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

gamma-ray 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 gamma-rays 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 resolution-variance 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 Compton-scattering,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 signal-to-noise ratio (SNR)

deteriorates dramatically when a continuous background is in-

cluded in the field-of-view (FOV).However,the coded aper-

ture may still be attractive in some particular situations.For ex-

ample,when the non-FOV count-rate 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 (e-mail: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 so-called

time-modulated 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?

0018-9499/03$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 position-sensitive 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

two-dimensional (2-D) 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.Variance-Resolution Tradeoff

In gamma-ray 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 resolution-variance 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 full-width at half-max-

imum (FWHM),full-width at one-tenth 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 region-of-interest (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 re-state some of the key

steps and the final results.Given a measured data set

,the

log-likelihood 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 first-order

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

shift-invariant [18].Under this assumption,the matrix

has a block-Toeplez structure.It can be inverted approxi-

mately using Fourier Transform,which leads to the closed-form

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 shift-invariant 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 resolution-variance 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 non-Compton 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 low-resolution 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 resolution-variance curve close to that

of detector with HC (with 121 pinholes).Although not offering

superior image quality,the use of the collimator reduces the

gamma-ray flux reaching the Compton camera.This may help

Fig.7.Pointwise SNRas a function of CRCwith 200 keVgamma-rays,same

measuring time and the 49 pinhole collimator.

Fig.8.Pointwise SNRas a function of CRCwith 200 keVgamma-rays,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 count-rates 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 resolution-variance

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 gamma-ray 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 gamma-rays.

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 resolution-variance

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

resolution-variance 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-

noise-energy (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 resolution-variance 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 wide-angle gamma-ray vision system, IEEE Trans.Nucl.

Sci.,vol.42,pp.668674,1995.

[3] D.Ramsden,A.J.Bird,M.J.Palmer,and P.J.Durrant,Gamma-ray

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 gamma-ray 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 Tc-99m 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 gamma-ray 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-

nalized-likelihood image reconstruction:Space-invariant 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

CramerRao-type 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 3-D PET, IEEE Trans.Med.Imaging,vol.19,

pp.493506,2000.

[18] J.A.Fessler and S.D.Booth,Conjugate-gradient preconditioning

methods for shift-variant pet image reconstruction, IEEE Trans.Image

Processing,vol.8,pp.688699,1999.

[19] L.Parra and H.H.Barrett,List-mode likelihood:EM algorithm and

image quality estimation demonstrated on 2-DPET, IEEE Trans.Med.

Imaging,vol.17,pp.228235,1998.

[20] Numerical Recipes in C.Cambridge,MA:Cambridge Univ.Press,

1997,ch.7.

## Comments 0

Log in to post a comment