The Compton camera is used for imaging the distributions of gamma ray direction in a gamma ray telescope for astrophysics and for imaging radioisotope distributions in nuclear medicine without the need for collimators. The integration of gamma rays on a cone is measured with the camera, so that some sort of inversion method is needed. Parra found an analytical inversion algorithm based on spherical harmonics expansion of projection data. His algorithm is applicable to the full set of projection data. In this paper, six possible reconstruction algorithms that allow image reconstruction from projections with a finite range of scattering angles are investigated. Four algorithms have instability problems and two others are practical. However, the variance of the reconstructed image diverges in these two cases, so that window functions are introduced with which the variance becomes finite at a cost of spatial resolution. These two algorithms are compared in terms of variance. The algorithm based on the inversion of the summed back-projection is superior to the algorithm based on the inversion of the summed projection.
Compton cameras have been developed for use in gamma-ray astronomy and nuclear medicine. Their defining merit is that they do not need collimators; however, on the demerit side, they need inversion procedures for image reconstruction, since a measured datum is proportional to the integration of incident gamma rays along a cone surface with the same Compton scattering angle. First, an iteration method was adopted for this task. Later, analytical methods were found under restricted conditions. Parra (2000 IEEE Trans. Nucl. Sci. 47 1543-50) deduced a purely analytical reconstruction algorithm for a complete set of scattering-projection data that include data at all the scattering angles. Tomitani and Hirasawa (2002 Phys. Med. Biol. 47 2129-45) found that by making a slight modification, Parra's algorithm could be extended to the scattering-projection data in limited scattering angles. However, their algorithm neglected the effects of practical problems that cause the degradation of spatial resolution. Sources of degradation were identified as noise in the energy signal of their front detector and the Doppler effect in the scattering process. In this paper, we first analyse the effects of these sources on the angular resolution of the scattering-projection data and then present a revised reconstruction algorithm in which these two factors are incorporated. Simulation studies on digital phantoms reveal that the algorithm can reconstruct images even when these two factors are included.
Since the integration of gamma-rays on a cone is measured with Compton cameras, some sort of image reconstruction method is necessary. Parra developed an analytical reconstruction algorithm based on a spherical harmonics expansion of projection data that covers the entire scattering-angle range. The measurable scattering angle range is limited due to the electrical noise of the detector and to the finite detector configuration. In this article, a reconstruction algorithm from the projection of a limited scattering angle range is presented. The algorithm was based on an inversion of the summed backprojection. The variance of the reconstructed image of a uniform source was calculated and found proportional to the third power of the number of expansion terms. Thus, window functions were introduced to suppress noise at the cost of spatial resolution. The dependency of the spatial resolution on the number of expansion terms was analyzed. From this, the dependency of the variance on the spatial resolution was found to be variance (spatial resolution) 3 .
In Compton cameras, the measured scattering angle is associated with an uncertainty which becomes larger as the incident gamma-ray energy decreases. Since this uncertainty degrades the spatial resolution of reconstructed images, Hirasawa and Tomitani (2003 Phys. Med. Biol. 48 1009-26) previously revised their analytical reconstruction algorithm to compensate for it. As the new algorithm improved the spatial resolution in effect, they expected an enhancement of the statistical noise. In this paper, the effect of this compensation has been analysed in view of spatial resolution (the FWHM of the noise-free reconstructed image for a point source distribution), statistical noise (the relative standard deviation of reconstructed images for an isotropic source distribution) and image quality (the roughness of reconstructed images for a phantom). The results describe not only the effect of the compensation, but also the relation between the statistical noise and three parameters, i.e., the incident gamma-ray energy, the spatial resolution and the measured total event numbers, in reconstruction with compensation. This relation should be taken into account for the design of Compton cameras with good quality images, i.e., useful image, output.
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