IEEE TRANSACTIONS ON NUCLEAR SCIENCE,VOL.50,NO.4,AUGUST 2003 1103
Feasibility Study of Using Hybrid Collimation for
Nuclear Environmental Imaging
L.J.Meng and D.K.Wehe
Abstract This paper presents a feasibility study of a
gammaray imager using a hybrid collimation (HC) scheme.
This detector is based on the use of a multiple pinhole collimator
and a position sensitive scintillation detector with Anger logic
readout.A pixelated semiconductor detector,located between the
collimator and the scintillation detector,is used as a scattering
detector.For gammarays scattered in the first detector and then
stopped in the second detector,an image can be built up based on
the joint probability of their passing through the collimator and
falling into a broadened conical surface,defined by the detected
Compton scattering event.Since these events have a much smaller
angular uncertainty,they provide more information per photon
compared with using solely the mechanical or electronic collima
tion.Therefore,the overall image quality can be improved.This
feasibility study used a theoretical approach based on analyzing
the resolutionvariance tradeoff in images reconstructed using
maximuma posteriori (MAP) algorithms.The effect of the detector
configuration,Doppler broadening,the energy resolution of the
scattering detector and mechanical aperture design are studied.
The results showed that the combined collimation leads to a
significant improvement in image quality at energies below 300
keV.However,due to the mask penetration,the performance of
such a detector configuration is worse than a standard Compton
camera above this energy.
Index Terms Comptonscattering,hybrid collimation.
I.I
NTRODUCTION
O
VER the past few decades,knowledge of the spatial and
energy distribution of radioactive contamination has been
shown to be of great importance for cleanup and decommis
sioning of nuclear sites.A number of groups have been de
veloping systems for mapping the distributions of radioactive
isotopes for industrial applications [1],[2].Among these ef
forts,pinhole imagers such as EPSLON[3] have been proven to
be capable of providing good angular resolution,while having
drawbacks such as limited viewangle and sensitivity [4].Coded
aperture has been studied,as a possible replacement of the pin
hole,in an effort to improve systemsensitivity [5].Although the
raw sensitivity,in terms of the number of counts collected,can
be improved,it was shown that the signaltonoise ratio (SNR)
deteriorates dramatically when a continuous background is in
cluded in the fieldofview (FOV).However,the coded aper
ture may still be attractive in some particular situations.For ex
ample,when the nonFOV countrate is too high to be effec
tively reduced by the shielding,the use of a coded aperture with
Manuscript received December 2,2002;revised July 7,2003.
The authors are with the Department of Nuclear Engineering and Radiolog
ical Sciences,University of Michigan,Ann Arbor,MI 48109 USA (email:lj
meng@umich.edu).
Digital Object Identifier 10.1109/TNS.2003.815135
Fig.1.Detector design using combined mechanical and electronic
collimation.
relatively large open fraction may be used to counteract the ef
fect of shielding penetration and,therefore,improve the overall
SNR [3].In order to achieve a good compromise between an
gular resolution and sensitivity,several designs using socalled
timemodulated collimation have also been applied [6].
The concept of hybrid collimation has been introduced
by several authors [7],[8] in which a mechanical collimator
is placed in front of a Compton scattering camera.In this
detector design,the angular uncertainty of a detected photon is
constrained not only by the Compton scattering information,
but also by the multiple pinhole aperture.This improves
the information content conveyed by each detected photon.
However,this improvement is achieved at the cost of raw
detection sensitivity.The key question is Would the increase in
information content per photon be sufficient for compensating
the loss in the number of detected photons? To answer this
question,Uritani [7] and Smith et al.[9] have experimentally
evaluated prototype detectors based on this concept.These
studies,however,were limited by the availability of a suitable
scattering detector and,therefore,left many important issues
to be addressed.In this paper,we present a theoretical study
of using this detector concept in environmental imaging
applications.It is based on recent advances in understanding
the properties of images reconstructed using MAP algorithms
[10],[11].This approach allows one to study the effects of
common physical aspects of the detector design based on
realistic detector geometries.Several key design issues,which
we addressed through this study,are outlined as follows.
Would this detector concept offer an improved image
quality and what is the suitable energy range for this
detector configuration?
00189499/03$17.00 © 2003 IEEE
1104 IEEE TRANSACTIONS ON NUCLEAR SCIENCE,VOL.50,NO.4,AUGUST 2003
TABLE I
P
ARAMETERS
U
SED IN THE
S
IMULATION
What is the effect of the amount of multiplexing on the
imaging performance?
What is the effect of the achievable energy resolution in
the scattering detector on image quality?This would also
help to determine the best scattering material to use.
II.D
ETECTOR
The proposed detector using hybrid collimation is shown in
Fig.1.It consists of a multiple pinhole collimator,a semicon
ductor first detector and a positionsensitive scintillation de
tector as the secondary detector.The basic detector configura
tions used in this study is shown in Table I.In order to study
the effect of amount of multiplexing,four pinhole configura
tions were used,with 25,49,121,and 225 pinholes,respec
tively.All the pinholes were arranged in a square pattern and
the pinhole distances were 2.0,1.5,1.0,and 0.75 cm.A48
48
twodimensional (2D) source object was modeled using
32
32 square pixels.It is located 25 cm away from the mul
tiple pinhole aperture.
III.T
HEORY
A.VarianceResolution Tradeoff
In gammaray imaging applications,the reconstructed image
is a biased estimate of the true object due to the presence of sta
tistical noise and imperfections in the systemmodel.This is true
for the popular maximum likelihood expectation maximization
(MLEM),maximum a posteriori (MAP) algorithms,and ana
lytical reconstruction methods such as filtered backprojection.
It is,therefore,important to compare the detector performance
or the image quality as a function of the bias.Many methods
have been developed for this purpose [12][15].In this study,
we used the resolutionvariance tradeoff as the index for image
quality.The basic idea is that a better detector is the one that
provides images having lower variance at the same spatial res
olution.
One difficulty in using this approach is defining the spatial
resolution.Many quantities,such as fullwidth at halfmax
imum (FWHM),fullwidth at onetenth maximum (FWTM),
and contrast recovery coefficient (CRC) [16] have been used in
the past.However,none of them can fully quantify the spatial
resolution property in reconstructed images.In principle,since
the system response to a unit disturbance in the object (also
called local impulse response function or LIR) is a multivariate
function,it should not be represented by a single index.In
this study,CRC was used because of its simplicity and its
known correlation to the ability in quantifying the activity
concentration in a preset regionofinterest (ROI).
B.Variance and Resolution With MAP Reconstruction
The most accurate method for calculating the resolution
and variance is Monte Carlo simulation,if a sufficiently large
number of events can be used.However,for system having a
large number of detector bins and source pixels,generating
a large number of realizations of data is extremely time
consuming.In this study,we adapted a theoretical approach
proposed by Fessler et al.[10],[11] and Qi et al.[16],[17],
which analyzes the image properties based on MAP recon
struction method.Here,we briefly restate some of the key
steps and the final results.Given a measured data set
,the
loglikelihood of an estimator
of the underlying object is
(1)
where
is the unknown image and
is the
measured data.The mean of the data is related to the image
through transformation
(2)
where
is the detector response function and
is the mean
contribution fromobject scattering events and background radi
ation.In MAP reconstruction,the solution achieved is also in
fluenced by the a posteriori information about the object,rep
resented by the function
.In order to control the amount of
influence of such information on the final solution,a Lagrange
multiplier
is introduced,which results in an object function
(3)
For simplicity reason,we only focused on the quadratic
roughness penalty with the form
(4)
MENG AND WEHE:FEASIBILITY STUDY OF USING HYBRID COLLIMATION FOR NUCLEAR ENVIRONMENTAL IMAGING 1105
where
is the weighting factor that takes into account the 26
neighbors and
(5)
The MAP estimator can be achieved by maximizing this ob
jective function
(6)
For a nonlinear estimator,one can use the local impulse re
sponse (LIR) as a measure of the spatial resolution property.For
the
th voxel it is defined as
(7)
where the
is the expectation operator.Using the firstorder
Taylor expansion and chain rule,one can approximate the local
impulse response by its linearized representation
(8)
where
(9)
is the Fisher information matrix (FIM) given the measurement
and
is the transpose of the matrix
.One can similarly show
that the covariance of MAP reconstruction can be approximated
as
(10)
By using the recipe presented in [10],one can calculate the
local impulse response function and variance at a certain point
by calculating a row of the inversed matrix
.
Although this method is independent of the particular opti
mising algorithmand agrees well with the Monte Carlo results,
the computational cost remains high.This is because it involves
the inversion of a Hessian matrix or solving related linear equa
tions.To make the calculation computationally practical,one
may further assume that the proposed detector systemis locally
shiftinvariant [18].Under this assumption,the matrix
has a blockToeplez structure.It can be inverted approxi
mately using Fourier Transform,which leads to the closedform
expression of variance and LIR
(11)
(12)
where the
and
are the eigenvalues of matrix
and
,
which are derived approximately using FFT.
and
are the
unitary Fourier transformoperator and its transpose.The
th el
ement of
is defined as the contrast recovery coefficient
(CRC).This quantity has been shown to have a strong correla
tion to the FWHMof LIR [16].
Based on these results,one can also calculate the pixel wise
SNR in the reconstructed image as
(13)
C.Monte Carlo Integration (MCI)
In order to derive the approximations of resolution and vari
ance at a fixed point using (10) and (11),one needs the
th
column of the FIM.For system with a large number of de
tector bins,this is a very computationally expensive task.In
the proposed detector,there are 1024 pixels in the image and
256
(14)
Since calculating the
through multidimensional integra
tion is not practical for a complicated systemconfiguration,one
can use a Monte Carlo integration instead [19],[20]
(15)
with
the actual measured events.If the FIMis
corresponding to a measured data with
detected events,while
we use
randomly generated (and detected) events in MCI
(16)
where
is the probability of detecting an event
,given
a photon is emitted from source pixel
and
is the number
of emitted photons frompixel
during the period of measure
ment.
cor
responding to
measured counts can be calculated by using
events and a better accuracy can be achieved when
is
larger than
.
Calculating
using Monte Carlo integration (MCI)
requires
floating point calculations.For a detector
system with
detector bins,this leads to a factor of
reduction in computation compared with using (9).
Another important feature is that the MCI calculation of FIM
requires only very modest amount of memory space.The re
ductions in both computation and memory space required were
proven to be the key in the theoretical performance assessment
for Compton camera related detector designs.
1106 IEEE TRANSACTIONS ON NUCLEAR SCIENCE,VOL.50,NO.4,AUGUST 2003
Fig.2.Comparing standard deviation as a function of beta
derived using
(central solid line) and
.The ten thin lines correspond to
ten realizations of
.The circles with error bars are the empirical values
from 100 Monte Carlo simulations.
IV.D
ESIGN
S
TUDY AND
R
ESULTS
A.Monte Carlo Verification
The CRC and variance approximations (11) and (12) have
been carefully studied for many imaging applications.They
generally showed very good accuracy for systems that are rea
sonably close to the shiftinvariant approximation.In this study,
we concentrated on verifying the accuracy of approximating
the true FIMwith
and
.Note that the feature shown contains a set of 11 individual
curves,which include one derived using
and ten others calculated
using
.
Fig.4.(a) Phantomand (b) reconstructed images using data collected with the
detector with hybrid collimation;(c) mechanically collimated detector;and (d)
Compton camera.All data contain the same 0.25 Mevents.
In order to reduce the bias that might be introduced by using
a particular collimator,all comparisons were made using two
different pinhole configurations,with 49 and 121 pinholes re
spectively.
The resolutionvariance curves for the Compton camera,me
chanically collimated detector and the proposed detector at 200
keV,are shown in Figs.5 and 6.The curves are normalized to
the same measuring time.For the proposed detector with hy
brid collimation,one can choose to use either the Compton scat
tered events only or both Compton and nonCompton events.
These results showed that the detector with hybrid collimation
(HC) outperforms both Compton camera and mechanically col
limated detector at 200 keV.Using both Compton and non
Compton events resulted in the lowest standard deviation and
therefore highest SNR,when CRC is larger than 0.3 (Figs.7
and 8).It is worth noting that at low spatial resolution region
MENG AND WEHE:FEASIBILITY STUDY OF USING HYBRID COLLIMATION FOR NUCLEAR ENVIRONMENTAL IMAGING 1107
Fig.5.Standard deviation as a function of CRC for the four detector
configurations.Results are normalized to the same measuring time.The 49
pinhole collimator was used.
Fig.6.Standard deviation as a function of CRC for the four detector
configurations.Results are normalized to the same measuring time.The 121
pinhole collimator was used.
(
),Compton camera provided the lowest noise in
the images.This,however,has no practical significnce because
these lowresolution reconstructions do not provide useful im
ages.
At higher energies,the mechanical collimation becomes less
effective,while the performance of Compton camera is much
improved due to the reduced effect of Doppler broadening.In
this case,the increase in the information content per detected
event,through adding the collimator,may not be able to com
pensate for the reduction in sensitivity.As a result,although
the hybrid collimation greatly reduces the variance compared
with using mechanically only,it becomes less advantageous
when compared with Compton camera.These conclusions are
supported by the results presented in Figs.912.At 400 keV,
Compton camera has a resolutionvariance curve close to that
of detector with HC (with 121 pinholes).Although not offering
superior image quality,the use of the collimator reduces the
gammaray flux reaching the Compton camera.This may help
Fig.7.Pointwise SNRas a function of CRCwith 200 keVgammarays,same
measuring time and the 49 pinhole collimator.
Fig.8.Pointwise SNRas a function of CRCwith 200 keVgammarays,same
measuring time and the 121 pinhole collimator.
Fig.9.Pointwise SNR as a function of CRC at 400 keV with the same
measuring time and the 49 pinhole collimator.
to reduce the challenge in handling high countrates as in
standard Compton cameras.At 662 keV,Compton camera by
1108 IEEE TRANSACTIONS ON NUCLEAR SCIENCE,VOL.50,NO.4,AUGUST 2003
Fig.10.Pointwise SNR as a function of CRC at 400 keV with the same
measuring time and the 121 pinhole collimator.
Fig.11.Pointwise SNR as a function of CRC at 662 keV with the same
measuring time and the 49 pinhole collimator.
itself performs so much better in terms of resolutionvariance
tradeoff and adding a collimator in front becomes redundant.
These results demonstrated that the HCis useful mainly at lower
energies,where the mechanical collimation is effective,while
Compton cameras are less useful.
It is important to note that the above analysis did not take
into account the photon penetration and scattering in the aper
ture.At gammaray energies above 400 keV,these effects are
expected to be significant and induce further degradation in de
tector performance.Therefore,we expect that the performance
differences between the hybrid detector and standard Compton
camera to be even larger than what shown in Figs.912.
C.Effect of Multiplexing
In designing a detector using HC,an important and com
plicated question is what is the optimum collimator to use
with the Compton camera.Here,we choose to study the
effect of the amount of multiplexing.The issue of optimum
coding scheme will be left for future research.In this study,
Fig.12.Pointwise SNR as a function of CRC at 662 keV with the same
measuring time and the 121 pinhole collimator.
Fig.13.Pointwise SNRas a function of CRC.The detector configurations are
exactly the same except with different multiple pinhole apertures.The data sets
contained no Compton scattering information collimator.to the same measuring
time.
we kept most of the detector configurations unchanged,while
modifying the number of pinholes and pinhole distance on
the collimator (as described in Section II).We focused on the
detector performance for 200 keV gammarays.
The relative performance of detectors using these four aper
tures without using Compton scattering information are shown
in Fig.13.It is interesting to see that although more informa
tion per photon is provided by aperture with less pinholes (the
25 pinhole collimator in this case),the best resolutionvariance
tradeoff for the same measuring time was achieved with the
49 pinhole aperture.This indicated that even when continuous
background is presented,a collimator with a modest amount of
multiplexing is desired.
After adding the Compton scattering information,the best
resolutionvariance tradeoff was offered by the 225 pinhole
aperture.The difference between 121 and 225 pinhole apertures
was relatively small (Fig.14).For detector using HC,the best
tradeoff between the information per detected photon and
MENG AND WEHE:FEASIBILITY STUDY OF USING HYBRID COLLIMATION FOR NUCLEAR ENVIRONMENTAL IMAGING 1109
Fig.14.Pointwise SNRas a function of CRC.The detector configurations are
exactly the same except with different multiple pinhole apertures.All events
used contained Compton scattering information.
Fig.15.Pointwise SNR as a function of CRC for several electronic noise
levels.
sensitivity is achieved with a collimator having a relatively
large open fraction,while this benefit becomes saturated after
the amount of multiplexing reaching a certain threshold.
D.Effect of Electronic Noise
For Compton camera working at lowenergies,the energy res
olution of the scattering detector plays an important role in the
angular uncertainty provided by detected Compton scattering
events.For example,in Si,Doppler broadening contributes
FWHM for a 90
scattering.Reducing the effective
noiseenergy (ENE) from2 to 1 keV FWHMwould reduce the
overall energy uncertainty from2.38 to 1.64 keV FWHM.This
significantly improves the amount of high spatial frequency in
formation provided per detected photon and,therefore,offers
better image quality.Here,we studied the effect of electronic
noise on the performance of the proposed detector using HC.
The 225 pinhole aperture was used and the electronic noise was
varied from 0.5 to 4 keV.Fig.15 showed the SNR as a func
tion of resolution at a center pixel for several given values of
ENE.The same measuring time was used and all data sets con
tained only Compton scattered events.Clearly,the detector per
formance,in terms of achievable pointwise SNR,is very sensi
tive to the noise level on the scattering detector.This result is not
surprising given the large number of pinholes in the collimator.
Having better angular information fromthe Compton scattering
events is crucial for assigning a probability to each pinhole.For
the proposed detector design,this result also indicated the im
portance of improving the overall energy resolution achievable
with the scattering detector.Using detector material with less
Doppler broadening is also desired.Therefore,as in designing
a conventional Compton camera for low energy applications,
Silicon would be the most appropriate scattering detector mate
rial for the proposed detector design,amongst commonly used
semiconductors.
V.C
ONCLUSION AND
D
ISCUSSIONS
In this study,we applied a theoretical approach to evaluate
a detector design that makes use of combined mechanical and
electronic collimation.The results are summarized as follows.
The combination of mechanical and electronic collimation
results in a superior imaging performance at lowenergies.
This performance benefit will be limited to below300 keV.
For higher energies,a standard Compton camera would be
a better choice in terms of resolutionvariance tradeoff.
The proposed detector design works best with a collimator
having a relatively large open fraction.
It is important to optimize the achievable energy resolution
for the scattering detector.For low energies,Si would be
the best detector material to use for this detector design.
These analytical approximations for variance and resolution
provided an accurate method for evaluating detector perfor
mance,without doing the extremely time consuming Monte
Carlo simulations.This is very valuable for feasibility study
involving a complicated detector configuration and multiple
design variables.
It has been shown that the proposed detector works best with
an appropriate amount of multiplexing.However,howthe mul
tiplexing should be provided or,in other words,what is the op
timumcoding scheme to use with Compton camera,was beyond
the scope of this investigation.This issue need to be addressed
in our future work.Practically,the proposed detector requires
a relatively large amount of Si detectors and a large number of
readout electronics channels.The construction of a prototype
detector remains a very challenging issue.
A
CKNOWLEDGMENT
The authors would like to thank Prof.J.Fessler,Department
of Electrical Engineering and Computer Science,University of
Michigan,Ann Arbor,for the very valuable discussions and sug
gestions.
R
EFERENCES
[1] S.V.Guru,Z.He,D.K.Wehe,G.F.Knoll,R.H.Redus,and M.R.
Squillante,Portable high energy gamma ray imagers, Nucl.Instrum.
Methods,vol.A378,pp.612619,1996.
1110 IEEE TRANSACTIONS ON NUCLEAR SCIENCE,VOL.50,NO.4,AUGUST 2003
[2] Z.He,S.Guru,D.K.Wehe,G.F.Knoll,A.Truman,and D.Ramsden,
Portable wideangle gammaray vision system, IEEE Trans.Nucl.
Sci.,vol.42,pp.668674,1995.
[3] D.Ramsden,A.J.Bird,M.J.Palmer,and P.J.Durrant,Gammaray
imaging systems for the nuclear environment, Nucl.Energy J.British
Nucl.Energy Soc.,vol.35,pp.353359,1996.
[4] R.Redus,M.Squillante,J.S.Gordon,P.Bennett,G.Entine,G.F.Knoll,
D.K.Wehe,and S.Guru,An imaging nuclear survey system, IEEE
Trans.Nucl.Sci.,vol.43,pp.18271831,1996.
[5] P.T.Durrant,M.Dallimore,I.D.Jupp,and D.Ramsden,The applica
tion of pinhole and coded aperture imaging in the nuclear environment,
Nucl.Instrum.Methods,vol.A422,pp.667671,1999.
[6] S.V.Guru,Z.He,D.K.Wehe,and G.F.Knoll,Monte Carlo modeling
of a multiple pinhole collimator for high energy gammaray imaging,
IEEE Trans.Nucl.Sci.,vol.41,pp.898902,1994.
[7] A.Uritani,N.H.Clinthorne,J.E.Gormley,J.W.LeBlanc,W.L.
Rogers,D.K.Wehe,and S.J.Wilderman,An electronically collimated
gamma camera with a parallel plate collimator for Tc99m imaging,
IEEE Trans.Nucl.Sci.,vol.44,pp.894898,1997.
[8] L.E.Smith,Z.He,D.K.Wehe,G.F.Knoll,and S.J.Wilderman,De
sign and modeling of the hybrid portable gamma camera system, IEEE
Trans.Nucl.Sci.,vol.45,pp.963969,1998.
[9] L.E.Smith,C.Chen,D.K.Wehe,and Z.He,Hybrid collimation for
industrial gammaray imaging:Combining spatially coded and compton
aperture data, Nucl.Instrum.Methods,vol.A462,pp.576587,2001.
[10] J.A.Fessler,Mean and variance of implicitly defined biased estimator
(such as penalized maximum likelihood):Application to tomography,
IEEE Trans.Image Processing,vol.5,pp.493506,1996.
[11] J.A.Fessler and W.L.Rogers,Spatial resolution properties of pe
nalizedlikelihood image reconstruction:Spaceinvariant tomographs,
IEEE Trans.Image Processing,vol.5,pp.13461358,1996.
[12] H.H.Barrett,J.Denny,R.Wagner,and K.Myers,Objective assessment
of image quality II:Fisher information,fourier crosstalk and figure of
merit for task performance, J.Opt.Soc.Amer.A,vol.12,pp.834852,
1995.
[13] H.H.Barrett,T.Gooley,and K.Girodias,Linear discriminants and
image quality, Image Vision Computing,vol.10,pp.451460,1992.
[14] A.O.Hero,J.A.Fessler,and M.Usman,Exploring estimator bias
variance tradeoffs using the uniform CR bound, IEEE Trans.Signal
Processing,vol.44,pp.20262041,1996.
[15] A.O.Hero and J.A.Fessler,A recursive algorithm for computing
CramerRaotype bounds on estimator covariance, IEEE Trans.In
form.Theory,vol.40,pp.12051212,1994.
[16] J.Y.Qi,A theoretical study of the contrast recovery and variance of
MAP reconstructions from PET data, IEEE Trans.Med.Imaging,vol.
18,pp.293305,1999.
[17] J.Y.Qi and R.M.Leahy,Resolution and noise properties of MAP
reconstruction for fully 3D PET, IEEE Trans.Med.Imaging,vol.19,
pp.493506,2000.
[18] J.A.Fessler and S.D.Booth,Conjugategradient preconditioning
methods for shiftvariant pet image reconstruction, IEEE Trans.Image
Processing,vol.8,pp.688699,1999.
[19] L.Parra and H.H.Barrett,Listmode likelihood:EM algorithm and
image quality estimation demonstrated on 2DPET, IEEE Trans.Med.
Imaging,vol.17,pp.228235,1998.
[20] Numerical Recipes in C.Cambridge,MA:Cambridge Univ.Press,
1997,ch.7.
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