A family of π-scheme exponential Radon transforms and the uniqueness of their inverses

We propose a family of π-scheme exponential Radon transforms (ERTs) of an object function, which characterize the data functions that arise in π-scheme single-photon emission computed tomography (SPECT) with uniform attenuation. Single-interval 180° acquisition in SPECT can be interpreted as a special case of the proposed π-scheme SPECT. We show the existence and uniqueness of the inverse π-scheme ERT and provide an iterative algorithm for obtaining the object function from the π-scheme ERT. Numerical results in our simulation studies confirm the mathematical results.

[1]  S. Bellini,et al.  Compensation of tissue absorption in emission tomography , 1979 .

[2]  Frédéric Noo,et al.  Image reconstruction in 2D SPECT with 180° acquisition , 2001 .

[3]  Michael A. King,et al.  Investigation of causes of geometric distortion in 180o and 360o angular sampling in SPECT , 1989 .

[4]  S. Helgason.  The Radon Transform , 1980 .

[5]  K Kose,et al.  Image reconstruction algorithm for single-photon-emission computed tomography with uniform attenuation. , 1989, Physics in medicine and biology.

[6]  Thomas F. Budinger,et al.  The Use of Filtering Methods to Compensate for Constant Attenuation in Single-Photon Emission Computed Tomography , 1981 .

[7]  N C Yang,et al.  The circular harmonic transform for SPECT reconstruction and boundary conditions on the Fourier transform of the sinogram. , 1988, IEEE transactions on medical imaging.

[8]  Donald C. Solmon,et al.  Filtered-backprojection and the exponential Radon transform , 1989 .

[9]  Thomas M. Cover,et al.  An algorithm for maximizing expected log investment return , 1984, IEEE Trans. Inf. Theory.

[10]  X Pan,et al.  Minimal-scan filtered backpropagation algorithms for diffraction tomography. , 1999, Journal of the Optical Society of America. A, Optics, image science, and vision.

[11]  Xiaochuan Pan,et al.  Optimal noise control in and fast reconstruction of fan-beam computed tomography image. , 1999, Medical physics.

[12]  Frank Natterer,et al.  Inversion of the attenuated Radon transform , 2001 .

[13]  X. Pan,et al.  Unified reconstruction theory for diffraction tomography, with consideration of noise control. , 1998, Journal of the Optical Society of America. A, Optics, image science, and vision.

[14]  C. Metz,et al.  The exponential Radon transform , 1980 .

[15]  C. Metz,et al.  /spl pi/-scheme short-scan SPECT and image reconstruction , 2001, 2001 IEEE Nuclear Science Symposium Conference Record (Cat. No.01CH37310).

[16]  M A King,et al.  Investigation of causes of geometric distortion in 180 degrees and 360 degrees angular sampling in SPECT. , 1989, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[17]  David H. Feiglin,et al.  Clinical Evaluation of 360° and 180° Data Sampling Techniques for Transaxial SPECT Thallium-201 Myocardial Perfusion Imaging , 1985 .

[18]  K. Lange,et al.  EM reconstruction algorithms for emission and transmission tomography. , 1984, Journal of computer assisted tomography.

[19]  Xiaochuan Pan,et al.  Analysis of noise properties of a class of exact methods of inverting the 2-D exponential radon transform [SPECT application] , 1995, IEEE Trans. Medical Imaging.

[20]  Xiaochuan Pan,et al.  A unified analysis of exact methods of inverting the 2-D exponential radon transform, with implications for noise control in SPECT , 1995, IEEE Trans. Medical Imaging.

[21]  D. Parker.  Optimal short scan convolution reconstruction for fan beam CT , 1982 .

[22]  L. Shepp,et al.  Maximum Likelihood Reconstruction for Emission Tomography , 1983, IEEE Transactions on Medical Imaging.

[23]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[24]  Leonid Kunyansky.  A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula , 2001 .