Appl. Radiar. hf. Vol. 41, No. 10/l I, pp. 975-979, In!. J. Radial. Appl. Inslrum. Parr A Printed in Great Britain. All rights reserved
1990 Copyright
0883-2889/90 $3.00 + 0.00 0 1990 Pergamon Press plc
Scattering and Attenuation Correction in Emission Tomography in the Nuclear Industry D. J. GENTLE,’
‘Department
N. M. SPYROU,’ A. DHANI,’ I. G. HUTCHINSON’ and J. HUDDLESTONE’
of Physics, University 2Harwell Laboratory,
of Surrey, Guildford, Surrey GU2 SXH, U.K. and Harwell, Oxford OX11 ORA, U.K.
(Received July 1989; in revised form November
1989)
A multi-energetic y-ray source has been used in conjunction with titanium and water-bath phantoms to simulate the scattering and attenuation effects encountered in nuclear industrial applications. The effect of scattered photons on the image reconstructions has been investigated by the subtraction of varying fractions of the scattered counts from the photopeak counts. A dual energy attenuation correction also been applied to the data. In order to evaluate the effects of these corrections a “fidelity” factor been calculated.
Introduction
Experiments
and Background
Transmission and emission tomography are now established techniques for imaging internal body structure and the study of body function in the medical field. The success of these tomographic methods has led to the investigation of their suitability in industrial non-destructive testing resulting in a diversity of non-medical applications (Davis et al., 1986; Reimers et al., 1984). Our particular area of interest centres on the use of tomographic techniques in the examination of materials in the nuclear industry e.g. determining the spatial distribution of fission products within a fuel rod or the distribution of radionuclides in waste containers. In order to carry out simulation experiments a prototype scanning rig has been constructed. The design of the rig allows a certain amount of flexibility with regard to its use with a variety of detector, source and collimater configurations in both emission and transmission tomography (Sanders, 1982). Equivalent systems exist at both Harwell Laboratories and the University of Surrey. The scanning rig is controlled via a microcomputer and data is acquired through single or multichannel analysers. Reconstruction of the images is performed using a filtered backprojection program on the microcomputer though access to the University mainframe computer allows the use of more sophisticated computer codes for image processing and analysis.
has has
and Methods
To simulate the attenuation effects that would be experienced in nuclear industrial applications we used a “Se source in combination with titanium and water phantoms. 75Se emits y rays with the following energies and yields: 121 keV (15.8%), 136 keV (SS%), 265 keV (58%), 280 keV (25.9%) and 401 keV (11.6%). Titanium was chosen as a phantom material since it would attenuate the y-rays of 75Se to a similar extent as would a fuel pin attenuate y-rays from radionuclides of interest. The water bath phantom was also used for comparison in applications where attenuation was less severe, as for example in certain types of fuel storage containers for radioactive waste. Both sodium iodide (NaI(T1)) and high purity germanium (HPGe) detectors were used in the experiments. Hence comparisons could be made between results obtained by the high resolution HPGe detector and the poor resolution high efficiency scintillation detector, where it would be expected that scattered photons would contribute significantly to the full energy photopeak counts. With the HPGe detector the full energy photopeak counts will be isolated to a greater extent and the contribution of Compton scattered events to the counts in the area described by the full energy photopeak will be reduced. Data was acquired in two energy windows, one centered on the full energy photopeak and another directly below it to record scattered photons. The width of each energy window was a function of the 975
D. J.
976
GENTLEet al.
full width at half maximum (FWHM) and generally taken to be equal to the full width at one tenth maximum (FWTM). Two methods of scatter correction could be employed. The first requires the reconstruction from the raw data of both a scatter image and a full energy photopeak image. It has been shown that in medical emission tomography images can be improved by the subtraction of a fraction of the scattered image from the photopeak image. Empirically the optimum fraction has been found to be one half (Jasczcek et al., 1984). Alternatively one can subtract a fraction of the scattered counts from the full energy photopeak counts before reconstruction, this being the method used in this study. Correction for the scattering component becomes more important when dealing with multi-energetic y-ray spectra where the contribution from higher energies may also be significant. Attenuation correction was carried out using a dual energy correction method (Cline, 1972). This technique makes use of the multiple energy y-rays emitted by “Se and the differential absorption between two y-ray energies in the material. If we consider two different energy photons from the same radionuclide we can write Z, = (Z,)Oexp(-p,x)
(1)
Zz= (Z?& exp(-p,x)
(2)
This “fidelity” follows:
factor
~ = , _ F where
(Linfoot,
W(iJ)
1960) is defined
- D’(i,j))2
YLD’(i,j)2
as
(5)
is the pixel value in the “ideal” image, pixel value and the summation is over all pixels in the image. The titanium and water phantoms are shown in Figs 1 and 2 respectively. The concentration of “Se was the same in all the individual volumes. A multichannel analyser was used to collect the emitted spectrum, but only the counts in the regions of interest in the spectrum, i.e. the full energy photopeaks and associated scattering windows, were recorded because of the limited data storage capacity of the microcomputer and the time taken to transfer the data serially to it. The collimator size employed was of 2 mm diameter and a stepping interval of 1.5 mm was used to obtain projections of length 61.5 mm. Sixty projections were taken through 360”. D’(i,j)
D(i,j)is the reconstructed
Results and Discussion Tomographic images of the “Se activity distribution were reconstructed for each phantom (water and titanium) using both the NaI(T1) and HPGe
where I, and Z2 are the observed intensities, (I,), and (I,), are the emitted intensities, p, and p(2 are the attenuation coefficients at the energies 1 and 2 respectively and x is the average distance that both the y-rays have to travel in the medium before reaching the detector. Taking the ratio of these intensities and rearranging gives 1 X =- PI -& substituting
in equation
1
ln (Z, IZ, )ll [ (Z,lZ*) (1) therefore
(I,), = Z, exp{&ln[$$$]}.
gives (4)
This method can most effectively be employed on data collected by the HPGe detector since the energy resolution of the NaI(T1) detector does not allow the individual peaks of “Se to be resolved (e.g. the 265 and 280 keV y-ray lines and the 121 and 136 keV y-ray lines respectively are not separately resolved) and so average values of the attenuation coefficient over the broadened energy range have to be used. There are several methods described in the literature which attempt to evaluate image quality but no single method has gained general acceptance. In order to evaluate the effectiveness of the scatter and attenuation correction techniques a quantitative measure of the accuracy of the reconstructed image compared to the “ideal” image is adopted.
Fig. 1. Titanium phantom. Diameter, 52mm; hole diameters, 10, 5 and 2 mm.
Fig. 2. Water-bath phantom. Diameter, containers, all diameters,
57 mm; active liquid 6 mm.
Scattering and attenuation correction detectors. The full energy photopeak windows employed when a spectrum was obtained using the NaI(T1) detector were 121 and 136 keV combined, 265 and 280 keV combined and 401 keV. Single y-ray energies were used when the spectrum was obtained with the HPGe detector. The images were reconstructed with the raw data (i.e. counts in the full energy photopeak region) and also with data resulting from subtraction of a fraction of the scattered window counts from the full energy photopeak window counts. This fraction, SFS (scatter fraction subtracted) for example, took values of 0.25, 0.5, etc. In addition, images were also reconstructed from data which had scattered counts subtracted and an attenuation correction applied. The attenuation corrections were determined from combinations of pairs of y-ray energies. For each reconstruction a fidelity was calculated for the whole image and for the regions of interest where the sources of activity were located. A noise-to-signal ratio was also calculated for these images using the following equation (Budinger et al., 1978)
917
0.90
0.85
0.80
b ;; e 0.75 B. c zi 0 IL 0.70
0.65
(d) noise = 1.2 (total no of image elements)3’4 signal
0.60 0.25
x (total no of detected
events)m”2.
Figure 3 summarizes our results by showing the effect of scatter fraction correction on the fidelity factor of the image for the two detectors and the two phantoms at the y-ray energies of 121 keV and 136 keV. Note that for the results obtained with the NaI(T1) detector the combined energies are included in the energy window. The fidelity factor was calculated for the region in the images where the “Se activity was known to be located. The variation of the fidelity factor with an increasing proportion of the scattered counts subtracted from the photopeak window shows an increase in the value of the former with increasing scatter fraction subtracted, although the improvement in the fidelity is not significant beyond a value of about 0.5 SFS for the NaI(T1) detector for either of the phantoms. In the case of the HPGe detector the fidelity factor decreases when the value of the SFS is greater than 0.5 in water for both the 121 keV and the 136 keV y-ray energies. Beyond the value of 0.5 the SFS over-compensates for the contribution of scattered photons to the full energy photopeak window and the image deteriorates. The fidelity factor for the images reconstructed using the 136 keV y-ray energy data is superior to that obtained from the 121 keV y-ray energy window because of better counting statistics and a smaller background underlying the full energy photopeak. This also explains why there is a significantly greater improvement when scattered counts are subtracted from the full energy photopeak area of the 121 keV y line and also its rapid deterioration when Door statistics come into nlav. 1
1
i
0.50
0.75
Fig. 3. Graph showing variation of the fidelity factor with fraction of scatter counts recorded, removed from the peak counts. (a) NaI(T1) detector water-bath phantom (121 + 136) keV. (b) NaI(TI) detector titanium phantom (121 + 136) keV. (c) HPGe detector water-bath phantom 136 keV. (d) HPGe detector water-bath phantom 121 keV. (e) HPGe detector titanium phantom 121 keV.
The value of 0.5 for SFS, up to which the fidelity factor increases for both NaI(T1) and HPGe detectors in water, is one that has been adopted in many clinical applications (Jasczcek et al., 1984) for 99mTc (140 keV) imaging. No improvement in the fidelity factor is discernible for the reconstructed images when the SFS is increased in the case of” Se (12 1 keV) distributed in the titanium phantom using the HPGe detector. In this higher atomic number matrix the effect of the absorption of photons is considerably more significant than the relative contribution of scattered photons to the full energy photopeak and the implication is that no scatter correction needs to be performed with a high resolution detector in these circumstances. It is interesting to note that when the attenuation correction, using the dual energy method, was applied to the data with a value of 0.5 SFS and reconstructed, the fidelity factor decreased for all cases. On examination of the images obtained, this was attributed to the over suppression of the source on the periphery of the object. However the variation in the fidelity factor was dependent on the two energies being used for calculating the attenuation correction. Despite the poor statistics associated with the 401 keV y line, the fidelity factor obtained using
D. J. GENTLE et al.
978
Fig. 4. Reconstructed image of water-bath phantom. equal to 0.5 and using attenuation
NaI(TI) detector (121 + 136) keV data correction by (265 + 280) keV.
with SFS
4.0
Fig. 5. Reconstructed
image of water-bath phantom. HPGe detector 136 keV data with SFS equal and attenuation correction using 265 keV data.
to 0.5
1.0
0.0 0.6 0.4 0.2 0
4.0
Fig. 6. Reconstructed
image of water-bath phantom. Nal(T1) detector (121 + 136) keV data equal to 0.5 and attenuation correction using 401 keV data.
with SFS
Scattering and attenuation correction this energy to correct
for attenuation
of the 121 keV
and 136 keV y-rays was higher than that resulting from the use of the 265 keV and 280 keV y-rays for the same purpose. Figures 4 and 5 show the attenuation corrected images in water with SFS value of 0.5 for the NaI(T1) and HPGe detectors using the (265 + 280) keV energy window and the 265 keV y-ray energy, respectively. For comparison Fig. 6 shows the attenuation corrected image obtained using the NaI(T1) detector for the same SFS value and 401 keV energy window. The improvement in the image is clearly visible and is confirmed by a significantly higher fidelity value.
Conclusions The fidelity factor has allowed us to evaluate the effect on a tomographic image of the scatter contribution to the full energy photopeak window used for obtaining the data for reconstruction. We have shown that a value of 0.5 SFS is optimum in this respect. However we have also ascertained that when the radionuclide is distributed in a higher atomic number matrix and a high resolution detector is used, the need for scatter subtraction is greatly reduced.
979
The changes in the value of the fidelity factor were found to be useful in evaluating the method of attenuation correction employed and confirming that the dual energy correction method is best applied when the energy separation between the y-rays of interest is greatest. Acknowledgements-We
would like to thank the Science and Engineering Research Council and Harwell Laboratory for the support through a CASE research studentship awarded to one of the authors (DJG).
References Budinger T. F., Derenzo S. E., Greenberg W. L., Gullherg G. T. and Huesman R. H. (1978) J. Nuci. Med. 19, 309. Cline J. (1972) Aerojet Nuclear Company Report 1055 Idaho Falls, U.S.A. Davies G., Spyrou N. M., Hutchinson I. G. and Huddleston J. (1986) Nucl. Instrum. Methodr A242, 615. Jasczcak R. J., Greer K. L., Floyd C. E., Harris C. C. and Coleman R. E. (1984) J. Nucl. Med. 25, 897. Linfoot E. H. (1960) Fourier Methods in Oprical Image Evaluafion. Focal Press, London. Reimers P., Goebbels J., Weise H. P. and Wilding K. (1984) Nucl. Insrrum. Methods 221. 201. Sanders J. (1986) Ph.D. Thesis, University of Surrey.
1990 Copyright
0883-2889/90 $3.00 + 0.00 0 1990 Pergamon Press plc
Scattering and Attenuation Correction in Emission Tomography in the Nuclear Industry D. J. GENTLE,’
‘Department
N. M. SPYROU,’ A. DHANI,’ I. G. HUTCHINSON’ and J. HUDDLESTONE’
of Physics, University 2Harwell Laboratory,
of Surrey, Guildford, Surrey GU2 SXH, U.K. and Harwell, Oxford OX11 ORA, U.K.
(Received July 1989; in revised form November
1989)
A multi-energetic y-ray source has been used in conjunction with titanium and water-bath phantoms to simulate the scattering and attenuation effects encountered in nuclear industrial applications. The effect of scattered photons on the image reconstructions has been investigated by the subtraction of varying fractions of the scattered counts from the photopeak counts. A dual energy attenuation correction also been applied to the data. In order to evaluate the effects of these corrections a “fidelity” factor been calculated.
Introduction
Experiments
and Background
Transmission and emission tomography are now established techniques for imaging internal body structure and the study of body function in the medical field. The success of these tomographic methods has led to the investigation of their suitability in industrial non-destructive testing resulting in a diversity of non-medical applications (Davis et al., 1986; Reimers et al., 1984). Our particular area of interest centres on the use of tomographic techniques in the examination of materials in the nuclear industry e.g. determining the spatial distribution of fission products within a fuel rod or the distribution of radionuclides in waste containers. In order to carry out simulation experiments a prototype scanning rig has been constructed. The design of the rig allows a certain amount of flexibility with regard to its use with a variety of detector, source and collimater configurations in both emission and transmission tomography (Sanders, 1982). Equivalent systems exist at both Harwell Laboratories and the University of Surrey. The scanning rig is controlled via a microcomputer and data is acquired through single or multichannel analysers. Reconstruction of the images is performed using a filtered backprojection program on the microcomputer though access to the University mainframe computer allows the use of more sophisticated computer codes for image processing and analysis.
has has
and Methods
To simulate the attenuation effects that would be experienced in nuclear industrial applications we used a “Se source in combination with titanium and water phantoms. 75Se emits y rays with the following energies and yields: 121 keV (15.8%), 136 keV (SS%), 265 keV (58%), 280 keV (25.9%) and 401 keV (11.6%). Titanium was chosen as a phantom material since it would attenuate the y-rays of 75Se to a similar extent as would a fuel pin attenuate y-rays from radionuclides of interest. The water bath phantom was also used for comparison in applications where attenuation was less severe, as for example in certain types of fuel storage containers for radioactive waste. Both sodium iodide (NaI(T1)) and high purity germanium (HPGe) detectors were used in the experiments. Hence comparisons could be made between results obtained by the high resolution HPGe detector and the poor resolution high efficiency scintillation detector, where it would be expected that scattered photons would contribute significantly to the full energy photopeak counts. With the HPGe detector the full energy photopeak counts will be isolated to a greater extent and the contribution of Compton scattered events to the counts in the area described by the full energy photopeak will be reduced. Data was acquired in two energy windows, one centered on the full energy photopeak and another directly below it to record scattered photons. The width of each energy window was a function of the 975
D. J.
976
GENTLEet al.
full width at half maximum (FWHM) and generally taken to be equal to the full width at one tenth maximum (FWTM). Two methods of scatter correction could be employed. The first requires the reconstruction from the raw data of both a scatter image and a full energy photopeak image. It has been shown that in medical emission tomography images can be improved by the subtraction of a fraction of the scattered image from the photopeak image. Empirically the optimum fraction has been found to be one half (Jasczcek et al., 1984). Alternatively one can subtract a fraction of the scattered counts from the full energy photopeak counts before reconstruction, this being the method used in this study. Correction for the scattering component becomes more important when dealing with multi-energetic y-ray spectra where the contribution from higher energies may also be significant. Attenuation correction was carried out using a dual energy correction method (Cline, 1972). This technique makes use of the multiple energy y-rays emitted by “Se and the differential absorption between two y-ray energies in the material. If we consider two different energy photons from the same radionuclide we can write Z, = (Z,)Oexp(-p,x)
(1)
Zz= (Z?& exp(-p,x)
(2)
This “fidelity” follows:
factor
~ = , _ F where
(Linfoot,
W(iJ)
1960) is defined
- D’(i,j))2
YLD’(i,j)2
as
(5)
is the pixel value in the “ideal” image, pixel value and the summation is over all pixels in the image. The titanium and water phantoms are shown in Figs 1 and 2 respectively. The concentration of “Se was the same in all the individual volumes. A multichannel analyser was used to collect the emitted spectrum, but only the counts in the regions of interest in the spectrum, i.e. the full energy photopeaks and associated scattering windows, were recorded because of the limited data storage capacity of the microcomputer and the time taken to transfer the data serially to it. The collimator size employed was of 2 mm diameter and a stepping interval of 1.5 mm was used to obtain projections of length 61.5 mm. Sixty projections were taken through 360”. D’(i,j)
D(i,j)is the reconstructed
Results and Discussion Tomographic images of the “Se activity distribution were reconstructed for each phantom (water and titanium) using both the NaI(T1) and HPGe
where I, and Z2 are the observed intensities, (I,), and (I,), are the emitted intensities, p, and p(2 are the attenuation coefficients at the energies 1 and 2 respectively and x is the average distance that both the y-rays have to travel in the medium before reaching the detector. Taking the ratio of these intensities and rearranging gives 1 X =- PI -& substituting
in equation
1
ln (Z, IZ, )ll [ (Z,lZ*) (1) therefore
(I,), = Z, exp{&ln[$$$]}.
gives (4)
This method can most effectively be employed on data collected by the HPGe detector since the energy resolution of the NaI(T1) detector does not allow the individual peaks of “Se to be resolved (e.g. the 265 and 280 keV y-ray lines and the 121 and 136 keV y-ray lines respectively are not separately resolved) and so average values of the attenuation coefficient over the broadened energy range have to be used. There are several methods described in the literature which attempt to evaluate image quality but no single method has gained general acceptance. In order to evaluate the effectiveness of the scatter and attenuation correction techniques a quantitative measure of the accuracy of the reconstructed image compared to the “ideal” image is adopted.
Fig. 1. Titanium phantom. Diameter, 52mm; hole diameters, 10, 5 and 2 mm.
Fig. 2. Water-bath phantom. Diameter, containers, all diameters,
57 mm; active liquid 6 mm.
Scattering and attenuation correction detectors. The full energy photopeak windows employed when a spectrum was obtained using the NaI(T1) detector were 121 and 136 keV combined, 265 and 280 keV combined and 401 keV. Single y-ray energies were used when the spectrum was obtained with the HPGe detector. The images were reconstructed with the raw data (i.e. counts in the full energy photopeak region) and also with data resulting from subtraction of a fraction of the scattered window counts from the full energy photopeak window counts. This fraction, SFS (scatter fraction subtracted) for example, took values of 0.25, 0.5, etc. In addition, images were also reconstructed from data which had scattered counts subtracted and an attenuation correction applied. The attenuation corrections were determined from combinations of pairs of y-ray energies. For each reconstruction a fidelity was calculated for the whole image and for the regions of interest where the sources of activity were located. A noise-to-signal ratio was also calculated for these images using the following equation (Budinger et al., 1978)
917
0.90
0.85
0.80
b ;; e 0.75 B. c zi 0 IL 0.70
0.65
(d) noise = 1.2 (total no of image elements)3’4 signal
0.60 0.25
x (total no of detected
events)m”2.
Figure 3 summarizes our results by showing the effect of scatter fraction correction on the fidelity factor of the image for the two detectors and the two phantoms at the y-ray energies of 121 keV and 136 keV. Note that for the results obtained with the NaI(T1) detector the combined energies are included in the energy window. The fidelity factor was calculated for the region in the images where the “Se activity was known to be located. The variation of the fidelity factor with an increasing proportion of the scattered counts subtracted from the photopeak window shows an increase in the value of the former with increasing scatter fraction subtracted, although the improvement in the fidelity is not significant beyond a value of about 0.5 SFS for the NaI(T1) detector for either of the phantoms. In the case of the HPGe detector the fidelity factor decreases when the value of the SFS is greater than 0.5 in water for both the 121 keV and the 136 keV y-ray energies. Beyond the value of 0.5 the SFS over-compensates for the contribution of scattered photons to the full energy photopeak window and the image deteriorates. The fidelity factor for the images reconstructed using the 136 keV y-ray energy data is superior to that obtained from the 121 keV y-ray energy window because of better counting statistics and a smaller background underlying the full energy photopeak. This also explains why there is a significantly greater improvement when scattered counts are subtracted from the full energy photopeak area of the 121 keV y line and also its rapid deterioration when Door statistics come into nlav. 1
1
i
0.50
0.75
Fig. 3. Graph showing variation of the fidelity factor with fraction of scatter counts recorded, removed from the peak counts. (a) NaI(T1) detector water-bath phantom (121 + 136) keV. (b) NaI(TI) detector titanium phantom (121 + 136) keV. (c) HPGe detector water-bath phantom 136 keV. (d) HPGe detector water-bath phantom 121 keV. (e) HPGe detector titanium phantom 121 keV.
The value of 0.5 for SFS, up to which the fidelity factor increases for both NaI(T1) and HPGe detectors in water, is one that has been adopted in many clinical applications (Jasczcek et al., 1984) for 99mTc (140 keV) imaging. No improvement in the fidelity factor is discernible for the reconstructed images when the SFS is increased in the case of” Se (12 1 keV) distributed in the titanium phantom using the HPGe detector. In this higher atomic number matrix the effect of the absorption of photons is considerably more significant than the relative contribution of scattered photons to the full energy photopeak and the implication is that no scatter correction needs to be performed with a high resolution detector in these circumstances. It is interesting to note that when the attenuation correction, using the dual energy method, was applied to the data with a value of 0.5 SFS and reconstructed, the fidelity factor decreased for all cases. On examination of the images obtained, this was attributed to the over suppression of the source on the periphery of the object. However the variation in the fidelity factor was dependent on the two energies being used for calculating the attenuation correction. Despite the poor statistics associated with the 401 keV y line, the fidelity factor obtained using
D. J. GENTLE et al.
978
Fig. 4. Reconstructed image of water-bath phantom. equal to 0.5 and using attenuation
NaI(TI) detector (121 + 136) keV data correction by (265 + 280) keV.
with SFS
4.0
Fig. 5. Reconstructed
image of water-bath phantom. HPGe detector 136 keV data with SFS equal and attenuation correction using 265 keV data.
to 0.5
1.0
0.0 0.6 0.4 0.2 0
4.0
Fig. 6. Reconstructed
image of water-bath phantom. Nal(T1) detector (121 + 136) keV data equal to 0.5 and attenuation correction using 401 keV data.
with SFS
Scattering and attenuation correction this energy to correct
for attenuation
of the 121 keV
and 136 keV y-rays was higher than that resulting from the use of the 265 keV and 280 keV y-rays for the same purpose. Figures 4 and 5 show the attenuation corrected images in water with SFS value of 0.5 for the NaI(T1) and HPGe detectors using the (265 + 280) keV energy window and the 265 keV y-ray energy, respectively. For comparison Fig. 6 shows the attenuation corrected image obtained using the NaI(T1) detector for the same SFS value and 401 keV energy window. The improvement in the image is clearly visible and is confirmed by a significantly higher fidelity value.
Conclusions The fidelity factor has allowed us to evaluate the effect on a tomographic image of the scatter contribution to the full energy photopeak window used for obtaining the data for reconstruction. We have shown that a value of 0.5 SFS is optimum in this respect. However we have also ascertained that when the radionuclide is distributed in a higher atomic number matrix and a high resolution detector is used, the need for scatter subtraction is greatly reduced.
979
The changes in the value of the fidelity factor were found to be useful in evaluating the method of attenuation correction employed and confirming that the dual energy correction method is best applied when the energy separation between the y-rays of interest is greatest. Acknowledgements-We
would like to thank the Science and Engineering Research Council and Harwell Laboratory for the support through a CASE research studentship awarded to one of the authors (DJG).
References Budinger T. F., Derenzo S. E., Greenberg W. L., Gullherg G. T. and Huesman R. H. (1978) J. Nuci. Med. 19, 309. Cline J. (1972) Aerojet Nuclear Company Report 1055 Idaho Falls, U.S.A. Davies G., Spyrou N. M., Hutchinson I. G. and Huddleston J. (1986) Nucl. Instrum. Methodr A242, 615. Jasczcak R. J., Greer K. L., Floyd C. E., Harris C. C. and Coleman R. E. (1984) J. Nucl. Med. 25, 897. Linfoot E. H. (1960) Fourier Methods in Oprical Image Evaluafion. Focal Press, London. Reimers P., Goebbels J., Weise H. P. and Wilding K. (1984) Nucl. Insrrum. Methods 221. 201. Sanders J. (1986) Ph.D. Thesis, University of Surrey.
