skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry

Abstract

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.

Authors:
;  [1];  [2]
  1. Department of Biomedical Engineering, Beijing Institute of Technology, Beijing, 100081 (China) and Department of Radiology, State University of New York, Stony Brook, New York 11794 (United States)
  2. (United States)
Publication Date:
OSTI Identifier:
20775098
Resource Type:
Journal Article
Resource Relation:
Journal Name: Medical Physics; Journal Volume: 33; Journal Issue: 3; Other Information: DOI: 10.1118/1.2170596; (c) 2006 American Association of Physicists in Medicine; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
62 RADIOLOGY AND NUCLEAR MEDICINE; ACCURACY; ATTENUATION; BRAIN; COLLIMATORS; EFFICIENCY; FILTERS; IMAGE PROCESSING; IMAGES; PHANTOMS; RADON; SINGLE PHOTON EMISSION COMPUTED TOMOGRAPHY

Citation Formats

Wen Junhai, Liang Zhengrong, and Department of Radiology, State University of New York, Stony Brook, New York 11794. An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry. United States: N. p., 2006. Web. doi:10.1118/1.2170596.
Wen Junhai, Liang Zhengrong, & Department of Radiology, State University of New York, Stony Brook, New York 11794. An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry. United States. doi:10.1118/1.2170596.
Wen Junhai, Liang Zhengrong, and Department of Radiology, State University of New York, Stony Brook, New York 11794. Wed . "An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry". United States. doi:10.1118/1.2170596.
@article{osti_20775098,
title = {An inversion formula for the exponential Radon transform in spatial domain with variable focal-length fan-beam collimation geometry},
author = {Wen Junhai and Liang Zhengrong and Department of Radiology, State University of New York, Stony Brook, New York 11794},
abstractNote = {Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.},
doi = {10.1118/1.2170596},
journal = {Medical Physics},
number = 3,
volume = 33,
place = {United States},
year = {Wed Mar 15 00:00:00 EST 2006},
month = {Wed Mar 15 00:00:00 EST 2006}
}
  • A system design has been proposed for fast sequential SPECT/transmission CT on a three-headed SPECT camera. The design consists of a long focal length (114 cm) asymmetric fan beam collimator and a stationary transmission line source. The advantages of this system include: increased field-of-view, high sensitivity to the transmission line source, low sensitivity to radionuclide within the patient (reduced cross-talk effects), high spatial resolution, easy transition between SPECT and transmission acquisition, and no moving line source. An iterative ML-EM reconstruction algorithm for transmission data was implemented and adapted to the proposed acquisition geometry. Evaluations were performed using Monte Carlo simulatedmore » data. With a 40 cm detector width, transmission reconstructions of a 38 x 26 cm elliptical body were effectively artifact-free. Reconstructions of a 46 x 31 cm body contained only minor truncation artifacts that did not substantially affect the attenuation compensated SPECT image. Contamination of the transmission data from radionuclide within the patient can be substantially reduced with this system by increasing the resolution of the fan beam collimator. We conclude that long focal length, asymmetric fan beam collimation, combined with iterative reconstruction, offers a promising approach for fast sequential SPECT/TCT acquisition on a three-headed SPECT camera.« less
  • Single-photon emission-computed tomography (SPECT) imaging of deep brain structures is compromised by loss of photons due to attenuation. We have previously shown that a centrally peaked collimator sensitivity function can compensate for this phenomenon, increasing sensitivity over most of the brain. For dual-head instruments, parallel-hole collimators cannot provide variable sensitivity without simultaneously degrading spatial resolution near the center of the brain; this suggests the use of converging collimators. We have designed collimator pairs for dual-head SPECT systems to increase sensitivity, particularly in the center of the brain, and compared the new collimation approach to existing approaches on the basis ofmore » performance in estimating activity concentration of small structures at various locations in the brain. The collimator pairs we evaluated included a cone-beam collimator, for increased sensitivity, and a fan-beam collimator, for data sufficiency. We calculated projections of an ellipsoidal uniform background, with 0.9-cm-radius spherical lesions at several locations in the background. From these, we determined ideal signal-to-noise ratios (SNR{sub CRB}) for estimation of activity concentration within the spheres, based on the Cramer-Rao lower bound on variance. We also reconstructed, by an ordered-subset expectation-maximization (OS-EM) procedure, images of this phantom, as well as of the Zubal brain phantom, to allow visual assessment and to ensure that they were free of artifacts. The best of the collimator pairs evaluated comprised a cone-beam collimator with 20 cm focal length, for which the focal point is inside the brain, and a fan-beam collimator with 40 cm focal length. This pair yielded increased SNR{sub CRB}, compared to the parallel-parallel pair, throughout the imaging volume. The factor by which SNR{sub CRB} increased ranged from 1.1 at the most axially extreme location to 3.5 at the center. The gains in SNR{sub CRB} were relatively robust to mismatches between the center of the brain and the center of the imaging volume. Artifact-free reconstructions of simulated data acquired using this pair were obtained. Combining fan-beam and short-focusing cone-beam collimation should greatly improve dual-head brain SPECT imaging, especially for centrally located structures.« less
  • Fan-beam collimators are used in single photon emission computed tomography to improve the sensitivity for imaging of small organs. The disadvantage of fan-beam collimation is the truncation of projection data surrounding the organ of interest or, in those cases of imaging large patients, of the organ itself producing reconstruction artifacts. A spatially varying focal length fan-beam collimator has been proposed to eliminate the truncation problem and to maintain good sensitivity for the organ of interest. The collimator is constructed so that the shortest focal lengths are located at the center of the collimator and the longest focal length is locatedmore » at the periphery. The variation of the focal length can have various functional forms but in the authors work it is assumed to increase monotonically toward the edge of the collimator. The authors have derived a reconstruction algorithm for this type of fan-beam collimation. The algorithm is expressed as an infinite series of convolutions followed by one backprojection. However, simulations show that only a small number of N terms in the series are needed to obtain high quality reconstructions. The weighting and convolution are executed N times, then N convolved projections are summed up and one backprojection is performed to obtain the final reconstructed image. The algorithm was tested for two spatially varying focal length formulations. Through computer simulations it is found that if the focal length function is not smooth, a singular artifact is seen in the reconstruction, whereas for smooth functions, the reconstructions are free of artifacts.« less
  • In deriving algorithms to reconstruct single photon emission computed tomography (SPECT) projection data, it is important that the algorithm compensates for photon attenuation in order to obtain quantitative reconstruction results. A convolution backprojection algorithm was derived by Tretiak and Metz to reconstruct two-dimensional (2-D) transaxial slices from uniformly attenuated parallel-beam projections. Using transformation of coordinates, this algorithm can be modified to obtain a formulation useful to reconstruct uniformly attenuated fan-beam projections. Unlike that for parallel-beam projections, this formulation does not produce a filtered backprojection reconstruction algorithm but instead has a formulation that is an inverse integral operator with a spatiallymore » varying kernel. This algorithm thus requires more computation time than does the filtered backprojection reconstruction algorithm for the uniformly attenuated parallel-beam case. However, the fan-beam reconstructions demonstrate the same image quality as that of parallel-beam reconstructions.« less
  • The objective of this work is to increase system sensitivity in cardiac single-photon emission-computed tomography (SPECT) studies without increasing patient imaging time. For imaging the heart, convergent collimation offers the potential of increased sensitivity over that of parallel-hole collimation. However, if a cone-beam collimated gamma camera is rotated in a planar orbit, the projection data obtained are not complete. Two cone-beam collimators and one fan-beam collimator are used with a three-detector SPECT system. The combined cone-beam/fan-beam collimation provides a complete set of data for image reconstruction. The imaging geometry is evaluated using data acquired from phantom and patient studies. Formore » the Jaszazck cardiac torso phantom experiment, the combined cone-beam/fan-beam collimation provided 1.7 times greater sensitivity than standard parallel-hole collimation (low-energy high-resolution collimators). Also, phantom and patient comparison studies showed improved image quality. The combined cone-beam/fan-beam imaging geometry with appropriate weighting of the two data sets provides improved system sensitivity while measuring sufficient data for artifact free cardiac images.« less