ARTICLE IN PRESS

0168-9002/$ - se

doi:10.1016/j.ni

$This work

NNSA NA-22�CorrespondE-mail addr

hezhong@umic1Present add

2104, USA.

Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109

www.elsevier.com/locate/nima

Gamma-ray energy-imaging integrated spectral deconvolution$

D. Xu�, Z. He1

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, Michigan, USA

Received 6 September 2006; received in revised form 29 December 2006; accepted 25 January 2007

Available online 11 February 2007

Abstract

In conventional Compton camera systems, the image reconstruction is performed only in two-dimensional or three-dimensional spatial

coordinates for a specific gamma-ray energy. By doing so, a priori knowledge of the incident gamma-ray energy is required, and usually

an energy window is applied to select full energy deposition events. In some other applications, spectral-deconvolution algorithms were

developed to estimate the incident gamma-ray spectrum by deconvolving the observed energy-loss spectrum. However, usually the

spectral system response function of a non-spherical detector depends on the incident gamma-ray’s direction, which cannot be modeled

by those spectral-deconvolution algorithms. In this paper, we propose a new energy-imaging integrated spectral-deconvolution method,

which utilizes both the Compton imaging and the spectral-deconvolution techniques. In the new method, the deconvolution takes place

in a integrated spatial and energy space. This technique eliminates the requirement of knowing the gamma-ray energy in the imaging

part, and removes the directional dependence in the spectral-deconvolution part. The deconvolved result provides the image at any

specific energy, as well as the spectrum at any specific direction. The deconvolution method is based on the maximum likelihood

expectation maximization (MLEM) algorithm, which is popular in reconstructing photon-emission images. Since the ML solution

estimates the true incident gamma-ray intensity, the deconvolved energy spectrum at the source location is free of Compton continuum.

To truthfully reconstruct the source distribution from the observation data, the accuracy of the system response function tij , i.e. the

probability for a photon from source pixel j to be observed as event i, is the most crucial information. Because of the large number of

pixels in the energy-imaging integrated space, and the very large number of possible measurement events, it is impossible to pre-calculate

the system response function tij by simulations. In this paper, an analytical approach is introduced so that the system response function

can be calculated during the reconstruction process. In order to perform Compton imaging, gamma-ray detectors are required to have

position-sensing capability. The energy-imaging integrated deconvolution algorithm is applied to a three-dimensional position-sensitive

CdZnTe gamma-ray imaging spectrometer, which can provide not only the energy-deposition information, but also the position

information of individual gamma-ray interactions. The results demonstrate that the technique is capable of deconvolving the energy

spectrum and of reconstructing the image simultaneously.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Compton imaging; Spectral deconvolution; MLEM; System response function

1. Introduction

A gamma-ray Compton camera captures two or moreinteractions of an incident photon and records the positionand energy information of each interaction. The scattering

e front matter r 2007 Elsevier B.V. All rights reserved.

ma.2007.01.171

was supported in part by the U.S. Department of Energy/

Office under Grant No. DE-FG52-01-NN20122.

ing author.

esses: xud@umich.edu, xud@ge.com (D. Xu),

h.edu (Z. He).

ress: 2355 Bonisteel Boulevard, Ann Arbor, MI 48109-

angle y can be obtained from the deposited energies by theCompton scattering formula of

cos y ¼ 1�mec

2E1

E0ðE0 � E1Þ(1)

where mec2 is the rest mass energy of an electron, E0 is the

initial gamma-ray energy, and E1 is the energy deposited inthe Compton scattering. The direction of the scatteredphoton can be obtained from the interaction positions.Therefore, the direction of each incident photon isconstrained on the surface of a cone with its vertex placedat the first interaction location, and the axis of the cone is

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 99

determined by the first and the second interactionpositions, as shown in Fig. 1. Since the back-projectioncone of each event passes the true source location, thesource image can be reconstructed when many conesoverlap with each other.

In conventional Compton imaging systems, it is some-times assumed that the incident gamma-ray energy isknown a priori. To reduce the Compton background, animage is typically reconstructed only from those eventswithin a narrow energy window. However, in manyapplications, the energy of the incident gamma-ray isunknown, or the energy spectrum is continuous, and it isunlikely to set an energy window to select the full energyevents. Therefore, the reconstructed image is degraded bythe Compton background, in which only a fraction of theincident photon’s energy is deposited within the detector.Even if the incident gamma-ray energy is known, in thecase of multiple sources, the low-energy photopeak iscontaminated by the Compton background from the high-energy sources, and the reconstructed image of the low-energy gamma-rays is affected by the distribution of thehigh-energy gamma-rays. Furthermore, although the back-projection rings of the Compton background do not passthe true source direction, those background events stillcontain information about the source distribution. There-fore, there will be a waste in efficiency if only full energyevents are reconstructed.

In order to obtain the true incident gamma-ray spectragiven the measured energy loss spectra and the detectorsystem response function, spectral deconvolution methodswere developed for gamma-ray spectroscopy [1,2]. Thosedeconvolution methods are performed only on energyspectra. Since the responses of most actual gamma-raydetectors depend on the incident direction of gamma-rays,those methods are fundamentally vulnerable to spatiallydistributed sources.

As the advances in modern gamma-ray detector devel-opments, more and more detector systems begin to haveposition-sensing capabilities. The interaction of positioninformation can provide extra information about thedirection of the incident gamma-rays by means ofCompton imaging. Therefore, the detector system responsefunction can be modeled as a function of the incident

Ω

θ

First Scattering

Second Scattering orPhoton AbsorptionE2

E1

*

Fig. 1. If a gamma-ray interacts at least twice in the detector, the direction

of the incident gamma-ray can be constrained on the surface of a cone.

The half angle is determined by the energy losses, the cone vertex is placed

at the first interaction position, and the cone axis is defined by the first and

the second interaction positions.

gamma-ray’s direction for those measured events withmultiple interactions. In this paper, we describe a newenergy-imaging integrated deconvolution algorithm inwhich the techniques of Compton imaging and spectraldeconvolution are integrated together. The deconvolutionspace is defined as a combined space of both energy andspatial dimensions. By this means, this method cansimultaneously provide the source image at any gamma-ray energy, as well as the gamma-ray energy spectrum atany incident direction. By applying the new deconvolutionalgorithm, the Compton continuum in the spectrum willcontribute to the full energy peak, and a priori knowledgeof the incident gamma-ray’s energy is no longer required.Since the directional dependence can be modeled by thesystem response function, the new algorithm can be appliedto spatially distributed sources.Among various spectral-deconvolution algorithms, the

maximum likelihood expectation maximization (MLEM)algorithm was proven to be superior [2]. In the proposedenergy-imaging integrated deconvolution algorithm, a pixelj is defined in a combined space of both spatial and energydimension, and a measurement i is defined by two sets ofposition and energy information. Suppose the position isdefined on an 11� 11� 10 grid, and the energy is dividedinto 1000 channels, as a result, there will be more than 106

bins in each set of position and energy information. ACompton scattered event requires at least two sets ofposition and energy information. As a result, 1012 bins arerequired to store the measurement output, which is beyondthe limit of any memory system currently available. Toovercome the difficulty, list-mode MLEM is implementedinstead of general MLEM algorithm. The list-mode MLEMalgorithm is performed using the iterative equation of [3–5]:

lnþ1j ¼

lnj

sj

XN

i¼1

tijPktikl

nk

(2)

in which lnj is the estimated value of pixel j at nth iteration,

the sensitivity image sj is the probability of a gamma-rayemission from pixel j to be detected, N is the total numberof measured events, and the system response function tij isthe probability of a gamma-ray emission from pixel j to beobserved as measurement i.Because of the huge number of possible measurement

events, it is impossible to pre-calculate the system res-ponse function tij by simulations. In this paper, Section 2describes an analytical model to calculate the systemresponse function for each measured event during thereconstruction process. This analytical approach considersthe binning process and the uncertainties in an actualdetector system. Since the imaging space includes theenergy dimension, there are two possibilities for a photonfrom an imaging pixel to create a measured event with twointeractions. The first possibility is that the photon depositsall its energy in the detector by a Compton scatteringfollowed by a photoelectric absorption. The other possibi-lity is that the photon deposits part of its energy by two

ARTICLE IN PRESS

Z2

Z1

Cathode

Incident gamma-ray

Anode1

Anode2

Fig. 2. Three-dimensional position-sensitive CdZnTe detector. The size of

the detector is 15� 15� 10mm3.

D. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109100

Compton scatterings and the scattered photon escapes. Thesystem response function model should account for bothpossibilities. This way, the measured Compton backgroundwill contribute to the photopeak. With an accurate systemresponse function, the deconvolved energy spectrum will bethe estimation of the true intensity of the incident gamma-rays, thus free of Compton scattering continuum.

The sensitivity image sj can be obtained by summing therows of the system response function tij, i.e.

sj ¼X

i

tij. (3)

However, because of the huge number of possiblemeasurement outputs, the direct approach of Eq. (3) isnot practical. In this work, the sensitivity image sj isobtained by simulations.

Three-dimensional position-sensitive CdZnTe detector isa novel room-temperature semiconductor gamma-rayspectrometer. It was initially developed to achieve goodenergy resolution even if the detector material is notuniform and has charge trapping problems. This is becausethe true energy deposition can be obtained by correctingthe measured pulse amplitude as a function of the three-dimensional position of interactions [6,7]. However, thecapability of position sensing, as well as the excellentenergy resolution, enables three-dimensional CdZnTedetector systems to perform Compton imaging [8,9]. Thesize of the detector is 15� 15� 10mm3. The anode of theCdZnTe detector is divided into an 11� 11 pixel arrayFig. 2. The lateral coordinates are given by the location ofindividual anode pixels collecting electrons, and theinteraction depth is measured by the drift time of electronsfrom the interaction position to the collecting anode. Thedepth resolution is about 0.5mm, which makes the positionresolution of the CdZnTe detector about 1mm3. If agamma-ray interacts multiple times in the detector, theindividual interactions will be registered by different pixels.Therefore, each measured event may have differentnumbers of pixels which have energy depositions. Themeasured events are grouped into single-pixel events, two-pixel events, three-pixel events, etc. With the capability ofregistering multiple interactions, a single detector iscapable of performing Compton imaging. A 15� 15�10mm3 CdZnTe detector was applied to demonstrate thatthe energy-imaging integrated algorithm is capable ofdeconvolving the energy spectrum and of reconstructingthe image simultaneously. Since there is no mechanicalcollimator required, the spatial imaging space is the 4pangular space around the detector.

2. Energy-imaging integrated system response function

The system response function tij is defined as theprobability of a photon emitted from pixel j to be observedas a measured event i. Here, a pixel j is defined in thecombined spatial and energy space. The system responsefunction tij therefore can be described as the probability of

a photon with a certain energy emitted from a certainspatial direction to be observed as event i.The measurement data are usually binned due to

pixellation or digitization. Therefore, the measurementresult is actually a small bin volume around point i in themeasurement space, which is defined by a set of positionand energy attributes given by the detector system. Beforethis binning process, the imperfect detector system willintroduce uncertainties to the measurement quantities dueto noise or Doppler broadening. Therefore, the systemresponse function can be written as

tij ¼

ZDVi

di

Zf ðij~iÞf ð~ijjÞd~i (4)

in which, f ð~ijjÞ is the probability density function for aphoton from pixel j to create a ‘real’ event ~i, which consistsof the actual interaction positions and deposited energies,and f ðij~iÞ is the probability for the detector system togenerate a response of i due to uncertainties given the realevent ~i. DVi is the bin volume around measurement i.Our model is based on the following procedure to

calculate the system response function. First, we derive theprobability density function f ð~ijjÞ for a photon emittedfrom pixel j to create a real event ~i. Then we calculate theprobability density function f ðij~iÞ for the detector system tooutput this real event ~i as a measured event i in themeasurement space due to uncertainties, assuming all theuncertainties are Gaussian. Finally, to obtain tij, theprobability density function is integrated over the binvolume DVi around i.In this work, only two-interaction events are modeled.

The system response function modeling for three or moreinteraction events can follow the same procedures butusually is much more complicated. The spatial domain inthe imaging space is the 4p angular space around thedetector, thus a source pixel in the energy-imagingintegrated space is defined by ðE0;X0Þ. The deconvolvedimage gives the incident gamma-ray intensity from a

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 101

certain direction, which is irrelevant to the distancebetween the source and the detector. Therefore, it isreasonable to assume that the source is distributed on thesurface of a sphere with radius R which is much greaterthan the dimension of the detector. As a result, a sourcepixel can also be described by ðE0; r0Þ, in which r0 ¼ RX0.The measurement event ~i and i can be representedby ð ~E1; ~r1; ~E2; ~r2Þ and ðE1; r1; E2; r2Þ, respectively. Fig. 3illustrates a two-interaction event. d1 is the distance thatthe incident photon travels before reaching the firstinteraction position, d is the distance between the two-interaction positions, and d2 is the distance that theescaping photon travels before leaving the detector.

2.1. Probability density function of f ð~ijjÞ

In a measured event, if two interactions are observed bythe detector system, the first interaction is assumed to be aCompton scattering. For the second interaction, there aretwo possibilities, which are another Compton scattering ora photoelectric absorption. The general case is discussedbelow.

The directions of the incident and scattered photons aredefined by ~X1 ¼ ð~r1 � r0Þ=j~r1 � r0j, and ~X2 ¼ ð~r2 � ~r1Þ=j~r2 � ~r1j. We have the following conditional probability:

f ð ~X2jE0; ~E1; ~X1Þ ¼dðyr � yeÞ2p sin ye

(5)

in which ye is the scattering angle determined by energiesE0 and E1 (Eq. (1)), yr is the angle between ~X1 and ~X2, anddðyr � yeÞ is the Dirac delta function. Eq. (5) means thatgiven the direction and the initial energy of the incidentphoton, and the energy deposited in the first scattering, thesecond interaction must occur on the surface of a cone withhalf angle determined by Compton scattering kinematics(here, coherent scattering, Doppler broadening and polar-ization are neglected).

We introduce t as the distance that the scattered photontravels before the second interaction. If t is known, the

Detector

(E2, r2)

R

dd1

d 2

(E1, r1)

Imaging sphere

j(E0, r0)

Fig. 3. A photon from pixel j creates a two-interaction event i, which

consists of energy depositions of E1 and E2 at positions r1 and r2,

respectively.

second scattering must occur on a ring which is on the conedefined by Eq. (5). Since ~X1 and ~X2 are defined by ~r0, ~r1and ~r2, from Eq. (5) we have

f ð~r2jE0; r0; ~E1; ~r1; tÞ ¼dðyr � yeÞdðd � tÞ

2pt2 sin ye. (6)

The probability density function for a photon with energyE0 from r0 to interact at position ~r1 is

f ð~r1jE0; r0Þ ¼1

4pR2mE0

e�mE0d1 (7)

in which 1=4pR2 is the solid angle and mE is the linearattenuation coefficient for gamma-rays at energy E.The probability density function for a photon from

(E0, r0) to deposit ~E1 in a Compton scattering given thecondition that the photon interacts at position ~r1 is

f ð ~E1jE0; r0; ~r1Þ ¼1

stðE0Þ

dscðE0Þ

dO

����~E1

dO

d ~E1

����~E1

¼N

mE0

dscðE0Þ

dO

����~E1

dO

d ~E1

����~E1

ð8Þ

in which, stðEÞ is the total cross-section at energy E, N isthe number of nuclei per unit volume, and dscðE0Þ=dO isthe differential scattering cross-section defined by theKlein–Nishina formula.The probability density function for the scattered photon

to travel a distance of t before the second interaction is

f ðtjE0; r0; ~E1; ~r1Þ ¼ mE0�E1e�mE0�E1

t. (9)

From Eqs. (7), (8), and (9), we can obtain

f ð ~E1; ~r1; tjE0; r0Þ

¼ f ðtjE0; r0; ~E1; ~r1Þf ð ~E1jE0; r0; ~r1Þf ð~r1jE0; r0Þ

¼1

4pR2e�mE0

d1NdscðE0Þ

dO

����~E1

dO

d ~E1

����~E1

mE0� ~E1e�mE0�

~E1tð10Þ

which is the probability density function that a photon withenergy E0 and from r0 creates the first scatteringinteraction of ð ~E1; ~r1Þ and the scattered photon travels adistance of t before the second interaction.From Compton scattering kinematics, we have

dO ¼ 2p sin ydy ¼2pmec

2

ðE0 � E1Þ2dE1. (11)

From Eqs. (6), (10), and (11), we obtain

f ð ~E1; ~r1; t; ~r2jE0; r0Þ

¼ f ð~r2jE0; r0; ~E1; ~r1; tÞf ð ~E1; ~r1; tjE0; r0Þ

¼1

4pR2e�mE0

d1NdscðE0Þ

dO

����~E1

2pmec2

ðE0 � ~E1Þ2

� mE0� ~E1e�mE0�

~E1t�dðyr � yeÞdðt� dÞ

2pt2 sin ye. ð12Þ

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109102

Therefore, given the initial photon has an energy of E0 andis from r0, the probability that the first interaction deposits~E1 at ~r1 and the second interaction happens at ~r2 is

f ð ~E1; ~r1; ~r2jE0; r0Þ

¼

Z 10

f ð ~E1; ~r1; t; ~r2jE0; r0Þdt

¼1

4pR2e�mE0

d1NdscðE0Þ

dO

����~E1

2pmec2

ðE0 � ~E1Þ2

� mE0� ~E1e�mE0�

~E1d�dðyr � yeÞ

2pd2 sin ye. ð13Þ

We will discuss two cases here depending on the type of thesecond interaction.

(A) Photopeak events: The second interaction is a photoabsorption, thus the full energy of the incident gamma-rayis deposited in the detector (ðE0 � E1 þ E2Þ). f ð~ijjÞ becomes

f ð~ijjÞ ¼ f ð ~E1; ~r1; ~E2; ~r2jE0; r0Þ

¼ f ð ~E1; ~r1; ~r2jE0; r0Þf ð ~E2jE0; r0; ~E1; ~r1; ~r2Þ

¼ f ð ~E1; ~r1; ~r2jE0; r0Þspð ~E2ÞdðE0 � ~E1 � ~E2Þ

stð ~E2Þ

¼ f ð ~E1; ~r1; ~r2jE0; r0ÞNspð ~E2ÞdðE0 � ~E1 � ~E2Þ

mE0� ~E1

ð14Þ

in which spð ~E2Þ is the photoelectric cross-section at energy~E2.(B) Compton continuum: The second interaction is a

Compton scattering and the scattered photon escapes thedetector, thus only partial energy of the incident gamma-ray is deposited in the detector (ðE04E1 þ E2Þ). f ð~ijjÞbecomes

f ð~ijjÞ ¼ f ð ~E1; ~r1; ~E2; ~r2jE0; r0Þ

¼ f ð ~E1; ~r1; ~r2jE0; r0Þf ð ~E2jE0; r0; ~E1; ~r1; ~r2Þ

¼ f ð ~E1; ~r1; ~r2jE0; r0ÞN

mE0� ~E1

dscðE0 � ~E1Þ

dO

����~E2

�2pmec

2

ðE0 � ~E1 � ~E2Þ2e�mE0�

~E1�~E2

d2 . ð15Þ

2.2. Uncertainties

Due to detector uncertainties, the observed event isdifferent from the actual event. The probability for aphoton emitted from pixel j to create an observed event i

can be obtained by the integral of

f ðijjÞ ¼

Zf ðij~iÞf ð~ijjÞd~i. (16)

Specifically, Eq. (16) can be written as

f ðE1; r1; E2; r2jE0; r0Þ

¼

Zf ðE1; r1; E2; r2j ~E1; ~r1; ~E2; ~r2Þf ð ~E1; ~r1; ~E2; ~r2jE0; r0ÞdV

ð17Þ

in which, dV is the integral volume in the measurementspace defined by ~E1; ~r1; ~E2 and ~r2.Suppose the measurements of the energy E and the

position ðx; y; zÞ all follow Gaussian distribution withuncertainties of s

E, sx, sy, and sz, respectively. The

probability density function of observing E given theactual energy deposition of ~E is

f EðEj~EÞ ¼

1ffiffiffiffiffiffiffiffiffiffiffi2ps2E

p e�ððE�~EÞ2=2s2

EÞ. (18)

Similarly, we have

f xðxj ~xÞ ¼1ffiffiffiffiffiffiffiffiffiffi2ps2x

p e�ððx� ~xÞ2=2s2xÞ ð19Þ

f yðyj ~yÞ ¼1ffiffiffiffiffiffiffiffiffiffi2ps2y

q e�ððy� ~yÞ2=2s2yÞ ð20Þ

f zðzj~zÞ ¼1ffiffiffiffiffiffiffiffiffiffi2ps2z

p e�ððz�~zÞ2=2s2z Þ. ð21Þ

As a result, f ðij~iÞ ¼ f ðE1; r1; E2; r2j ~E1; ~r1; ~E2; ~r2Þ is a jointGaussian distribution.If the uncertainties in energy and position are small

enough so that f ð ~E1; ~r1; ~E2; ~r2jE0; r0Þ varies slowly aroundðE1; r1; E2; r2Þ, f ð ~E1; ~r1; ~E2; ~r2jE0; r0Þ can be regarded as aconstant and taken out of the integral in Eq. (17). SinceR

f ðE1; r1; E2; r2j ~E1; ~r1; ~E2; ~r2ÞdV ¼ 1, we can approximatef ðE1; r1; E2; r2jE0; r0Þ by replacing ð ~E1; ~r1; ~E2; ~r2Þ withðE1; r1; E2; r2Þ in Eqs. (14) and (15).The above approximation is usually valid when

~E1 þ ~E2oE0, which means the second interaction is aCompton scattering. However, if ~E1 þ ~E2 ¼ E0, whichmeans the second interaction is a photoelectric absorption,f ð ~E1; ~r1; ~E2; ~r2jE0; r0Þ has a term of a delta function andcannot be regarded as a constant. In this case, the deltafunction should be integrated over ~E1 and ~E2.(A) Photopeak events: f ð~ijjÞ is obtained by

f ðE1; r1; E2; r2jE0; r0Þ

¼ f ðE1; r1; r2jE0; r0ÞNspðE2Þ

mE0�E1

�

ZZd ~E1 d ~E2dðE0 � ~E1 � ~E2Þ

2psE1sE2

�e�ððE1� ~E1Þ

2=2s2E1Þ�ððE2� ~E2Þ

2=2s2E2Þ

¼ f ðE1; r1; r2jE0; r0ÞNspðE2Þ

mE0�E1

�

Zd ~E1

2psE1sE2

e�ððE1� ~E1Þ

2=2s2E1Þ�ððE0�E2� ~E1Þ

2=2s2E2Þ

¼f ðE1; r1; r2jE0; r0ÞNspðE2Þ

mE0�E1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðs2E1

þ s2E2Þ

q e�ððE0�E1�E2Þ

2=2ðs2E1þs2

E2ÞÞ.

ð22Þ

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 103

(B) Compton continuum: From Eq. (15), f ð~ijjÞ is obtainedby

f ðE1; r1; E2; r2jE0; r0Þ

¼ f ðE1; r1; r2jE0; r0ÞN

mE0�E1

dscðE0 � E1Þ

dO

�����E2

�2pmec

2

ðE0 � E1 � E2Þ2e�mE0�E1�E2

d2 . ð23Þ

In Eqs. (22) and (23), f ðE1; r1; r2jE0; r0Þ is the same asEq. (13), except ~E1, ~r1, and ~r2 are replaced by E1, r1, and r2.

r 2 r1

d

θe

ΔV2 ΔV1

z

S

r0

R0

R0

Fig. 4. The bin volumes in the measurement space are approximated by

two spheres in calculating the system response function.

2.3. Binning

The binning process is to integrate the probabilitydensity function over the bin volume, i.e.

tij ¼

ZDVi

f ðijjÞdi ¼

ZDVi

f ðE1; r1; E2; r2jE0; r0Þdi (24)

in which di ¼ dE1 dr1 dE2 dr2.If the bin volume is small enough such that all terms

except the delta function in f ðE1; r1; E2; r2jE0; r0Þ varyslowly within the bin volume, the integral can beapproximated by moving those terms out of the integral.

(A) Photopeak events: The system response function is

tij ¼

ZDVi

f ðE1; r1; E2; r2jE0; r0Þdi

¼1

4pR2e�mE0

d1NdscðE0Þ

dO

����E1

2pmec2

ðE0 � E1Þ2

� e�mE0�E1d 1

2pd2 sin ye

NspðE2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pðs2E1

þ s2E2Þ

q

� e�ððE0�E1�E2Þ

2=2ðs2E1þs2

E2ÞÞ�

ZDVi

dðyr � yeÞdi. ð25Þ

(B) Compton continuum: The system response function is

tij ¼

ZDVi

f ðE1; r1; E2; r2jE0; r0Þdi

¼1

4pR2e�mE0

d1NdscðE0Þ

dO

����E1

2pmec2

ðE0 � E1Þ2

� e�mE0�E1d 1

2pd2 sin ye

�NdscðE0 � E1Þ

dO

����E2

2pmec2

ðE0 � E1 � E2Þ2

� e�mE0�E1�E2d2

ZDVi

dðyr � yeÞdi. ð26Þ

In both cases, there is an integral ofRDVi

dðyr � yeÞdi whichneeds to be solved. Considering that this integral dependson r0, r1, r2, E0 and E1, and the integral volume is arectangular parallelepiped due to pixellation and digitiza-tion, it is formidable to obtain an analytical solution for

this integral. Here, we make the following assumptions toease the calculation:

(1)

r0 is on the back-projection cone defined by r1, r2, E0

and E1. A Gaussian function with its standarddeviation equal to the angular uncertainty will be usedto approximate the system response for pixels not onthe back-projection cone.

(2)

The integral volumes around r1 and r2 have the samevolume, and are approximated by two spheres withequivalent radius of R0 as illustrated in Fig. 4.

(3)

ye is constant in the integral region around E1.

Under the above assumptions, the integral ofRDVi

dðyr � yeÞdi can be calculated byZDVi

dðyr � yeÞdi

¼ DE1DE2

ZDV1

dr1

ZDV2

dr2dðyr � yeÞ

¼ DE1DE2

ZDV1

dr1

Z R0

�R0

dz

ZS

dsdðyr � yeÞ. ð27Þ

The integral region S consists of those points which satisfyyr ¼ ye. Strictly speaking, S is not a plane. However, whendbR0, S can be approximated by a plane. Also, when dbR0,dz ¼ d � dyr. As a result, we obtainZDVi

dðyr � yeÞdi

¼ DE1DE2d

ZDV1

S dr1

¼ DE1DE2d

Z p

0

pðR0 sin yÞ2pðR0 sin yÞ

2R0 sin ydy

¼16

15p2R5

0 dDE1DE2. ð28Þ

2.4. Final result

For source pixel j of which the gamma-ray initialposition is on the back-projection cone defined by r1, r2,

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109104

E0 and E1, the system response function is(A) Photopeak events:

tij ¼2R5

0DE1DE2N2

15R2d sin yee�mE0

d1dscðE0Þ

dO

����E1

2pmec2

ðE0 � E1Þ2

� e�mE0�E1d spðE2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pðs2E1þ s2E2

Þ

q e�ððE0�E1�E2Þ

2=2ðs2E1þs2

E2ÞÞ.

ð29Þ

(B) Compton continuum:

tij ¼2R5

0DE1DE2N2

15R2d sin yee�mE0

d1dscðE0Þ

dO

����E1

2pmec2

ðE0 � E1Þ2

� e�mE0�E1ddscðE0 � E1Þ

dO

����E2

2pmec2

ðE0 � E1 � E2Þ2

� e�mE0�E1�E2d2 . ð30Þ

For other source pixels not on the back-projection cone,their system responses are approximated by a Gaussianfunction with its standard deviation equal to the angularuncertainty determined by the detector’s geometry andenergy uncertainties.

In Eqs. (29) and (30), e�mE0d1 is the probability for the

incident photon to reach the first interaction position,dscðE0Þ=dOjE1

2pmec2=ðE0 � E1Þ

2 represents the probabil-ity for the incident photon to deposit E1 in the scattering,e�mE0�E1

d is the probability for the scattered photon to reachthe second interaction position, spðE2Þ represents theprobability for the scattered photon to be photoelectricallyabsorbed, dscðE0 � E1Þ=dOjE2

2pmec2=ðE0 � E1 � E2Þ

2 re-presents the probability for the scattered photon to depositE2 in the second scattering, and e�mE0�E1�E2

d2 is theprobability for the escaping photon to leave the detector.The 1= sin ye term represents the probability that r2 is onthe direction of the scattered photon. In the Comptoncontinuum case, the lack of the energy Gaussian spreadshown in the photopeak case is due to the facts that thesystem response function in the Compton continuum caseis a slow-changing continuous function of the incidentgamma-ray energy E0, and a Gaussian spread with a smalluncertainty will not affect the system response functionvery much. This is the direct result of the approximationdone in Eq. (23).

We notice that R0, DE1, DE2, N and R are all constants fora specific detector system, and d is a fixed value for a specificevent. From Eq. (2), the iteration is not sensitive to the scalingfactor in the system response function since tij appears in boththe numerator and the denominator. Therefore, we canignore those constants to simplify the system responsefunction for the two considered cases as follows:

(A) Photopeak events:

tij ¼1

sin yee�mE0

d1dscðE0Þ

dO

����E1

1

ðE0 � E1Þ2

� e�mE0�E1d spðE2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pðs2E1þ s2E2

Þ

q e�ððE0�E1�E2Þ

2=2ðs2E1þs2

E2ÞÞ.

ð31Þ

(B) Compton continuum:

tij ¼1

sin yee�mE0

d1dscðE0Þ

dO

����E1

1

ðE0 � E1Þ2

� e�mE0�E1ddscðE0 � E1Þ

dO

����E2

2pmec2

ðE0 � E1 � E2Þ2

� e�mE0�E1�E2d2 . ð32Þ

There are several details that need to be addressed:

(A) Total cross-section: The total cross-section may notinclude the coherent scattering (Rayleigh scattering) cross-section because the photon does not lose energy in coherentscattering process. The coherent scattering is usuallyunimportant in modeling gamma-ray transportations.However, when the gamma ray energy is low (below afew hundred keV), the cross-section and average deflectionangle of the coherent scattering become non-trivial, and thecoherent scattering should be taken into account in acomplete model.(B) Sequence ambiguity: Because of the poor timing

resolution of the three-dimensional position-sensitiveCdZnTe detector, it is difficult to tell which interactionoccurs first. Various sequencing algorithms have beendeveloped to reconstruct the interaction sequences basedon the position and energy information of the interactions[8,10,11]. However, those algorithms can only estimatewhich sequence has the highest probability. They cannoteliminate the sequence ambiguity. An accurate systemresponse model should consider all the possible sequences.Particularly, for a two-interaction event, each interactioncan be the first interaction. Therefore, there are two back-projection rings at each energy in our model of the systemresponse function.(C) Conventional Compton cameras: For conventional

Compton cameras with known incident gamma ray energyE0 and known interaction sequence, the system responsefunction can be greatly simplified due to the insensitivity ofthe MLEM algorithm to the scaling factors in the systemresponse function. In this case, Eqs. (31) and (32) can bothbe reduced to e�stðE0Þd1 , which is the only term in the systemresponse function related to pixel j. If the size of thedetector system is small, this term can be further reduced.Therefore, the system response function for conventionalCompton cameras is constant on the back-projection conewith a spread defined by the angular uncertainties.(D) Variation in the binning volume size: The anode

surface of the three-dimensional position-sensitive CdZnTedetector is pixellized into an 11� 11 array. The effectivevolume of the peripheral pixels is slightly larger than thatof the center pixels. This variation in the binning volumesize will affect the integral of

RDVi

dðyr � yeÞdi in thederivation of the system response function. However, this

ARTICLE IN PRESS

475keV, 90°

650keV, 180°

825keV, 270°

300keV, 0°

1MeV, 360°CdZnTe

Detector

Fig. 5. Source distribution in the Geant4 simulation. The source is

uniformly distributed around the sides of the detector. The source energy

increases from 300 keV to 1MeV linearly as a function of the rotational

angle from 0� to 360�.

Counts

Energy (keV)

100

80

60

40

20

0

Reconstr

ucte

d inte

nsity

Simulated two-pixel spectrum

Deconvolved spectrum

Fig. 6. Simulated two-pixel spectrum and deconvolved spectrum.

D. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 105

integral does not depend on the energy or direction of theincident gamma-ray, thus is a constant to all pixels. Again,due to the insensitivity of the MLEM algorithm to thescaling factors in tij , this integral can be ignored. Therefore,the variation in the binning volume size will not affect thedeconvolution process.

3. Performance

The energy-imaging integrated deconvolution algorithmwas applied to both simulated and measured data. In thesystem response function model of Eq. (31), the energyresolution is modeled by

DoverallðEÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2noise þ 2:352FEW

q(33)

in which Doverall and Dnoise are the full width at halfmaximum (FWHM) of the overall and electronic noise, E

is the gamma-ray energy, the electronic noise Dnoise is set at5 keV, the Fano factor F is assumed to be 1, and theaverage ionization energy W is 5 eV for CdZnTe. Thespatial imaging space of the 4p sphere was divided into64� 64 pixels, and the energy space was divided into 500bins from 0 to 2MeV. The MLEM algorithm was stoppedat the 24th iteration for all calculations.

3.1. Simulation

Ideally, the energy-imaging integrated deconvolutionalgorithm estimates the true incident gamma-ray intensity.A Monte Carlo simulation using Geant4 [12] wasperformed with a gamma-ray source uniformly distributedfrom 300 keV to 1MeV. The source is also uniformlydistributed on a circle around the detector, and the gamma-ray energy is a linear function of the source direction, asshown in Fig. 5. By deconvolving the simulation data, wecan examine both the energy and spatial uniformity of thedeconvolution method.

Fig. 6 shows the simulated two-pixel spectrum and thedeconvolved spectrum. About 226k two-pixel events wereused in the deconvolution. The simulated raw spectrumdoes not imply that the source is uniform between 300 keVand 1MeV. This is because the detection efficiencies atdifferent energies are different, and the low-energy part ofthe spectrum is contaminated by the Compton backgroundfrom the high-energy gamma-rays. The simulated rawspectrum also shows a Compton background below300 keV although there is no incident gamma-ray withenergy below 300 keV. However, the deconvolved spectrumrepresents the true energy distribution of the incidentgamma-rays, which is uniform from 300 keV to 1MeV.Fig. 7 shows the deconvolved energy spectra at differentgamma-ray incident directions. The deconvolved spectrumin Fig. 6 still shows some Compton background below300 keV. This background is caused by the low-detectionefficiency for gamma-rays with energy lower than 200 keV,and the MLEM algorithm tends to amplify the statistical

noise if the detection efficiency is low. However, in Fig. 7we can see that the Compton background around 200 keVis distributed across all angles and can be ignored for eachdirection. Although the detection sensitivity varies accord-ing to the incident directions because of the asymmetry ofthe geometry of the detector, the deconvolved spectracorrectly show that the source is spatially uniform.

3.2. Experiment

A single three-dimensional CdZnTe detector was used inthe experiment. A 137Cs, a 22Na, and a 133Ba source wereplaced at three different sides of the three-dimensionalCdZnTe detector as shown in Fig. 8. The measured rawtwo-pixel spectrum is shown in the top figure in Fig. 10, inwhich a Compton continuum is clearly present. About 41ktwo-pixel events were measured. After the energy-imagingintegrated deconvolution, the three sources were wellresolved. Fig. 9 shows the images at the three characteristicenergies of the gamma-ray sources, which are 356, 511 and662 keV. It clearly shows the locations of the three sources.If we look at the directions of the three sources, as shownin Fig. 10, the deconvolved spectra only show the trueincident gamma-ray spectra which are free of Comptoncontinuum.Although the spectral-deconvolution method can re-

move the Compton continuum which is caused by the

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109106

scatters within the detector, it is not able to remove thebackground in which the scatters occur outside thedetector, since those scattered gamma-rays are ‘‘true’’

200 400 600 800 1000 1200

Energy (keV)

360

315270

225180

13590

45D

irect

ion

(deg

rees

)

40

35

30

25

20

15

10

5

0

Inte

nsity

Fig. 7. Deconvolved spectra at different directions. The spectra show that

the source is not only uniformly distributed in energy from 300 keV to

1MeV, but it is also uniformly distributed across spatial directions.

133Ba

137Cs

22Na

180°

270°

90°

CdZnTe

Detector

0°

Fig. 8. Experiment setup. Three point sources (a 137Cs, a 22Na and a133Ba) were placed at three different sides of the three-dimensional

CdZnTe detector.

energy=356keV energy=

Fig. 9. The deconvolved images at the photopeak energies of the three sources

and the polar angles of the sphere.

incident gamma-rays and cannot be distinguished from theunscattered photons. For those events, the gamma-rayfrom the source is first scattered by the materialssurrounding the detector, then the scattered photon entersthe detector and is recorded. Since the deconvolved spectrarepresent the intensity of the incident gamma-rays, whichinclude the scattered photons from the surroundingmaterials, we expect to see a distribution of those scatteredphotons, especially a backscatter peak. Fig. 10 also showsthe deconvolved spectrum from all directions, which stillpresents a continuous background. However, because thescattered photons are spatially distributed, their distribu-tion is not prominent in the localized spectrum shown inFig. 10.In order to verify that the deconvolved spectrum

represents the true intensity of the incident gamma-rays,the relative strength of the four 133Ba gamma lines to the356 keV line is listed in Table 1. The relative intensities arecalculated with background subtracted. As we can see, thedeconvolved spectrum represents the true relative inten-sities much better than the measured raw spectrum.

3.3. Comparison with conventional Compton imaging and

spectral deconvolution

Conventional Compton imaging was performed bysetting an energy window around the full energy peak.By doing so, the true full energy deposition events as wellas the Compton backgrounds from higher energy gamma-rays are selected. The Compton backgrounds are usuallyspatially distributed. As a result, the Compton back-grounds have little effect on the angular resolution of thereconstructed image of a point source. However, as shownin Fig. 11(a), the Compton backgrounds do introducenoises in the reconstructed image because they are fromrandom directions. Fig. 11(b) shows the energy-imagingintegrated deconvolved image summed in the same energywindow. The Compton backgrounds are correctly placedin the energy bin of the original gamma-rays. As a result,the Compton backgrounds introduce less noises comparingwith the conventional Compton imaging.

511keV energy=662keV

. The 4p imaging space is projected onto a plane defined by the azimuthal

ARTICLE IN PRESSC

ou

nts

Inte

ns

ity

Inte

ns

ity

Energy (keV)

Fig. 10. The upper figure is the measured two-pixel spectrum. The middle

figure shows the deconvolved spectra at the three source directions. The

lower figure is the deconvolved spectrum from all directions.

Table 1

Relative intensities of the four 133Ba lines to the 356 keV line

276 keV 302keV 356 keV 384 keV

Relative source intensity (%) 11.55 29.54 100 14.41

Measured spectrum (%) 8.87 26.70 100 11.39

Deconvolved spectrum (%) 12.30 29.65 100 14.27

Azimuthal angle (degree)

Pola

r angle

(degre

e)

0 45 90 135 180 225 270 315 360

Azimuthal angle (degree)

0 45 90 135 180 225 270 315 360

90

45

0

-45

-90

Pola

r angle

(degre

e)

90

45

0

-45

-90

Fig. 11. Images of the 384 keV line from the 133Ba source: (a)

Reconstructed image by setting an energy window from 370 to 390 keV.

(b) Energy-imaging integrated deconvolved image summed from 370 to

390keV.

D. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 107

The energy-imaging integrated algorithm uses all mea-sured events and puts the Compton backgrounds into theright energy bin. As a result, the image obtained at a fullenergy peak is from both the true full energy depositionevents and the Compton backgrounds. The Comptonbackgrounds usually have larger angular uncertainties thanthe full energy events. Therefore, the imaging spatialresolution of the new imaging-energy integrated deconvo-lution algorithm is not superior compared to conventionalCompton imaging. Fig. 12 shows the images obtained bythe two methods for the 511 keV gamma-rays from a 22Nasource. It can be seen that the energy-imaging integrateddeconvolved image has a spatial resolution slightly worsethan the conventionally reconstructed image.

Energy spectral deconvolution was also applied to themeasurement. Because it does not require the directionalinformation of the incident gamma-rays, single-pixel eventscan also be used in the deconvolution. The system response

function for the spectral-only deconvolution was obtainedby Geant4 simulations. Fig. 13 shows the measured rawspectrum of all events and the deconvolved spectrum. Inthe spectrum, it can be seen that the 80 keV photopeakfrom the 133Ba source is also present. In the current system,the energy threshold for individual pixels is about 60 keV.Therefore, for two-pixel events, the minimal detectableenergy is 120 keV, which is the reason why the 80 keV peakis absent in the spectra in Fig. 10. Comparing the energy-imaging integrated deconvolved spectrum in Fig. 10 withthe spectral deconvolved spectrum in Fig. 13, we can seeboth spectra show backgrounds due to photons that werescattered outside the detector. Because both methods didnot model the surrounding materials, the scattering back-ground cannot be removed by either method. By carefullymodeling the surrounding materials, the energy spectraldeconvolution method can provide a more detailed systemresponse function and can remove the scattering back-ground. However, this practice is not possible for hand-held detectors of which the operation environment changesfrequently. In the energy-imaging integrated deconvolvedspectrum, because the scattering background is spatiallydistributed, the scattering background is not present in the

ARTICLE IN PRESS

Azimuthal angle (degree)

Pola

r angle

(degre

e)

0 45 90 135 180 225 270 315 360

Azimuthal angle (degree)

0 45 90 135 180 225 270 315 360

90

45

0

-45

-90

Pola

r angle

(degre

e)

90

45

0

-45

-90

Fig. 12. Images of the 511 keV line from the 22Na source: (a)

Reconstructed image by setting an energy window from 490 to 520 keV.

(b) Energy-imaging integrated deconvolved image summed from 490 to

520 keV.

Co

un

tsIn

ten

sit

y

Energy (keV)

Fig. 13. Energy spectral deconvolution on the measured spectrum. The

top figure shows the measured raw spectrum of all events. The bottom

figure shows the deconvolved spectrum.

D. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109108

localized spectra at the true source directions. As a result,the signal-to-noise ratio can be improved at the true sourcedirections.

4. Conclusions

Traditional spectral-deconvolution methods are per-formed only on energy spectra. Since the gamma-rayresponse of any non-spherical detector depends on theincident direction of the gamma-rays, those algorithmscannot truthfully deconvolve the spectra of spatiallydistributed sources. Even if the detector is spherical,different surrounding materials will affect the back-scattering and will result in different spectrum responsesto the same gamma-ray source. In this work, an imagingprocedure is introduced into the spectral-deconvolutionprocess by means of Compton imaging. Modern position-sensitive gamma-ray detector systems can give positioninformation about the gamma-ray interactions in thedetector. If more than one interaction is observed in asingle event, extra information about the incident gamma-ray’s direction can be derived. Therefore, the detectorresponse function can be established as a function of theincident gamma-ray’s direction. The proposed energy-imaging integrated deconvolution algorithm takes theadvantage of this directional dependence in the systemresponse function, and performs the deconvolution processin a combined spatial and energy space, thus overcomingthe difficulties in traditional spectral-deconvolution meth-ods. The deconvolved spectrum is a function of the incidentgamma-ray’s direction; therefore, it can provide the energyspectrum at any specific direction, as well as the image atany specific energy. This energy-imaging integrated decon-volution algorithm does not need to know the energy of theincident gamma-rays, which is required information inmost Compton camera systems.The MLEM algorithm, which is popular in solving

photon-emission image reconstruction problems, is appliedin the deconvolution process. Due to the large number of binsof the measurement output, the deconvolution is performedin list-mode. The number of elements in the system responsefunction is so large that it is impossible to pre-calculate thesystem response function by simulations. An analyticalapproach is derived to allow the calculation of the systemresponse function during the reconstruction process.The three-dimensional position-sensitive CdZnTe detector

is an instrument which can provide both the position and theenergy information of individual gamma-ray interactions.Although the size of the detector is relatively smallð15� 15� 10mm3Þ, the detection efficiency as a Comptonimager is much higher than a conventional two-detectorCompton imaging system because there is no separationbetween the first and the second detector. The proposedenergy-imaging integrated deconvolution algorithm is appliedto the three-dimensional position-sensitive CdZnTe detectorwith both simulated and measured data. The results showthat the new algorithm is capable of deconvolving the energyspectrum and reconstructing the image simultaneously. Thedeconvolved spectra, as expected, are free of Comptoncontinuum and are the estimations of the true intensities ofthe incident gamma-rays.

ARTICLE IN PRESSD. Xu, Z. He / Nuclear Instruments and Methods in Physics Research A 574 (2007) 98–109 109

The energy-imaging integrated deconvolution algorithmis not limited to three-dimensional position-sensitiveCdZnTe detectors. This algorithm can be applied to anygamma-ray detector system with three-dimensionalposition-sensitive capability, such as three-dimensionalposition-sensing germanium, stacks of double-sidedsilicon strip detectors, depth of interaction (DOI) scintilla-tors, and gas or liquid Xenon time projection chambers(TPC).

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