Recent advances in imaging techniques and position-sensitive gamma-ray detectors

have made feasible hard x-ray and gamma-ray telescopes with arc-second resolution [ 1].

Above an energy of 100 keV, past instrumentation has been limited to a typical angular

resolution of a few degrees. A gamma-ray imaging device with 1 arc-second resolution

would be a dramatic improvement over conventional, non-imaging instrumentation

and have substantial new capabilities for observation of astrophysical gamma-ray sources.

The arc-second gamma-ray imager is based on the Fourier transform imaging technique

[2]. We briefly describe Fourier transform imaging and its application to hard x-ray

and gamma-ray imaging. This description is followed by an analysis of Fourier transform

imaging in the statistics limited regime. Computer simulations and laboratory

demonstrations of practical gamma-ray imaging systems are presented

Hurford, G. J.

Megeath, S. Thomas

Palmer, David

Prince, Thomas A.

Caltech Authors - Main

MD3-l 
Statistical Limits of Fourier Transform Imaging 
in the -y-ray Energy Range 
Gordon J. Hurford, S. Thomas Megeath, David Palmer, and Thomas A. Prince 
California Institute of Technology 
Pasadena, California 91125 
Summary 
Recent advances in imaging techniques and position-sensitive -y-ray detectors 
have made feasible hard x-ray and -y-ray telescopes with arc-second resolution [ 1]. 
Above an energy of 100 keY, past instrumentation has been limited to a typical angu-
lar resolution of a few degrees. A -y-ray imaging device with 1 arc-second resolution 
would be a dramatic improvement over conventional, non-imaging instrumentation 
and have substantial new capabilities for observation of astrophysical -y-ray sources. 
The arc-second -y-ray imager is based on the Fourier transform imaging technique 
[2]. We briefly describe Fourier transform imaging and its application to hard x-ray 
and -y-ray imaging. This description is followed by an analysis of Fourier transform 
imaging in the statistics limited regime. Computer simulations and laboratory 
demonstrations of practical -y-ray imaging systems are presented. 
1. Fourier transform Imaging at Hard X-ray and {ray :Energies 
In recent years, the first imaging -y-ray detectors have been developed for astro-
nomical observations in the energy range 50 keY to 10 MeV. These detectors have 
used coded-aperture imaging techniques to achieve sub-degree angular resolution 
[3]. However, in coded aperture-systems the angular resolution is limited. by the 
practically achievable spatial resolution of current photon detectors which is about 
1 em FWHM for 1 MeV -y-rays [ 4]. For typical coded-aperture systems, the angular 
resolution is "'tan-1(flx/L), where flx is the spatial resolution of the detectors and Lis 
the spacing between detector and coded aperture mask. 
Fourier transform imaging circumvents the limitation imposed by the spatial 
resolution of the detectors. In this technique [2], a position-sensitive -y-ray detector 
views a source field through a pair of widely separated collimator grids. The collima-
tor grids consist of parallel slits with a certain orientation 'I) and a certain slit spac-
ings (see Figure 1). The top and bottom grids differ slightly in either '1), or s, or both. 
The characteristic modulation pattern arises from this small difference in top and 
bottom grids . Each pair of collimator grids allows the measurement of the phase and 
amplitude of a particular Fourier component of the source angular distribution. In 
analogy to radio astronomy, each pair of collimators provides one "baseline" for 
measurement of the angular source distribution. The phase and amplitude informa-
tion from the individual -y-ray measurements is then combined to form images using 
algorithms identical to those of radio astronomy. 
Practical implementation of the Fourier transform technique at -y-ray energies 
requires large-area position sensitive -y-ray detectors . We have been active in the 
develo~ment of such detectors [ 4] . A FWHM position resolution of 0. 7 em over 
800 em was obtained, making it suitable for modulation detection techniques. We 
are currently collecting data with a position-sensitive Nai scintillation camera and a 
set of 7 pairs of Fourier transform grids. Modulation patterns and images obtained 
from this laboratory demonstration will be discussed. 
45 
EXTENDED SOURCE ... 
(FLARING ARCH 
ON THE SUN) 
MD3-2 
FOURIER CAMERA 
~~~ D/1 § ~ 
L~~~H 1 
SECOND 11/// ~ 
GRID+ 
DETECTOR 
PLANE 
FOURIER 
PROCESSING 
RECONSTRUCTED 
IMAGE 
2. Statistical Limits in Fourier 'l."ransform. Imaging 
Figure 1. 
We are in the process of a detailed analysis of the statistical limits of Fourier 
transform imaging. Whereas in multiple baseline radio astronomy the quantum limit 
is never approached, in -y-ray astronomy the low photon fiux and high background 
counting rates require a careful statistical analysis of imaging performance. 
To analyze these problems, consider a single point source. The distribution of 
counts in the detector can be approximated by: 
N(.,j)= 2:P ( 1 + Acos (.,j-9')) 
where 1) is the position on the detector plane, A is the signal to noise ratio, N is the 
total number of counts, N(-6) is the number of counts per unit length on the detec-
tor, P is length of one radian of modulation pattern, and 9' is the angular position of 
the source multiplied by a scaling constant. Now let us consider the Fourier integral 
of image intensity with respect to position: 
" V=P f N(-6)e''~d.-6 
-1r 
It we graph this on the complex plane, we tL."ld the complex exponential argument of 
Vis the phase position of the peak and its magnitude is one-half the total number of 
counts times the signal-to-noise ratio. If we plot the whole chain of events (i.e. plot 
the contribution to Von an event-by-event basis), the plot gives a random walk (Fig-
ure 2). 
46 
Random Walk 
-100 -50 
100 
50 
>-
c... 
ctJ 
c 
·r-1 0 
Cl 
ctJ 
E 
H 
-50 
-100 
-100 -50 
MD3-3 
of Fourier Transform 
0 50 100 
0 
Real 
50 
100 
50 
0 
-50 
-100 
100 
Figure 2. 
From the distribution of N(19-) and the random walk of V, the following equations 
for the standard deviation of rp and A can be derived for large N. 
u(rp)= .1_-./{ _g_) u(A)=..J{ £.(1- A2 )) A N' N 2 
The wedge shaped region in Figure 2 gives a(rp) and the concentric circles form the 
region for a(A). Each of these regions has a 68% confidence of the actual value being 
located in the region. 
The analysis summarized above is applicable to a single collimator grid pair and 
assumes N large. We are currently analyzing the statistical errors in combining the 
measurements of multiple collimator grid pairs to determine the angular position of 
a point source in one and two dimensions. We will also treat the problem of errors in 
resolving two point sources . 
Because basic information from a Fourier transform -y-ray telescope (i.e . phase 
and amplitude) is the same as that collected from multiple baseline radio interferom-
eters, we have applied software algorithms developed for radio astronomy to simu-
lated -y-ray measurements. We will present the results of these simulations with 
emphasis on the case of low statistics. These simulations will address the question of 
the number of collimator grid pairs needed to achieve a given image accuracy. 
47 
MD3-4 
References 
(1] C. J . Crannell. G. J . Hurford. L. E. Orwig and T. A Prince, "A Fourier Transform 
Telescope for Sub-arcsecond Imaging of X-rays and Gamma Rays", to be pub-
lished in Proc. of the Soc. of .Fhoto-Optical Elngi:neers, (1986). 
(2] G. J. Hurford and H.S. Hudson. "Fourier-Transform Imaging for X-Ray 
Astronomy" ,(BBS0#0180).(UCSD-SP-79-27), (1979) . 
(3] W. E. Althouse et al. . "A Balloon-Borne Imaging Gamma-Ray Telescope", Proc. of 
the 19th Int. Cosmic Ra.y ConJ .. 3, 299 (1985) . 
[ 4] W.R. Cook. M. Finger, and T.A. Prince. "A Thick Anger Camera for Gamma- Ray 
Astronomy" , IEEE 'Pransactions on Nuclear SciBnce, NS-32, 129 (1985) . 
48 


Statistical Limits of Fourier Transform Imaging in the Gamma-ray Energy Range

MD3-l 
Statistical Limits of Fourier Transform Imaging 
in the -y-ray Energy Range 
Gordon J. Hurford, S. Thomas Megeath, David Palmer, and Thomas A. Prince 
California Institute of Technology 
Pasadena, California 91125 
Summary 
Recent advances in imaging techniques and position-sensitive -y-ray detectors 
have made feasible hard x-ray and -y-ray telescopes with arc-second resolution [ 1]. 
Above an energy of 100 keY, past instrumentation has been limited to a typical angu-
lar resolution of a few degrees. A -y-ray imaging device with 1 arc-second resolution 
would be a dramatic improvement over conventional, non-imaging instrumentation 
and have substantial new capabilities for observation of astrophysical -y-ray sources. 
The arc-second -y-ray imager is based on the Fourier transform imaging technique 
[2]. We briefly describe Fourier transform imaging and its application to hard x-ray 
and -y-ray imaging. This description is followed by an analysis of Fourier transform 
imaging in the statistics limited regime. Computer simulations and laboratory 
demonstrations of practical -y-ray imaging systems are presented. 
1. Fourier transform Imaging at Hard X-ray and {ray :Energies 
In recent years, the first imaging -y-ray detectors have been developed for astro-
nomical observations in the energy range 50 keY to 10 MeV. These detectors have 
used coded-aperture imaging techniques to achieve sub-degree angular resolution 
[3]. However, in coded aperture-systems the angular resolution is limited. by the 
practically achievable spatial resolution of current photon detectors which is about 
1 em FWHM for 1 MeV -y-rays [ 4]. For typical coded-aperture systems, the angular 
resolution is "'tan-1(flx/L), where flx is the spatial resolution of the detectors and Lis 
the spacing between detector and coded aperture mask. 
Fourier transform imaging circumvents the limitation imposed by the spatial 
resolution of the detectors. In this technique [2], a position-sensitive -y-ray detector 
views a source field through a pair of widely separated collimator grids. The collima-
tor grids consist of parallel slits with a certain orientation 'I) and a certain slit spac-
ings (see Figure 1). The top and bottom grids differ slightly in either '1), or s, or both. 
The characteristic modulation pattern arises from this small difference in top and 
bottom grids . Each pair of collimator grids allows the measurement of the phase and 
amplitude of a particular Fourier component of the source angular distribution. In 
analogy to radio astronomy, each pair of collimators provides one "baseline" for 
measurement of the angular source distribution. The phase and amplitude informa-
tion from the individual -y-ray measurements is then combined to form images using 
algorithms identical to those of radio astronomy. 
Practical implementation of the Fourier transform technique at -y-ray energies 
requires large-area position sensitive -y-ray detectors . We have been active in the 
develo~ment of such detectors [ 4] . A FWHM position resolution of 0. 7 em over 
800 em was obtained, making it suitable for modulation detection techniques. We 
are currently collecting data with a position-sensitive Nai scintillation camera and a 
set of 7 pairs of Fourier transform grids. Modulation patterns and images obtained 
from this laboratory demonstration will be discussed. 
45 
EXTENDED SOURCE ... 
(FLARING ARCH 
ON THE SUN) 
MD3-2 
FOURIER CAMERA 
~~~ D/1 § ~ 
L~~~H 1 
SECOND 11/// ~ 
GRID+ 
DETECTOR 
PLANE 
FOURIER 
PROCESSING 
RECONSTRUCTED 
IMAGE 
2. Statistical Limits in Fourier 'l."ransform. Imaging 
Figure 1. 
We are in the process of a detailed analysis of the statistical limits of Fourier 
transform imaging. Whereas in multiple baseline radio astronomy the quantum limit 
is never approached, in -y-ray astronomy the low photon fiux and high background 
counting rates require a careful statistical analysis of imaging performance. 
To analyze these problems, consider a single point source. The distribution of 
counts in the detector can be approximated by: 
N(.,j)= 2:P ( 1 + Acos (.,j-9')) 
where 1) is the position on the detector plane, A is the signal to noise ratio, N is the 
total number of counts, N(-6) is the number of counts per unit length on the detec-
tor, P is length of one radian of modulation pattern, and 9' is the angular position of 
the source multiplied by a scaling constant. Now let us consider the Fourier integral 
of image intensity with respect to position: 
" V=P f N(-6)e''~d.-6 
-1r 
It we graph this on the complex plane, we tL."ld the complex exponential argument of 
Vis the phase position of the peak and its magnitude is one-half the total number of 
counts times the signal-to-noise ratio. If we plot the whole chain of events (i.e. plot 
the contribution to Von an event-by-event basis), the plot gives a random walk (Fig-
ure 2). 
46 
Random Walk 
-100 -50 
100 
50 
>-
c... 
ctJ 
c 
·r-1 0 
Cl 
ctJ 
E 
H 
-50 
-100 
-100 -50 
MD3-3 
of Fourier Transform 
0 50 100 
0 
Real 
50 
100 
50 
0 
-50 
-100 
100 
Figure 2. 
From the distribution of N(19-) and the random walk of V, the following equations 
for the standard deviation of rp and A can be derived for large N. 
u(rp)= .1_-./{ _g_) u(A)=..J{ £.(1- A
2 
)) 
A N' N 2 
The wedge shaped region in Figure 2 gives a(rp) and the concentric circles form the 
region for a(A). Each of these regions has a 68% confidence of the actual value being 
located in the region. 
The analysis summarized above is applicable to a single collimator grid pair and 
assumes N large. We are currently analyzing the statistical errors in combining the 
measurements of multiple collimator grid pairs to determine the angular position of 
a point source in one and two dimensions. We will also treat the problem of errors in 
resolving two point sources . 
Because basic information from a Fourier transform -y-ray telescope (i.e . phase 
and amplitude) is the same as that collected from multiple baseline radio interferom-
eters, we have applied software algorithms developed for radio astronomy to simu-
lated -y-ray measurements. We will present the results of these simulations with 
emphasis on the case of low statistics. These simulations will address the question of 
the number of collimator grid pairs needed to achieve a given image accuracy. 
47 
MD3-4 
References 
(1] C. J . Crannell. G. J . Hurford. L. E. Orwig and T. A Prince, "A Fourier Transform 
Telescope for Sub-arcsecond Imaging of X-rays and Gamma Rays", to be pub-
lished in Proc. of the Soc. of .Fhoto-Optical Elngi:neers, (1986). 
(2] G. J. Hurford and H.S. Hudson. "Fourier-Transform Imaging for X-Ray 
Astronomy" ,(BBS0#0180).(UCSD-SP-79-27), (1979) . 
(3] W. E. Althouse et al. . "A Balloon-Borne Imaging Gamma-Ray Telescope", Proc. of 
the 19th Int. Cosmic Ra.y ConJ .. 3, 299 (1985) . 
[ 4] W.R. Cook. M. Finger, and T.A. Prince. "A Thick Anger Camera for Gamma- Ray 
Astronomy" , IEEE 'Pransactions on Nuclear SciBnce, NS-32, 129 (1985) . 
48 


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Statistical Limits of Fourier Transform Imaging in the Gamma-ray Energy Range

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