Abstract
An algorithm for recovering a function from essentially localized values of its Radon transform and sparse nonlocal values was outlined in Reference 13. That algorithm utilized the time-frequency properties of wavelets, coupled with the range theorems for the Radon transform, to localize essentially the dependence of the Radon transform. In this paper we utilize alternative time-frequency projections which were introduced by Coifman and Meyer (4). We present evidence that these bases are optimal according to our criterion for localized tomography. These bases require significantly less data than the wavelet bases that were used in Reference 13. Finally, we present numerical results supporting this work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auscher, P., G. Weiss, and M. V. Wickerhauser. Local sine and cosine bases of Coifman and Meyer and construction of smooth wavelets, In:Wavelets, A Tutorial in Theory and Applications, edited by C. Chui. New York: Academic Press.
Walnut D. Applications of Gabor and wavelet expansions to the Radon transform, In:Probabilistic and Stochastic Methods in Analysis, with Applications, edited by J. S. Byrnes. Norwell, MA: Kluwer, 1992, pp. 187–205.
Berenstein, C., and D. Walnut. Local inversion of the Radon transform in even dimensions using wavelets. In: Proceedings of the Conference: 75 Years of the Radon Transform, Vienna, Austria, 1992. International Press, 1994, pp. 38–58.
Coifman, R. R., and Y. Meyer. Remarques sur l'analyse de fourier 'a fenêtre, sêrie I.C.R. Acad. Sci. Paris 312:259–261, 1991.
Donoho, D. L., and P. B. Stark. Uncertainty principles and signal recovery,SIAM J. Appl. Math. 49(3):906–931, 1989.
Faridani, A. Sampling and resolution in diffraction tomography II: an error analysis of the filtered backprojection algorithm.SIAM J. Appl. Math., in press.
Faridani, A., E. Ritman, and K. T. Smith. Examples of local tomography.SIAM J. Appl. Math. (4):1193–1198, 1992.
Faridani, A., F. Keinert, F. Natterer, E. L. Ritman, and K. T. Smith. Local and global tomography.Sign. Proc., IMA Vol. Math. Appl. 23:241–255, 1990.
Faridani, A., E. Ritman, and K. T. Smith. Local tomography.SIAM J. Appl. Math. 52(2):459–484, 1992.
Louis, A. K., and A. Reider. Incomplete data problems in X-ray computerized tomography.Numeriche Math. 56:371–383, 1989.
Maass, P. The interior Radon transform.SIAM J. Appl. Math. 52(3):710–724, 1992.
Natterer, F.The Mathematics of Computerized Tomography. Stuttgart: Wiley and Sons, 1986.
Olson, T., and J. DeStefano. Wavelet localization of the radon transform.IEEE Trans. Sign. Proc. August:2056–2067, Aug. 1994.
Radon, J. Über die Bestimmung von Funktionen durch ihre Integralwerte lángs gewisser Mannigfaltigkeiten.Berichte Sächsishe Akad. Wissenschaften, 69:262–267.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Olson, T. Optimal time-frequency projections for localized tomography. Ann Biomed Eng 23, 622–636 (1995). https://doi.org/10.1007/BF02584461
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02584461