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Performance evaluation of a sub-millimetre spectrally resolved CT system on high- and low-
frequency imaging tasks: a simulation
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2012 Phys. Med. Biol. 57 2373
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IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 57 (2012) 2373–2391
doi:10.1088/0031-9155/57/8/2373
Performance evaluation of a sub-millimetre spectrally
resolved CT system on high- and low-frequency
imaging tasks: a simulation
Moa Yveborg, Mats Danielsson and Hans Bornefalk
Department of Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden
E-mail: moay@kth.se, matsdan@kth.se and hansbor@kth.se
Received 9 June 2011, in final form 22 February 2012
Published 2 April 2012
Online at stacks.iop.org/PMB/57/2373
Abstract
We are developing a photon-counting silicon strip detector with 0.4×
0.5 mm2detector elements for clinical CT applications. Except for the limited
detection efficiency of approximately 0.8 for a spectrum of 80 kVp, the largest
discrepancies from ideal spectral behaviour have been shown to be Compton
interactions in the detector and electronic noise. Using the framework of
cascaded system analysis, we reconstruct the 3D MTF and NPS of a silicon
strip detector including the influence of scatter and charge sharing inside the
detector. We compare the reconstructed noise and signal characteristics with
a reconstructed 3D MTF and NPS of an ideal energy-integrating detector
system with unity detection efficiency, no scatter or charge sharing inside
the detector, unity presampling MTF and 1×1 mm2detector elements. The
comparison is done by calculating the dose-normalized detectability index for
someclinicallyrelevantimagingtasksandspectra.Thisworkdemonstratesthat
although the detection efficiency of the silicon detector rapidly drops for the
acceleration voltages encountered in clinical computed tomography practice,
and despite the high fraction of Compton interactions due to the low atomic
number, silicon detectors can perform on a par with ideal energy-integrating
detectors for routine imaging tasks containing low-frequency components. For
imaging tasks containing high-frequency components, the proposed silicon
detector system can perform approximately 1.1–1.3 times better than a fully
ideal energy-integrating system.
(Some figures may appear in colour only in the online journal)
1. Introduction
Spectral computed tomography (CT) is widely believed to be the next big leap in the
development of CT since the ‘slice race’. Various technical solutions whereby energy
information from x-rays is extracted exist. One is the dual source system which is already in
clinical practice (Flohr et al 2006). A second method is fast switching kVp systems (Kalender
0031-9155/12/082373+19$33.00© 2012 Institute of Physics and Engineering in MedicinePrinted in the UK & the USA 2373
Page 3
2374 M Yveborg et al
et al 1986) which is closely related to dual source systems from a physical image formation
point of view. Dual-layer detector systems (with a photon-counting top part and an energy-
integrating bottom part) (Roessl et al 2007) have been shown to be able to extract a substantial
amount of energy information despite their relatively simple construction (Alvarez 2010).
As the fourth and most technically advanced category are photon-counting systems with
pulse height discrimination, either based on cadmium(zinc)telluride (Roessl and Proksa 2007)
or silicon (Yoshida and Ohsugi 2005). Such systems can be used either for new imaging
applications such as K-edge imaging (Roessl and Proksa 2007, Schlomka et al 2008) or to
improvethesignal-to-noiseratio(imagequality)atanygivendosebytheapplicationofenergy
weighting schemes (Shikhaliev 2009, Schmidt 2009). Certainly the improved image quality
achievablebyspectralinformationcanbetradedforlowerdose,animportanttopicinpaediatric
imaging since infants and children are far more at riskfor developing radiation-induced cancer
than adults (Brenner et al 2001).
We are developing a photon-counting silicon strip detector for clinical CT applications
(Yveborg et al 2009, Bornefalk and Danielsson 2010, Xu et al 2011). Except for the limited
detection efficiency of approximately 0.8 for a spectrum of 80 kVp, the largest discrepancies
from ideal spectral behaviour have been shown to be Compton interactions in the detector
combined with electronic noise; in order to reject false counts originating from electronic
noise or induced from charges collected in neighbouring detector elements, a relatively high
lower threshold of 5 keV has to be applied (Bornefalk et al 2010) which discards a large
fraction of valuable counts emanating from Compton interactions. The study showed that,
taking these imperfections into account, such a silicon strip system can perform at 80–90%
of the signal-difference-to-noise ratio (SDNR) of a fully ideal energy-integrating system
(defined as a system with unity detection efficiency, no electronic noise, no charge loss or
scatter inside the detector). The SDNR in its simplicity is however an insufficient measure
of image quality as it does not take into account the spatial frequency dependence of signal
(MTF) and noise (NPS). Since the detector element (del) size in itself greatly affects the MTF,
and the proposed system has smaller dels than conventional energy-integrating systems, the
previous analysis is expanded in this paper to incorporate the spatial frequency dependence of
signal and noise. This is done by the application of cascaded system analysis (Cunningham
2000) to a model of our spectral CT system and also to an ‘ideal’ system, both of which are
described below. To separate the influence of spectral resolution and detector element size on
the detectability of the proposed silicon detector, we compare the performance of the silicon
detector weighing each photon equally independent of energy (i.e. a pure photon-counting
detector), with the performance using projection-based weighting. The method described by
Tward and Siewerdsen (2008) is used for the 3D reconstruction stage. Detectability indices d
for some clinically relevant imaging tasks and spectra, normalized with respect to dose, are
evaluated and compared.
2. Background
2.1. Model system overview
The detector geometry is described in previously referenced work and briefly recapitulated
here. The detector is illustrated in figure 1 and consists of silicon wafers stacked in two layers.
The depth of each wafer (in the direction of the incoming photons, the negative z-direction)
is approximately 30 mm and the thickness of each wafer is 0.5 mm in the x-direction). The
projection image is thus formed in the x–y plane. The active area of each wafer extends
20 mm in the y-direction. The backside of a wafer is coated by a 20 μm tungsten sheath and
Page 4
Performance evaluation of a sub-millimetre spectrally resolved CT system2375
(b) (a)
Figure 1. Illustration of the basic detector layout with two levels of stacked detectors.
the front side is segmented by means of 50 rows of collection electrodes, yielding a pixel
pitch of 0.4 mm. Each wafer is segmented along the direction of incident x-ray photons into
16segmentstoovercomethefluxrateproblem,withheightsexponentiallyincreasingtoensure
uniform count rates. Each segment has an individual charge sensing channel with eight energy
thresholds connected which are read out separately.
In a previous work (Xu et al 2012), we have presented data of the actual input
count rate per segment versus the observed output count rate per segment. The data
were measured in photon-counting mode on the detector ASIC and acquired by varying
the filament current of a tungsten x-ray tube operated at 120 kV potential. The count
rate linearity was shown to hold for up to approximately 2.5 × 106counts s−1per
segment, equal to 250 × 106photons s−1mm−2(resulting from 16 depth segments,
0.4 × 0.5 mm2large detector elements and assuming a detection efficiency of 0.8). Further,
the count rate linearity was in the same work shown to be kept for up to approximately
150 × 106photons s−1mm−2with retained energy information.
Only a small part of the lower level strips will be obscured by the tungsten shielding
of the upper level, resulting in a high geometric detection efficiency (GDE) of 0.98. No
scatter grid is applied, instead the intrinsic scatter rejection from the sheaths is relied upon
and in the simulations, it is assumed that no object scatter is detected. Reabsorbed scatter
from primary x-rays inside the detector is however modelled. The entire detector module will
rotate around an axis parallel to the y-axis, thus allowing acquisition of 50 slices with pitch
0.4 mm simultaneously. As can be seen from the schematics of figure 1, the detector wafers
are assembled to point towards the x-ray source, ensuring that primary x-rays hit the detector
surface orthogonally.
The detector-to-isocentre distance r is assumed to be 0.6 m and the modelled angular
sampling interval is such that oversampling with a factor of 2 is assumed, i.e. a del with width
dx= 0.5 mm will move 0.25 mm before readout. This indicates that the number of projections
m over 180◦is given by m = 2πr/dx. The readout electronics of the silicon detector distribute
the counts into eight energy bins. The thresholds Ti(in keV), where i = 1 − 8, for the eight
spectra (defined by kVp) used in the calculations in this work can be seen in table 1. The
specified thresholds are not optimized.
Page 5
2376M Yveborg et al
Table 1. Energy thresholds (keV) per kVp.
kVp
T1
T2
T3
T4
T5
T6
T7
T8
40
50
60
70
80
90
105
120
5
5
5
5
5
5
5
5
7 10
15
15
15
15
25
25
25
15
23
23
20
20
30
40
45
20
27
30
30
35
45
55
60
25
32
37
45
50
60
70
75
30
36
44
55
60
70
85
90
35
44
52
65
70
80
95
10
10
10
10
10
15
15 105
2.2. Ideal system description
The ‘ideal’ system is assumed to have unity detection efficiency and a unity presampling
MTF. The presampling NPS is assumed flat at a level determined solely by Poisson noise.
The detector is ideally energy integrating, i.e. the value in each del is incremented by a value
proportional to each interacting x-ray photon’s entire energy.
1×1 mm2dels are assumed, contrasting sharply with 0.4×0.5 mm2of the modelled
silicon system. The same detector-to-isocentre distance and oversampling by a factor of 2 are
assumed. Since the del of the ideal system is twice the width (2dx) in the rotational direction,
the number of projections will be halved, but with twice the fluence for each projection. For
identical kVps, the same spectrum and overall fluence are assumed for both systems.
2.3. Simulation of bin point spread functions
Simulated point spread functions (psfs) are needed for estimating both the MTF and NPS
(sections 3.1 and 3.2). Since we model a pulse height discriminating system where registered
counts are assigned to a certain energy bin depending on the detected energy, there will be
M separate psfs where M = 8 is the number of energy bins. In the calculation of the MTF,
these psfs are weighted together using either projection-based bin weights as calculated in
section 2.4, or using photon-counting weights, depending on which type of system
characteristics is to be evaluated.
Eight different tungsten x-ray spectra (Cranley et al 1997) of 40, 50, 60, 70, 80, 90, 105
and 120 kVp, filtered by 2 mm aluminium, 0.1 mm copper and 15 cm soft tissue are assumed
when simulating the bin-psfs, resulting in one psf for each spectrum. For each simulated
photon and spectrum, the initial photon energy is drawn from the spectrum and the interaction
location is randomly selected from a uniform distribution over the del. It is assumed that all
photons interact in a centre segment in the z- and y -direction. Primary photons interacting via
Compton scattering in the centre part of the detector in the z- or y-direction are more likely
to deposit secondaries inside the detector than primary photons interacting near the edges
of the detector. Since secondary interactions cause a smearing of the psf and subsequently
worsen the MTF, the assumption of a centre segment in the z- and y-direction will not cause
an overestimation of the MTF of the model system.
A model of tracing separate x-rays and their interactions in the detector module, including
fluorescence fromthetungstenshieldingandmultiplescattering,wasdeveloped andpresented
byBornefalkandDanielsson(2010)andappliedinthecurrentwork.Ifinteractionoccursneara
detectorelementborder,chargemightleaktoaneighbouringdelandresultinaregisteredpulse
at that location. Such charge sharing is also accounted for using the same model parameters as
thoseinBornefalkandDanielsson(2010).Withpropersettingsofthelowerthresholdandtime
constants of the electronic filters (for pulse shaping), no false noise counts will be detected and
Page 6
Performance evaluation of a sub-millimetre spectrally resolved CT system2377
x
y
Point spread function of B5
12345
5
10
15
20
25
30
35
40
45
50
0
200
400
600
800
1000
1200
1400
Figure 2. Spatial distribution of counts depositing energy in B5for the 80 kVp spectrum.
thus need not be simulated (Bornefalk et al 2010). The actual pulse height of detected events
will naturally be altered by the electronic noise but this is a small effect and neglected since
the mean RMS thermal (electronic) noise in earlier work (Xu et al 2011) has been determined
to be 0.77 keV.
For each of the 106simulated x-rays undergoing interaction in the detector, the del(s)
and energy bin(s) where energy is deposited are recorded, including scatter and reabsorption
inside the detector. The resulting bin psf for bin number 5 (detected energies in the range
35–50 keV) using an 80 kVp spectrum is shown in figure 2. For visualization purposes, the
del where x-rays are simulated to impinge (xdel= 1, ydel= 26) is set to zero. The actual value
is 121 205, indicating 1.2% charge sharing and Compton scatter in each of the positive and
negative y-direction (1457/121205 where 1457 is the maximum count in figure 2). To reduce
simulation time, the x-rays are assumed to impinge on the left end of the detector. The final
psf is then constructed by mirroring the result of the simulation. Due to this, photons that in
reality are scattered first to the left (i.e. outside of the simulation volume) and then change the
direction to the right due to a second scatter will not be detected.
Qualitatively, the other bin psfs are similar; they are highly non-isotropic and ‘striped’ in
the x-direction due to the two levels of detector wafers in the physical stacking of the detector
modules. Due to the tungsten shielding, the psfs drop off faster in the x-direction than in the
y-direction. For instance, for bin 5, the ratio between the number of counts in position (3, 26)
to that in position (1, 28), both 2 dels from the centre pixel but in different directions, is 0.67
and between (5, 26) and (1, 30) a mere 0.15. Although the spatial support of the simulation
geometry might not be sufficient in the x-direction, we take this as a pretext for keeping the
geometry used by Bornefalk and Danielsson (2010). For xdel= 1,3,5, the non-normalized
psf in the y-direction is shown in figure 3.
2.4. Bin weight factors
Projection-based weights have been proposed (Tapiovaara and Wagner 1985) in order to
maximize the signal-to-noise ratio (SNR) between the projections through a background
Page 7
2378M Yveborg et al
10
Pixel nr (y−direction)
20 304050
100
102
104
104
Bin 1
x=1
x=3
x=5
10
Pixel nr (y−direction)
203040 50
100
102
104
104
Bin 2
10
Pixel nr (y−direction)
2030 4050
100
102
104
Bin 3
10
Pixel nr (y−direction)
20304050
100
102
104
104
Bin 4
10
Pixel nr (y−direction)
20304050
100
102
104
Bin 5
10
Pixel nr (y−direction)
20304050
100
102
104
Bin 6
10
Pixel nr (y−direction)
20 30 4050
100
102
104
Bin 7
10
Pixel nr (y−direction)
20 3040 50
100
102
104
Bin 8
x=1
x=3
x=5
Figure 3. Non-normalized psfs for the 80 kVp spectrum in the y-direction (using a log scale). The
various values of x correspond to detector strips as in figure 2.
materialwithanembeddedcontrastelement.Thisrequirestheweightfactorstobeproportional
to the SNVR (signal-to-noise-variance ratio) of the projection data as expressed below, where
Nb(E) and Nt(E) are the number of detected Poisson distributed photons with energy E after
passage through only the background material (subscript b) or the background material with
an embedded contrast element (subscript t), respectively:
w(E) =Nt(E) − Nb(E)
Due to the use of silicon as a detector medium, there is however a large discrepancy between
actual photon energies and detected pulse heights. The expectation value of the number of
detected counts in each bin are estimated below by using the formalism presented by Roessl
and Herrmann (2009).
For a system with eight energy bins, events are allocated to bin Bi, i = 1,...,8, if the
registered signal corresponds to an energy between the thresholds Tiand Ti+1. In the article
by Bornefalk and Danielsson (2010), the inverse of the semi-ideal detector response function
(only including the effect of Compton scatter) was used to statistically assess the actual
photon energy from the deposited energy. Such a semi-ideal response function is depicted in
figure 4; for each actual x-ray with energy in the range 30–100 keV converting in the detector,
the distribution of deposited energies is depicted. For a given actual photon energy Eact, the
distribution is a probability density function where the probability of depositing energy in the
interval (E1,E2) is given by?E2
Si(E) =
Ti
the expectation value of the number of counts detected in bin Biin a detector element x?where
x-rays have traversed a path s through a cross section with a linear attenuation coefficient
μ(x,y;E), defined as ni, is given by
ni= N0(x?)
0
Nt(E) + Nb(E).
(1)
E1f(Edep,Eact)dEdepand?∞
f(E?,E)dE?,
0
f(Edep,Eact)dEdep= 1.
With
?Ti+1
(2)
?∞
?(E)D(E)Si(E)e−?
sμ(x,y;E)dE.
(3)
Page 8
Performance evaluation of a sub-millimetre spectrally resolved CT system 2379
20
40
60
80
100
30
40
50
60
70
80
90
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Deposited energy (keV)
Actual x−ray energy (keV)
Probability density (a.u.)
Figure4.Semi-idealresponsefunctionforsiliconyieldingthedistributionofdepositedenergiesas
a function of actual photon energy, f(Edep,Eact). The lower threshold T1is set to 5 keV. Deposited
energies under this level are not depicted.
Here N0(x?) is the number of photons in the unattenuated beam hitting a detector element x?
per projection, ?(E) is the energy probability distribution of the x-ray spectrum and D(E) is
the detection efficiency of the detector material.
Using (3), we now define ni,b and ni,t as the number of photons depositing energy
in bin i after passage through only the background material with thickness l and linear
attenuationcoefficientμb(E),andthebackgroundmaterialwithanembeddedcontrastelement
of thickness d and linear attenuation coefficient μt(E), respectively:
?∞
and
?∞
The weight wifor each bin is then calculated as (according to (1))
ni,b= N0(x?)
0
?(E)D(E)Si(E)e−μb(E)ldE,
(4)
ni,t= N0(x?)
0
?(E)D(E)Si(E)e−μt(E)d−μb(E)(l−d)dE.
(5)
wi=ni,t− ni,b
ni,t+ ni,b.
(6)
The square of the weight factors wiis normalized to sum to unity, i.e.?N
N
?
where Iiis the expected number of counts in the ith bin and N is the number of energy bins.
The photon-counting weight factors of each bin are identical independent of photon
energy, i.e. wi= 1. As for the projection-based weights, the square of the photon-counting
weight factors is normalized to sum to unity.
iw2
i= 1, and
the total projection image (in terms of expected pixel values) is given by
I(x,y) =
i=1
Ii(x,y)wi,
(7)
Page 9
2380M Yveborg et al
−10−50510
0
0.005
0.01
0.015
0.02
0.025
0.03
Pixel number
Weighted psf
p.b. x−direction
p.b. y−direction
p.c. x−direction
p.c. y−direction
0510
cycles/cm
152025
0
0.2
0.4
0.6
0.8
1
MTF
MTF(u,0) (p.b.)
MTF(0,v) (p.b.)
MTF(u,0) (p.c.)
MTF(0,v) (p.c.)
Figure5.Theleftimageshowsweightedpsfsforthemodelsystemusingprojection-basedweights
(denoted by p.b.), chosen as to optimize the SNR of a bony sphere of radius 1 mm embedded in
a background of soft tissue, and using photon-counting weights (denoted by p.c.), both for the
80 kVp spectrum and in the x- and y-directions. For visualization purposes, the centre value is set
to 0 and the other values are given relative to the original centre value. The right image shows 2D
MTF resulting from the weighted psfs in the left image.
3. Method
3.1. MTF
3.1.1. Modelled system.
the bin psfs of section 2.3 (using both the projection-based bin weights as calculated in (6),
and using photon-counting weights), psf =?
The psf and corresponding normalized MTF for the 80 kVp spectrum using projection-
based weights and photon-counting weights are shown in figure 5, assuming negligible signal
aliasing (Cunningham 2000). To be able to plot the psf, the centre value is again set to 0 in
the plot and the other values are given relative to the original centre value. In the construction
of the psf, the bin psfs (as psf5in figure 2) have been mirrored to be symmetric in x. After this
reflection, the centre coordinate is (5, 26) which for clarity is put equal to (0, 0) in figure 5.
For easier distinction between the psfs, only y-values between −10 and 10 are shown in the
left image of figure 5.
The 2D projection MTF for each kVp is obtained by weighting
iwipsfi, and then taking the magnitude of the
(discrete) Fourier transform: MTF(u,v) = |F{psf(x,y)}|.
3.1.2. Ideal system.
function and the presampling MTF flat at unity. This results in an aperture MTF as given
below, assuming that signal aliasing is negligible (Cunningham 2000):
????
For the ideal system, the 2D projection psf is by assumption a delta
MTF(u,v) =
sin(πaxu)
πu
sin(πayv)
πv
????,
(8)
where ax=1 mm and ay=1 mm (del size).
3.2. NPS
3.2.1. Modelled system.
andagivenkVpistodeterminethe2DprojectionNPSfromtheresultsofthesimulationofthe
The first step in determining the NPS of a reconstructed image slice
Page 10
Performance evaluation of a sub-millimetre spectrally resolved CT system2381
psf described in section 2.3. This is done by taking the Fourier transform of the autocovariance
function K,
K(rm,n,rm?,n?) = ?[I(rm,n) − ?I(rm,n)?][I(rm?,n?) − ?I(rm?,n?)?]?,
where ?? denotes the expectation value and I(rm,n) is the pixel value in the projection image
at location rm,n. Assuming shift invariance, the sample autocovariance function can then be
estimated as
M
?
where ?Ijis a zero-mean noise-only projection image constructed by randomly drawing with
replacement the photon tracks of a number of primary photons interacting in a specific del
from a setof simulated data points. The resampling data are constructed by letting 105primary
photons impinge onto a del. All primary x-rays and secondaries are traced and the relative
locations and deposited energies are recorded.
Hundred noise-only images are constructed and for each image (consisting of 9 × 21
dels),thenumberofprimaryphotonsinteractinginaspecificdel,Ndel,isdrawnusingaPoisson
distribution.ThePoissondistributionisassumedtohaveameanofN0×¯D,Ndel∼ Po(N0×¯D).
N0is the number of photons impinging on a detector element, here arbitrarily taken to be equal
to 400.¯D is the mean detection efficiency of the silicon detector wafer with linear attenuation
μSiand thickness tSiequal to 3.04 cm, a silicon oxide layer of thickness tSiOequal to 0.08 cm
and linear attenuation coefficient μSiO, as given by an application of Lambert–Beer’s law:
?
For an 80 kVp spectrum,¯D is approximately equal to 0.8.
Ndelprimaryphotontracksaresubsequentlyrandomlysampledwithreplacementfromthe
original simulated dataset. Before assigning the pixel value, depending on which NPS is to be
calculated, the photons are weighted according to either the projection-based weight factors as
calculated in section 2.4 or using photon-counting weights with w(E) = 1. The secondaries
originating from the sampled primaries are also weighted and distributed according to their
relative distance of interaction to the primary photon they originated from. Repeating this
scheme for all dels in the image and subtracting the mean results in a noise-only image with
mean zero. A summary of the resampling scheme can be seen in figure 6.
The noise consists of uncorrelated Poisson noise, but also correlated noise originating
from charge sharing and Compton scattering. The projection NPS is now determined as
NPS(u,v) = axay?|F{K(x,y)}|? where F is the (discrete) Fourier transform, the average is
taken over all 100 constructed images, and axand ayare the del sizes (0.4 mm and 0.5 mm,
respectively). K and the normalized NPS: nNPS = NPS = /¯ q2(¯ q is the mean detector signal),
where nNPS has units of mm2, for the modelled system using projection-based weights are
shown in figure 7. For ease of perception, the centre value of K is set to zero in the plot, and
the other values are relative to the original centre value. The NPS for 40, 50 , 60, 70, 80, 90,
105 and 120 kVp is calculated.
(9)
Kj(rm,n) =
1
MN
m?=1
N
?
n?=1
?Ij(rm?,n?)?Ij(rm?,n? − rm,n),
(10)
¯D =
?(E)e−μSiO(E)tSiO(1 − e−μSi(E)tSi)dE.
(11)
3.2.2. Ideal system.
uncorrelated noise, resulting in a flat presampling NPS. Hundred projection images are
reconstructed using the same method as for the modelled system. Since the ideal detector
has a unit detection efficiency and five times as large dels as the modelled system, the
number of primary photons interacting in each del, which is in fact the only interactions
For the ideal system, the noise-projection image consists only of
Page 11
2382M Yveborg et al
Let 105primary pho-
tons impinge onto one del.
Generation of resampling data.
start
Trace all primary x rays and secon-
daries. Determine relative locations and
deposited energies (Δxi, Δyi,Edep,i).
For each del (x,y), sample Ndel(x, y)
triplets (Δxi,Δyi,Edep,i) and incre-
ment Ij(x + Δxi, y + Δyi, Edep,i) .
Generation of synthetic image Ij.
j = j + 1
j = 100?
Define average I(x, y) =
1
M
j=1
ΔIj(x, y) = Ij(x, y) − I(x, y).
M
Ij(x, y) and set
Estimate autocovariance
Kj(rm,n) and construct the NPS:
NPS(u, v) =axay
K(x, y).
no
yes
Figure 6. Resampling scheme for generation of synthetic images.
assuming zero scatter and charge sharing, is drawn from a Poisson distribution with mean
N0× 5, Ndel ∼ Po(N0× 5), and N0 = 400. In each image, energy-integrating weights
are used, w(E) = E; the autocovariance is determined followed by the (discrete) Fourier
transform. As for the model system, the NPS for 40, 50, 60, 70, 80, 90, 105 and 120 kVp is
calculated.
Page 12
Performance evaluation of a sub-millimetre spectrally resolved CT system2383
Δ x
Δ y
Autocovariance matrix K
−4
−2024
−10
−5
0
5
10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
u (cycles/cm)
v (cycles/cm)
nNPS
−50510
−10
−5
0
5
10
7
8
9
10
11
12
x 10−6
Figure 7. Autocovariance function (with projection-based weights chosen as to maximize the SNR
of a bony sphere of radius 1 mm embedded in a background of 25 cm soft tissue) and 2D projection
NPS for the 80 kVp spectrum.
3.3. 3D image reconstruction
The three-dimensional NPS and MTF of the reconstructed image are determined according to
the method outlined by Tward and Siewerdsen (2008) and illustrated in the first figure of that
paper, both for the ideal and modelled systems. The method starts with log-normalization
of the projection NPS (the MTF remains unaffected). The resulting NPS is given by
NPSout= NPSin/¯ q2, where ¯ q is the mean detector signal. The normalization is followed by an
application of a ramp and an apodization filter whereby the NPS and MTF are transferred as a
deterministic convolution, i.e. NPSout= NPSinT2and MTFout= MTFinT at each stage (with
T being the filter expressed in the Fourier domain).
Anyinterpolationofthefilteredprojectionisalsotransferredasadeterministicconvolution
and in the evaluation we assume that bilinear interpolation is applied, i.e. after the application
of the ramp and apodization filters, T is now given by T = sinc2(πuau)sinc2(πvav)
where au and av are the sampling distances in the projection image (0.5 and 0.4 mm
respectively for the modelled system and 1 and 1 mm respectively for the ideal system).
Tward and Siewerdsen (2008) showed that the 3D reconstruction stage affects the NPS as
NPSout(f,w?) =
inanaxialsliceinthereconstructeddomain(u?,v?andw?arethespatialfrequencycoordinates
in the reconstructed 3D image). w?is the spatial frequency along the axis of rotation, M is the
magnification(forsimplicitytakenasequalto1)andmthenumberofprojections.Theeffectof
the3DreconstructionprocessontheMTFisgivenbyMTFout(f,w?) =
here using voxel-driven reconstruction.
As a final step, the NPS 3D voxel matrix is again sampled introducing aliasing in three
dimensions. We have selected to model the reconstruction using the natural voxel size, i.e. the
pixel size divided by the magnification M.
π
mfNPSin(u/M,v/M) where f =
?
(u?2+ v?2) is the radial spatial frequency
1
MfMTFin(u/M,v/M),
3.4. Detectability index and task functions
First it should be noted that there are two principally different methods of determining the
detectability index d of a given imaging task in CT: the 3D volumetric detectability index or
Page 13
2384M Yveborg et al
themorefamiliar2Ddetectabilityindexinasliceofthereconstructedimagedefinedaccording
to (12), where the integral is performed from the negative Nyquist frequency to the positive:
??
Which one of these methods best approximates a human observer is an open question
(Richard and Samei 2010), but we have selected the 2D formulation. Selecting a slice in
the spatial domain corresponds to the multiplication of the true object with a rect-function. In
the Fourier domain, this corresponds to convolution with a sinc-function, W(u?,v?,w?)out=
W(u?,v?,w?)in∗sinc(w?),whichwillintroducealiasinginthew?-direction.Ifhighfrequencies
arenearlyabsent,thenaliasingisnegligibleandW(u?,v?,w?)out≈ W(u?,v?,w?)in.Thecurrent
work assumes that the aliasing in the w?-direction is negligible and thus omits the convolution
of the NPS and task function with a sinc-function in the w?-direction.
ThetaskfunctionW(u?,v?,w?)isgivenbytheFouriertransformofthedifferencebetween
the hypotheses ‘target present’ and ‘target absent’ (ICRU 1995):
Ftask(u?,v?,w?) = N|F{h1(x,y,z) − h2(x,y,z)}|
W(u?,v?,w?) = C ×V × Ftask(u?,v?,w?),
where h1 and h2 are the signals in the spatial domain under the two hypotheses. N is a
normalization factor such that Ftask(0,0,0) is equal to 1, V is the volume of the lesion and C
is the difference in attenuation coefficient μ between the background and the target.
Had mono-energetic x-rays been used, the linear attenuation coefficient map of the
corresponding imaging task would be a natural choice for the contrast in (13b) since this
is what would be reconstructed by the inverse Radon transform. We however apply a spectrum
of x-ray energies which makes the average value of the linear attenuation coefficient over the
incident x-ray spectrum ? a more natural candidate, i.e. μave(x,y) =?μ(x,y;E)?(E)dE.
aimingtoreconstruct.Instead,forprojection-basedweighting,theattenuationcoefficient ˆ μfor
a material with thickness l using the weight wias determined in section 2.4 can be estimated
as
??M
i
where niand n0
material with thickness l and the expected number of photons in the unattenuated beam being
detected in the energy bin i, respectively, calculated using (3).
The reconstructed linear attenuation coefficient using projection-based weighting, ˆ μt, for
a target embedded in a background (denoted by b) is given by (using (14))
?
i
where ni,t is the expected number of photons detected in bin i after passage through the
background with thickness l and an embedded target material of thickness d, as given by (5).
The first imaging task (bony spheres of radii r = 1 and 2 mm embedded in soft
tissue, denoted as ‘small bone’ and ‘large bone’, respectively) was selected for its relatively
high-frequency components, whereas the second, blood in a background of brain matter
(approximatingcerebralhaemorrhage),wasselectedforitslow-frequencycontent:thefraction
of blood is 100% in the middle and then decays like Gaussian spheres with σr= 0.5 and 1 cm
(denoted as ‘small blood’ and ‘large blood’, respectively). With linear attenuation coefficients
d2=
|?MTF(u?,v?,w?)W(u?,v?,w?)dw?|2
?NPS(u?,v?,w?)dw?
du?dv?.
(12)
(13a)
(13b)
This is however not the entity that the projection-based weighting scheme of section 2.4 is
ˆ μ = −ln
i=1wini
?M
i=1win0
?
×1
l,
(14)
iare the expected number of photons detected in bin i after passage through the
ˆ μt=−ln
??M
i=1wini,t
?M
i=1win0
?
− ˆ μb(l − d)
?
×1
d,
(15)
Page 14
Performance evaluation of a sub-millimetre spectrally resolved CT system2385
024
cycles/cm
6810
0
0.005
0.01
0.015
0.02
Abs W (a.u.)
Bony spheres
01234
2
4
6
8
10
12
14
16
x 10−4
cycles/cm
Abs W (a.u.)
Gaussian blood spheres
small bone p.b.
large bone p.b.
small bone e.i.
large bone e.i.
small bone p.c.
large bone p.c.
small blood−in−brain p.b.
large blood−in−brain p.b.
small blood−in−brain e.i.
large blood−in−brain e.i.
small blood−in−brain p.c.
large blood−in−brain p.c.
Figure 8. Task functions for the 80 kVp spectrum where p.b. denotes projection-based weighting,
p.c. photon-counting and e.i. energy-integrating weighting.
obtained from the XCOM data base (Berger et al 2005) and the method of section 2.4
for finding weights, these are determined using (14) and (15) for both the model system
using projection-based weights and photon-counting weights, and the ideal energy-integrating
system.
b is set to 25 cm for bone-in-soft tissue and 15 cm for the blood-in-brain imaging
task. Figure 8 shows the magnitude of W(u?,v?,w?)|v?=0,w?=0at 80 kVp for the (spherically)
symmetric task functions for the modelled system constructed using projection-based and
photon-counting weighting, and for the ideal system using energy-integrating weighting. The
right image of figure 8 is for easier distinction between the different task functions only shown
for frequencies up to 4 cyclescm−1.
The detectability index d in the reconstructed CT image (axial plane) in the central slice
is determined for four imaging tasks, eight different spectra and the two different CT systems
(ideal energy integrating and the modelled silicon micro strip-based energy-resolved system
using projection-based weighting and photon-counting weighting).
3.5. Dose
ThedepositeddoseDkVp,taskforeachspectrum(kVp)andimagingtaskissimplyapproximated
as the energy difference (in keV) of the spectrum before and after the object:
?
where tbis the background thickness and ?(E) is the energy probability density function for
the x-rays impinging onto the object. Since the NPS for the two systems at different kVps is
calculatedassumingaconstantphotonfluenceof400 × 5photonspermm2aftertheobject,the
probability density function for the x-rays is normalized such that the x-ray fluence is identical
for all kVps after the object. The dose (in units of keV) as a function of peak kilovoltage is
plotted in figure 9.
DkVp,task=
E
E(1 − e−μbtb)?(E)dE,
(16)
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2386M Yveborg et al
40 6080
kVp
100120
0
0.5
1
1.5
2
2.5x 105
keV
DkVp,task
Figure 9. Deposited dose (in keV) as a function of peak kilovoltage, assuming a constant number
of photons after passage through matter.
4. Image simulation
Projections of the image task bony spheres of radii r = 1 are simulated with an x-ray tube
fluence of A0× 5 photons per projection and mm2, where A0= 560 and using an 80 kVp
spectrum. Images for both the silicon detector system using projection-based weights and the
ideal system are reconstructed using the MATLAB (2003, The MathWorks, Inc., Natick, MA)
routine iradon.
4.1. Model system
For the silicon detector system, m = 2πr/dx= 7539 projections are simulated (dx= 0.5 mm
and r = 0.6 m). Both Compton scattering and charge sharing are taken into account, but in
order to reduce simulation time each photon being Compton scattered is assumed to undergo
photoelectric absorption after having been scattered.
The number of primary photons impinging on a del with energy in bin i, Adel,i,
is drawn from a Poisson distribution with mean A0×?
the del after having passed the object. The number of photons in the del with energy in bin
i being scattered is sampled from the multinomial distribution with Adel,inumber of trials
and a probability of scattering to bin j calculated using the results from the simulations in
section 2.3. The scattered photons thus deposit energy in the bin belonging to the energy
difference Ei− Ejat the interaction site, followed by energy deposition in bin j at a position
different from the original site of interaction.
The number of photons in bin i being charge shared to the energy bin j is sampled from
the multinomial distribution with a probability calculated using the results in section 2.3. This
results in a deposition of energy in the bin belonging to the energy difference Ei− Ejin the
primary del of interaction and energy in bin j in a neighbouring del. Since charge sharing
only occurs in the y-direction, and only between two detector elements, the amount of theCi,j
ESi(E)?(E)delD(E)dE, Adel,i ∼
Po(A0×?
ESi(E)?(E)delD(E)dE). A0× ?(E)delis the part of the x-ray spectrum reaching
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