 103-106.
27. B. Rubin, Calderón-type reproducing formula, in: Encyclopaedia of Mathematics, Kluwer Academic, Dordrecht, 2000, pp. 104-105 (10 volumes, 1988-1994), Supplement II; Reprinted in: Fractional Calculus Appl. Anal. 3 (1) (2000) 103-106.
28. B. Rubin, Inversion formulas for the spherical Radon transform and the generalized cosine transform, Adv. in Appl. Math. 29(2002)471-497.
29. B. Rubin, Radon, cosine, and sine transforms on real hyperbolic space, Adv. in Math. 170 (2002) 206-223.
30. B. Rubin, Notes on Radon transforms in integral geometry, Fractional Calculus Appl. Anal. 6 (2003) 25-72.
31. B. Rubin, D. Ryabogin, The k-dimensional Radon transform on the n-sphere and related wavelet transforms, Contemp. Math. 278 (2001) 227-239.
32. D.C. Solmon, A note on k-plane integral transforms, J. Math. Anal. Appl. 71 (1979) 351-358.
33. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.
34. N.Ya. Vilenkin, A.V. Klimyk, Representations of Lie Groups and Special Functions, vol. 2, Kluwer Academic, Dordrecht, 1993.
35. D. Walnut, Applications of Gabor and wavelet expansions to the Radon transform, in: J. Byrnes, et al. (Eds.), Probabilistic and Stochastic Methods in Analysis, with Applications, in: NATO ASI Series, Kluwer Academic, Dordrecht, 1992, pp. 187-205.