PhotonCountingMethods

Application of photon counting multibin detectors
to spectral CT
Hans Bornefalk
March 23, 2015
Contents
1 Background and problems with standard CT 2
2 Raison d’être for multibin CT 6
2.1 Basis decomposition with data from a multi-bin photon count-
ing system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Method of basis decomposition . . . . . . . . . . . . . 6
2.1.2 What M and which bases to select in Eq. (7)? . . . . 7
2.1.3 Benefit of basis decomposition . . . . . . . . . . . . . 9
2.2 Energy weighting . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Energy weighting to optimize CNR for a given imaging
task . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Benefits and drawbacks with energy weighting . . . . 11
2.3 Basis decomposition by weighting . . . . . . . . . . . . . . . . 12
3 What is being displayed in spectral CT? 15
3.1 Bin images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Gray scale images from weighting . . . . . . . . . . . . . . . . 15
3.3 Basis images . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Synthetic mono-energetic images . . . . . . . . . . . . . . . . 16
4 Practical guide 18
5 Challenges for photon counting multibin systems 20
5.1 Pile-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.2 Scatter and charge sharing . . . . . . . . . . . . . . . . . . . . 22
5.3 Calibration issues . . . . . . . . . . . . . . . . . . . . . . . . . 23
1

6 How to evaluate 28
A Derivation of optimal weights 30
B Variance as a function of ROI size 34
C Constancy of c for different path lengths 34
1 Background and problems with standard CT
The principle of tomographic reconstruction relies on the Radon transform
[1] relating the set of line integrals of an object to an interior distribution
f(x, y). If the projection pl is defined as pl = l f(x, y)ds for the straight line
l, the set {pl} for all l ∈ R2 is the Radon transform of f(x, y), henceforth
denoted Rf(x, y).
In computed tomography, the goal is to reconstruct the interior distri-
bution of linear attenuation coefficients from projection data. This requires
solving the inverse Radon transform f(x, y) = R−1({pl}) and this is typ-
ically implemented by filtered back projection or iterative reconstruction.
The topic of reconstruction will not be dwelled further on here; the inter-
ested reader is referred to standard textbooks on the subject matter, for
instance [2, 3].
Conventional, or energy integrating CT, is based on indirect detectors
where a scintillator material (for example GOS) first converts the energy of
the x-ray quantum to visible light. This light is then converted to electric
charge in a photodiode and the generated current is collected (integrated)
for a given time period before the signal is digitized before further process-
ing. The following forward equation is constructed to capture the essential
physics of the imaging chain (neglecting electronic noise), and describes
many of the imperfections associated with energy integrating systems:
It,θ = n0 Φ(E)D(E)E exp −
l
µ(x, y; E)ds dE (1)
The parameters and variables of Eq. (1) are described in table 1 and depicted
in Fig.1.
One of the problems of conventional CT is that the inverse Radon trans-
form can only reconstruct the linear attenuation coefficient at an unknown
energy. To see why, let I0 = n0 Φ(E)D(E)EdE be the unattenuated signal
2

Symbol Quantity Unit Description
E photon energy keV -
n0 number of emitted photons
towards detector element t at
angle θ in one integration pe-
riod
- -
I energy integrated signal keV -
Φ(E) x-ray energy distribution
function
keV−1 Normalized such that
Φ(E)dE = 1
D(E) Detection efficiency - Convertion efficiency of
the detector times the
geometric efficiency
t detector coordinate/element - see fig 1
θ rotation angle of the projec-
tion
rad. see fig 1
l line connecting the x-ray
source and detector element t
at angle θ
- see fig 1
µ(x, y; E) linear attenuation coefficient cm−1 see fig 1
Table 1: Forward model parameters and variables.
µ(x,y)
t’
t
l
θ
detector
Object
x ray source
Figure 1: Projection geometry.
3

and define the set of projections {pt,θ} by
pt,θ = − log
It,θ
I0
= − log
n0 Φ(E)D(E)E exp − l µ(x, y; E)ds dE
n0 Φ(E)D(E)EdE
. (2)
By application of the mean value theorem for integrals, it follows that
pt,θ =
l
µ(x, y; E )ds (3)
where E is an interior energy of the spectrum Φ(E). E will depend on
the particular effective spectrum for the projection line l. If the effective
spectrum Φ is different in different views, for instance due to passing a
particulary radio-dense region such as bone, the linear attenuation coefficient
will be reconstructed at a higher energy. Since the linear attenuation tends
to decrease with energy, reconstruction at a higher effective energy results
in darker/lower values, which results in characteristic dark streaks in the
image. Even for homogenous objects this will be a problem due to the
different attenuation lengths which result in different effective spectra and
thereby different reconstruction energies. The result of this is the cupping
artifact, where the interior of the image appears darker. Both these problems
are referred to as beamhardening. This is one of the problem associated with
energy integrating CT systems, artifactual inhomogeneities within the image
due to beamhardening.
A related problem arises when one wants to compare CT-numbers (in
Hounsfield units) between different systems. In CT, it is customary to dis-
play the effective linear attenuation coefficient normalized to water:
HU(x, y) = 1000
µeff (x, y) − µH2O
eff
µH2O
eff − µair
eff
. (4)
In Eq. (4) the subscript ’eff’ denotes an effective linear attenuation coef-
ficient reconstructed by the inverse Radon transform of {pt,θ}. Clearly, if
any forward model parameter such as x-ray spectrum or detection efficiency
D(E) is different for different systems, the line integrals will correspond to
different energies E in Eq. (3) and thus CT-numbers will differ between
systems and depend on exposure conditions. Normalization with water is
partly used to compensate for this, but systematic differences still remain
between systems.[4, 5] This makes quantitative CT, i.e. the use of the abso-
lute reconstructed values for diagnostic purposes, difficult.
The third problem is perhaps more subtle: due to the decrease of linear
attenuation coefficients with energy, the contrast between tissues tend to
4

decrease with increasing energy. A system that, as in the case of energy
integrating systems, places more weight at high energy events is thus not
achieving ALARA1 since it is not utilizing the contrast information available
in the most optimal fashion.
Fourthly, if the linear attenuation coefficients of a particular target and
background when weighted over the spectrum Φ according to Eq. (1) are
similar, their contrast will cancel.[6] This is a clinically relevant issue that ex-
plains the observed loss of contrast for certain iodine concentrations2 against
plaque and bone.
A fifth problem with energy integrating detectors is that electronic noise
during the integration time is added to the signal. This makes real low-
dose acquisitions difficult as the performance of the system relative the dose
decreases with decreases dose as electronic noise becomes more and more
prominent.[7]
In summary, the problems with standard CT can be grouped in the
following categories:
1. Beamhardening
2. CT-numbers depend on parameters of the forward equation and thus
differ between systems, this makes quantitative CT difficult
3. The contrast and the contrast to noise ratio between two tissues in the
body is energy dependent, and low energy photons usually carry more
information. This information is lost in energy integrating detectors
and ALARA is not achieved.
4. Risk of contrast cancellation if linear attenuation curves overlap
5. Electronic noise is integrated into the signal preventing low-dose ex-
aminations
1
As Low As Reasonably Achievable; a doctrine stating that dose should be kept as low
as possible while still maintaining diagnostic quality.
2
Iodine contrast agents are very common in CT exams.
5

2 Raison d’être for multibin CT
All five limitations with energy integrating CT can be removed with photon
counting multibin CT. The solution to limitations 1 and 2 are based on the
method of basis decomposition and the solution to the third and fourth can
be based on energy weighting as well as well as on basis decomposition. The
problem with electronic noise integrated into the signal is removed by the
possibility of thresholding the signal. In this chapter, basis decomposition
and energy weighting will be described in detail.
2.1 Basis decomposition with data from a multi-bin photon
counting system
2.1.1 Method of basis decomposition
The description draws on the formalism presented by Roessl and Herrmann[46].
Suppose we have a photon counting multi-bin system with N energy bins
where events are allocated to bin Bj, j = 1, . . . , N, if the registered signal
corresponds to an energy between the thresholds Tj and Tj+1 (TN+1 =
∞). Further assume a detection efficiency D(E) and an energy response
function R(E, E ). The response function denotes the probability that an
x-ray quantum of energy E gives rise to a signal corresponding to E and is
normalized such that R(E, E )dE = 1. With
Sj(E) =
Tj+1
Tj
R(E, E )dE , (5)
the expected value of the number of counts in bin Bj in a detector element
t at a projection angle θ where x rays have traversed a path l with linear
attenuation µ(x, y; E) is given by
λ(t, θ; Bj) = λj = I0(t, θ)
∞
0
Φ(E)D(E)Sj(E)e− l
µ(x,y;E)ds
dE. (6)
The actual number of registered events in each bin, mj, will be Poisson
distributed with mean λj.
Equation (6) is of limited practical use since it does not allow any in-
ference about µ(E) to be made from the measurements mj. Therefore one
assumes that the unknown attenuation coefficient can be decomposed into
M bases with known energy dependencies:
µ(x, y; E) ≈
M
i=1
ai(x, y)fi(E). (7)
6

With
Ai(t, θ) =
l
ai(x, y)ds, (8)
the exponent l µ(x, y; E)ds in (6) can be approximated by i Ai(t, θ)fi(E)
allowing us to express the expected number of counts in each energy bin as
a function of the M parameters Ai:
λj(t, θ) = I0(t, θ)
∞
0
Φ(E)D(E)Sj(E)e− M
i=1 Ai(t,θ)fi(E)
dE, j = 1, . . . , N.
(9)
The final step of the method is to determine the line integrals Ai(t, θ)
that yields the best fit to the observed data mj(t, θ). A maximum likelihood
(ML) fit to the data can be performed. Since the measurements in the N
bins are independently Poisson distributed, the likelihood function can be
written
P(m1, . . . , mN |λ1, . . . , λN ) =
N
j=1
λj(A1, . . . , AM )mj
mj!
e−λj (A1,...,AM )
. (10)
Taking the negative logarithm of the likelihood function and dropping terms
not affected by the Ais yields
L(A1, . . . , AM ; m) =
N
j=1
λj(A1, . . . , AM ) − mj log λj(A1, . . . , AM ) . (11)
The maximum likelihood basis decomposition is now given by the Ais that
minimize (11) for the observed data m = (m1, . . . , mN ):
A∗
1, . . . , A∗
M = arg min L(A1, . . . , AM ; m). (12)
Once the line integrals A∗
i (t, θ) have been obtained, standard CT recon-
struction methods can be applied to determine âi(x, y) which then constitute
the decomposed cross sectional basis images.
2.1.2 What M and which bases to select in Eq. (7)?
The first questions to come to mind are which number of basis functions
M in Eq. (7) should be used and which the bases f(E) should be. It is
instructive to first consider the dependencies of the cross section of photon
interaction, σ(E; Z).
7

A common cross section parametrization is given by Rutherford et al.[8]:
σ(E; Z) = fph(E)Z4.62
+ finc(E)Z + fcoh(E)Z2.86
. (13)
If the dependencies of the cross section on atomic number Z and energy E
are separable as in (13) we can write
σ(E; Z) =
α
fα(E)gα(Z), α ∈ {ph, inc, coh}, (14)
and it follows from the mixture rule[9]that the space of linear attenuation
coefficients for bodily constituents (being mixtures of low Z elements) is also
spanned by the three energy basis functions fα(E):
µ(E) = ρNA
i
wi
massi
σ(E; Zi) = ρNA
α
fα(E)
i
wigα(Zi)
massi
. (15)
ρ is the density of the mixture, NA is Avogadro’s number, massi is the
atomic mass of element i and wi the fraction by weight for element i.
Although the above theoretical considerations conclude that the rank
of the linear attenuation space for human tissues at clinically relevant x-
ray energies should be at least three, possibly higher if the assumption on
separability must be waived, basis decomposition methods[10, 11, 12] as-
sume that the linear attenuation space, disregarding k-edges, is spanned
by only two basis functions. This is a reasonable assumption, since the
photoelectric effect and Compton scattering dominate over Rayleigh scat-
tering, i.e. for clinically relevant energies and atomic numbers it holds that
fcoh(E)gcoh(Z) fph(E)gph(Z) and fcoh(E)gcoh(Z) finc(E)ginc(Z) in
Eq. (14). Furthermore, any part of a third basis not being orthogonal to the
first two will be captured by the first two bases. Indeed such two function
decompositions work well; good basis decomposition results have been ob-
tained on real data by for instance Schlomka et al.[13]. Although theoretical
work on the dimensionality of the linear attenuation coefficient space has re-
sulting in a statistically significant dimension of four [14], the last two bases
are very weak and negligible for (most?) practical purposes – for naturally
occurring human tissues. When contrast agents with k-edge discontinuities
in the x-ray spectrum are used, the decomposition must be expanded to
account for this, using one additional basis for a each contrast compound
with a k-edge.
The two basis functions f1(E) and f2(E) go, they can either be en-
ergy bases, capturing the behavior of the photoelectric effect and Compton
8

scattering, or they can be selected to be material bases such as the linear at-
tenuation coefficients of bone and soft tissue. If a third base is used it should
be the linear attenuation coefficient of the corresponding k-edge element.
2.1.3 Benefit of basis decomposition
When we have the set of âi(x, y)’s for each image coordinate x, y, (see
Eq. (7)) the corresponding linear attenuation is derived as
ˆµ(x, y; E) =
i
âi(x, y)fi(E). (16)
By construction, due to the ML-solution taking the exponential nature of
photon attenuation into consideration, the linear attenuation coefficients are
reconstructed beam hardening free.
If the forward model is accurately specified (see also Sec. 5.3), the basis
decomposition results in unbiased estimates of the true linear attenuation co-
efficient at all energies. This allows quantitative CT, i.e. the use of absolute
image values for diagnostic purposes. Translation of radiologists’ rules-of-
thumb, such as “more than 2% higher attenuation in a liver cyst compared
to the background indicates. . . ”, between different systems will not be a
problem as long as the forward models of each system are known.
Basis decomposition also allows for generating a map of a k-edge contrast
agent concentration. This can be used for quantifying the contrast agent in
space and time and also for post-acquisition removal of the contrast agent,
thereby generating an artificial pre-contrast image. In diagnostic procedures
that require a pre- and a post-contrast image this is valuable not only be-
cause it lowers the dose since only one exposure is needed, but also because
any problem with missregistration due to patent movement between expo-
sures is eliminated.
Finally, if the linear attenuation coefficients of target and background
cross each other for certain energy they might cancel in energy integrating
imaging. However, as the entire energy dependence of the linear attenuation
coefficients is reconstructed, this method allows for the generation of syn-
thetic monoenergetic images that present the linear attenuation coefficient
at any energy. This ensures that cancellation does not occur as the linear
attenuation coefficients will not be equal for all energies.
2.2 Energy weighting
A more straight forward alternative to use the energy information is to
simply linearly weight the counts in each bin. While simple and intuitive,
9

such an approach does not eliminate beam hardening and does not result
in an image quantity which is easily interpreted physically. The weights are
normally selected to achieve one of two tasks: to maximize the contrast-
to-noise ratio (i.e. detectability) of a certain imaging task, for instance a
cyst against a homogenous liver background, or to approximate the basis
decomposition by determining the linear transform that most accurately
approximates the Ai’s of Eq. (12).
The weights can be applied either prior or after reconstruction, denoted
projection based weighting and image-based weighting respectively. Image-
based weighting was proposed by Gilat-Schmidt[15] and has some desirable
theoretical properties, but the difference to projection based weighting in
clinical applications appears to be small.[16] The mathematical treatment
is similar for the two methods. The only difference is whether the weights
operate on transmission intensities (I, see Eq. (1)) or on the images re-
constructed by each bin. We will only present a rigorous derivation of the
weights in the projection domain. Due to the linearity of the inverse Radon
transform, the image based weights are applied to the projections {pt,θ} for
each bin (see (Eq. (2)) and this eliminates the need for multiple reconstruc-
tions when performing image-based weighting.
2.2.1 Energy weighting to optimize CNR for a given imaging task
Consider the simplified object of Fig. 2. The expected number of counts in
each bin for rays 1,2 and b is given by:
n1j = I0
∞
0
Φ(E)D(E)Sj(E)e−(L−d)µb(E)−dµ1(E)
dE (17)
n2j = I0
∞
0
Φ(E)D(E)Sj(E)e−(L−d)µb(E)−dµ2(E)
dE (18)
nbj = I0
∞
0
Φ(E)D(E)Sj(E)e−Lµb(E)
dE (19)
In Appendix A we derive the weights with which each bin count should
be weighted in order to optimize the detectability of objects 1 and 2 against
the background. If nb is the vector with individual elements nbj and similar
for n1, the optimal weights are (proportional to):
w1 = Cov(nbj)−1
(n1 − nb) (20)
and similar for w2. If the crosstalk from charge sharing and scatter/escape
photons between energy bins is negligible, the covariance matrix Cov(nbj)
10

Figure 2: Simple geometry.
is diagonal with each diagonal element being the variance of the counts in
the corresponding bin: Cov(nbj) = diag(var(nbj)), or
w1j =
(n1j − nbj)
var(nbj)
. (21)
In the projections, due to the Poisson nature of photon counts, the vari-
ance of the counts is equal the expected value, and one can write w1j =
(n1j − nbj)/nbj. This is however not true in reconstructed CT images which
is why the more general form Eq. (21) must be used when weighting is
applied in the image domain.
2.2.2 Benefits and drawbacks with energy weighting
The main benefit with energy weighting is that it is straight forward and
that weights that optimize detectability of any known material can easily
be determined; either by use of forward model or by measurements over
regions-of-interest in images. Optimal weighting has been shown to increase
the signal-difference-to-noise ratio by 15-60% [15, 19]. Another benefit is
that it can be applied in both the image domain and projection domain; the
same general formula Eq. (21) holds.
A problem is that determination of the optimal weights require prior
knowledge of what one is searching for, i.e. knowledge of n1 and nb. This will
not always be the case as several such imaging task are likely to be relevant
11

in one and the same examination/image and the weights that are optimal for
enhancing a lung nodule are certainly not optimal when searching for small
cracks in the bone. If the task is unknown, all weights must be applied.
With N energy bins (usually between 5 and 8) this is a search over N − 1
dimensions as the entire surface of a hypersphere in RN must be covered.
This is not feasible and means that the displayed image has to be re-weighted
according to some pre-defined list constituting the imaging cases of interest,
but this again is not acceptable since one can never be sure that such a list
is indeed exhaustive. Furthermore, beam hardening can not be removed by
energy weighting.[41]
Despite these practical drawbacks, energy weighting plays an important
role when different system configurations are compared and it is reasonable
to perform such a comparison for a certain imaging case.
2.3 Basis decomposition by weighting
Basis decomposition by weighting, also denoted image domain basis decom-
position, is when the basis images are reconstructed via a linear approxima-
tion of the basis decomposition.
Here we give an example a linear approximation of basis decomposition
based on the phantom shown in Fig. 3. The example is taken from a 2014
SPIE contribution.[42]
Figure 3: Spectral CT phantom made of PMMA with inserts of (starting at
11 o’clock) water, plaster (high in calcium), gadolinium solution, iodine and
oil.
Consider the reconstructed image in the left panel of Fig. 4, denoted
12

photon counting since all bin images are just added together after recon-
struction. If the average bin counts of each ROI is plotted in the right
panel. Clearly, there is a distinct profile for each material.
1
2
3
4
5
6
HU
−1000 0 1000 2000 3000
1 2 3 4 5 6 7 8
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Energy bin number
AveragepixelvalueinROI,a.u.
1 Water
2 Calcium
3 Gadolinium
4 Iodine
5 Oil
6 Calcium border
Figure 4: Left panel: photon counting image of the cylindrical PMMA
phantom with water, calcium, gadolinium, iodine and oil. Also shown are
the ROIs used for measuring the image values for the different materials.
Right panel: Reconstructed image value in each energy bin, as measured in
the ROIs. A separate ROI was used for the border of the calcium insert,
where the energy dependence of the reconstructed values is slightly different,
presumably due to beam hardening.
Let m denote the [8×1]-vector with reconstructed values in each energy
bin and assume that a number of basis materials, with known linear attenu-
ation coefficient vectors µ1, . . . , µM have been chosen. Furthermore let a be
the [M × 1]-vector of densities of the different basis materials, normalized
to a unitless quantity in such a way that a component ai being equal to 1
corresponds to the same density as that of the basis material. The relation-
ship between m and a is nonlinear, because of the exponential form of x-ray
attenuation, but it can be approximated by the linear relationship
m = Ma (22)
where M = (m1, m2, . . . , mM ) is a [8 × M]-matrix built up from the basis
material attenuation profiles.
13

Calcium image Gadolinium image
Iodine image Color overlay
Figure 5: Basis images for imaged based decomposition according to
Eq. (23). The lower right panel shows the photon counting image with
a color overlay, where calcium is green, gadolinium is blue and iodine is red.
Since the number of bins exceeds the number of basis materials in this
case, Eq. (22) must be solved in the least squares sense, which gives an
estimate â of a as
â = MT
M
−1
MT
m (23)
where T denotes transpose. The different noise level in different bins could
be taken into account by using a weighted least squares solution instead,
but we have not done so here.
Eq. (23) maps the m vector at each pixel in the image to a set of basis
coefficients. In practice, the basis m vectors were measured as the averages
over regions of interest (ROIs) in the reconstructed images. The method is
illustrated in Fig. 5 where the materials are clearly separated.
14

3 What is being displayed in spectral CT?
In energy integrating CT, the effective attenuation µeff(x, y) normalized with
that of water is displayed, see Eq. (4). The choice of what to display in
spectral CT is much more complex. At least five different options exist
which will be briefly described below:
1. Bin images
2. Gray scale images from weighting
3. Basis images
4. Synthetic monoenergetic images
3.1 Bin images
In principle, one image could be presented for each energy bin. Based on
Eq. (6), this would constitute forming the set of projections
p(t, θ; Bj) =
I0(t, θ)
∞
0 Φ(E)D(E)Sj(E)e− l
µ(x,y;E)ds
dE
I0(t, θ)
∞
0 Φ(E)D(E)Sj(E)dE
(24)
and, via the inverse Radon transform, reconstruct µ(x, y; Ej) where Ej an
energy in the energy range of bin j. With N energy bins, this means N re-
constructions and therefore also an N-fold increase in computational time.
The images would have gray scale values with units cm−1 and be samples
of the linear attenuation coefficient at an effective energy inside the corre-
sponding bin. They could be normalized with the attenuation of water and
thus yield N different CT-number images in Hounsfield units.
It is very difficult to find any circumstance under which this approach
would be optimal. From the preceding chapters we know that a weight
factor w consisting of all but one zeros will not be optimal for realistic
backgrounds and targets if the goal is to maximize detectability (and a bin
image correspond to bin weight vector of the form [0, . . . , 0, 1, 0]).Such bin
images will thus be quite noisy and also qualitatively difficult to interpret
as rules of thumb such as fat exhibiting a HU value of −100 to −50 will be
different for different energy bins.
3.2 Gray scale images from weighting
We have shown how to determine an optimal weight vector w with which to
weight the bin signals m to get maximal signal to noise ratio and detectabil-
ity for any given imaging task. Unfortunately, this makes the display values
15

quantitatively useless. Even if the values are normalized with the corre-
sponding entity for water this would not allow for quantitative measure-
ments in the image as the values would depend on the weight vector which
in turn depends on the imaging task at hand. Thus, while intuitive and
good for detecting abnormalities, displaying optimally weighted gray scale
images does not allow quantitative CT.
3.3 Basis images
The reconstructed basis images âi(x, y) are scalar and can easily be displayed
for views. This can be highly beneficial if one particular base is chosen to
be clinically relevant, for instance being a contrast agent, in which case the
concentration can be readily determined, or calcium.
Displaying pure “photoelectric” and Compton cross sectional images add
little value. However, as shown by Alvarez[18] these bases can be linearly
combined to obtain the same signal to noise ratio as would optimal energy
weighting - but free of beam hardening. (In our opinion, this is the strongest
case for energy resolved CT; elimination of artifacts and the ability to se-
lectively enhance different sources of contrast.)
In conclusion, showing the basis images themselves, unless the bases are
selected to be physiologically relevant, is not recommended.
3.4 Synthetic mono-energetic images
Once the âi(x, y)’s are reconstructed, the scalar entity ˆµ(x, y; E) = i âi(x, y)fi(E)
can be displayed as a cross sectional image for any single energy E. These
images are beam hardening free and allow quantitative use of the image
data.
An added benefit is that for most imaging cases, i.e. a target volume
with µt(E) against a background µb(E)) as in Fig. 2, there exists an energy
E such that the signal-to-noise ratio in the mono-energetic image is equal
to the optimal achievable with energy weighting, i.e.
ˆµt(E ) − ˆµb(E )
varˆµb(E )
=
wt
T (λt − λb)
wt
T Cov(λb)wt
(25)
with wt according to Eq. (21). λt and λb being the vectors of expected
counts corresponding to a particular target and the background, see Eq. (6).
The nice thing is that Eq. (25) holds for most imaging cases. This means
that by just a one-dimensional search across the reconstruction energy one
can be assured that at one point or another, the displayed image will have
16

exhibited maximal CNR for any and all imaging cases. This reduces the
complexity of the search problem associated with finding the optimal weights
in RN since it can be implemented with a scroll wheel. This is illustrated in
Fig. 6 below and is analogous to the finding by Alvarez and Seppi[51] that
noise in monoenergetic images exhibit a clear minimum at a reconstruction
energy within the range of the x-ray spectrum.
In Fig. 6, the horizontal straight line is the right hand side of Eq. (25)
for a particular imaging case the specifics of which do not matter and the
curve is given by the expression on the left hand side, or, more explicitly, in
the following way:
1. The forward equation Eq. (6) is used to determine the expected num-
ber of counts in the background λb and λt for a specific target given
typical assumptions on the parameters of the forward model
2. Two basis functions spanning background and target are selected (f1(E)
and f2(E))
3. Eq. (12) is used to determine corresponding path length integrals A1
and A2
4. Given the known path lengths (L and t in Fig. 2) of the homogenous
materials, â1 and â2 are determined for target and background from
Â1 and Â2 for the target and background by division with the path
length
5. From â1 and â2, the linear attenuation coefficients are estimated:
ˆµj(E) = â1jf1(E) + â2jf2(E) where j ∈ {t, b}.
6. The left hand side of Eq. (25) is determined as
(a1t − a1b)f1(E) + (a2t − a2b)f2(E)
f2
1 (E)var(a1b) + 2Cov(a1b,2b )f1(E)f2(E) + f2
2 (E)var(a2b)
where Cov(a1b,2b ) (which will be negative) is determined using the
methods described by Alvarez[18].
17

0 20 40 60 80 100 120
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Reconstruction energy E(keV)
SdNR2
(arbitraryunits)
Figure 6: Illustration of how the signal (difference) to noise ratio of monoen-
ergetic reconstruction reaches the SdNR achieved by optimal weighting.
4 Practical guide
So - what should one do? Perform basis decomposition or energy weighting,
and in the latter case, in which domain; image domain or projection domain?
For some imaging tasks the choices do not matter, for other, the correct
selection is very important. This section is intended to bring some order to
the many different methods available.
The first question is what one wants to do?
1. Enhance some specific material?
2. Maximize CNR for a known target?
3. Maximize CNR for any unknown target (more realistic)?
4. Quantify a concentration or density (quantitative CT)?
5. Discriminate between materials?
Once this is determined, the different options below are available.
1. To enhance a specific material possible methods are:
(a) Basis decomposition in the projection domain and with the spe-
cific material as base fi(E). Then âi(x, y) is reconstructed from
the ML-estimates of Ai(t, θ) and displayed. This yields a beam
hardening free image of the selected material.
18

(b) Bin images can be reconstructed and then linearly weighted to
enhance speciﬁc materials, Eq. (23). This is in essence a linear
approximation of the above and will therefore not yield beam
hardening free images.[20, 21, 22]
2. To maximize CNR for a known target, possible methods are:
(a) Optimal weighting on projection data; requires knowledge of the
attenuating properties of target and background materials
(b) Optimal weighting of image data; requires same pre-reconstruction
knowledge as above.
(c) Basis decomposition and linear weighting of basis images; requires
pre-reconstruction knowledge
(d) Basis decomposition and display of monoenergetic image at en-
ergy E1; requires knowledge of reconstruction energy E1.
3. To maximize CNR for unknown target, the only feasible method is:
(a) Basis decomposition and display of monoenergetic image at all
energies E; requires scrolling the reconstruction energy E.
4. To quantify a concentration or a density there is only one feasible
method:
(a) Basis decomposition of projection data where one base is the
material for which the concentration should be determined.
5. For discrimination between materials
All combinations of domain and method work for discriminating be-
tween materials.
Note that for cases 3 and 4, basis decomposition in the projection domain
is the only option.
19

5 Challenges for photon counting multibin sys-
tems
The following are the main challenges for photon counting spectral systems
1. pile-up, associated with high x-ray flux
2. scatter/cross-talk from k-edge escape or Compton interactions
3. charge sharing between detector elements
4. accurate calibration
5.1 Pile-up
High flux results in an increased chance of overlapping pulses. This can
result in lost counts and distorted energy response functions. Due to the
exponentially distributed times between x-ray interactions pile-up to some
extent is always present.
No agreement exists within the community regarding how high x-ray
fluxes a photon counting detector must be designed for, but rates of 100
million x rays per square millimeter and second is generally considered suf-
ficient for scan protocols and x-ray tubes in use today. Worse is that no
agreement exists on how to present and compare the count rate capabilities
of different detectors. Several valid but not completely satisfactory measures
exist. Consider Fig. 7 showing two systems with roughly equal count rate
performance; for an input rate of 10 Mcps per channel3 (denoted segment
in the left panel) approximately 7.5 million event per second and channel
are counted. 25% of the counts are thus lost due to pile-up at an incident
conversion rate of 10 Mcps.
There is no agreement on how the performance depicted in Fig. 7 should
be reduced to a single scalar; is it the maximum count rate before 1% of the
counts are lost, 25% or 50%? From the figure it is also clear that the actual
count rate can be recouped from the output rate as long as the input rate-
output rate relationship is invertible (for the system in the right panel this
is valid up to 38 Mcps input rate). Statistical techniques[25, 26, 27] have
been derived for this conversion, but although the correct expected value
3
10 MCps might sound like a far cry compared to the required/desired flux rate capacity
of 100 Mcps/mm2
, but since the detector elements are smaller than a square millimeter,
and detector elements are distributed along the interaction depth in the silicon diode
system, both graphs translate to acceptable count rates
20

Figure 7: Left panel: Count rate performance for KTH silicon detector [23].
Right panel: count rate performance for ChromAIX[24].
of the transmission intensity can be recouped, the associated variance will
increase (and this translates to image noise). An issue of immense practical
importance is how the detector behaves in the time period following a peak
output count rate; does the detector break down or does it give reasonably
reliable readings in subsequent projections?
As noted above, pile-up not only results in lost counts but also decreased
energy resolution for the events that are actually detected. By scanning the
energy thresholds in a beam of synchrotron radiation the energy resolution
can be found as σ by fitting the complementary error function (1-error func-
tion where the error function is the integral of a Gaussian) to the counts[28],
see Fig. 8:
f(x; µn, σ, A, B) =
1
2
erfc
x − µn
√
2σ
(A(x − µn) + B). (26)
x is the detector threshold value (in mV) and µn and σ the parameters of
the underlying normal probability distribution function in the same units.
A captures charge sharing; for lower thresholds, more counts triggered by
charges that leak in from neighboring pixels are registered. The results in
the upper part of the s-curve not being flat, Fig. 8.
In Fig. 9, the resulting energy resolution is plotted vs. the count rate,
showing a clear decrease i.e. larger rms-values σ for increased flux. For this
reason it is important to state the count rate at which a reported energy
resolution has been obtained.
21

30 35 40 45 50 55 60
0
500
1000
1500
2000
2500
3000
x (mV)
counts
Figure 8: Registered counts vs. threshold value x and complementary error
function fitted to (26).
Figure 9: Energy resolution (rms of photo peak) as a function of flux.[23]
5.2 Scatter and charge sharing
An uncomfortable and sometimes overlooked truth regarding scatter and
charge sharing is that in photon counting mode, where all events are weighted
equally, a scatter-induced double count to a first approximation is just as
detrimental as a lost primary count. The same holds for the effect of object
scatter on image quality. To see this, consider Fig. 10 where it assumed
that scatter (S) is added homogenously to a projection image. Pre-scatter
contrast by definition is Cpre = |It−Ib|
Ib
. Using the same formula post-scatter
is
Cpost =
|(It + S) − (Ib + S)|
Ib + S
=
|It − Ib|
Ib
1
1 + S/Ib
. (27)
22

Since Ib is the primary signal (P) we get
Cpost = Cpre
1
1 + S/P
≈ Cpre(1 − S/P) (28)
where the approximation 1/(1 + x) ≈ 1 − x is valid for small scatter-to-
primary ratios S/P. Eq. (28) thus shows that 1% scattered radiation costs
as much as 1% missed counts. For this reason it is paramount to keep
scatter, both from the object and from within the detector, to a minimum
for instance by thresholding k-fluorescence rays in CdTe/CZT-detectors.
x
I0
Ib It Ib
Ib
Ib + S
It
It + S
Figure 10: Derivation of the contrast reduction factor.
The above pertains to photon counting mode. If pulse height determi-
nation is applied the energy resolution is also effected, leading to skewed
energy distributions towards lower energies. This skewness needs to be in-
corporated into the forward equation Eq. (6) if one intends to perform basis
material decomposition.
5.3 Calibration issues
Different detection efficiencies or energy response functions among individual
detector elements result in ring artifacts in the final image if not properly
23

compensated for. There are two distinct ways to approach the problem of
ring artifacts in multibin systems, depending on whether energy weighting
is applied or basis decomposition.
If weighting is applied, the total counts in each bin need to be identical
across detector elements in homogenous areas of the image, or rings will
occur. Since there is always some threshold dispersion between detector
channels this must be compensated for. For multibin systems, such com-
pensation schemes tend to be more complex than for energy integrating
systems; and will depend on imaging parameters. Nevertheless, a suggested
“reshuffling” of bin-counts by means of an affine transformation have shown
satisfactory results in simulations under realistic conditions.[29] For each
detector element j a N × N matrix Aj and an N × 1 vector bj is applied
to the bin counts mj to get the adjusted counts:
madj
j = Ajmj + bj. (29)
A and b are determined by measuring the response of each channel to combi-
nations of materials with known thickness to ensure that the method works
for the full range of different tissue combinations one can expect to encounter
in practice.
A second approach to eliminate the negative effects of threshold spread
is to perform material basis decomposition with separate forward models for
each detector element. The calibration process then focuses on determining
the forward model “accurately enough”. A reasonable question is just how
accurate such a calibration has to be. The question of how accurate the
parameters of the forward model must be known in order to perform material
basis decomposition in practice has been addressed by the authors[30] and
recapitulated here.
The figure of merit is based on the observation that when the param-
eters of the forward model are incorrect the estimates â1(r) and â2(r) (as
reconstructed from the ML-estimates of Eq. 12) will be biased. This bias
will add to the variance and result in a mean square error that is larger than
the variance of the unbiased estimate since:
MSE(â) = E (â − a)2 = E[â2] − 2E[â]a + a2 =
E[â2] − E[â]2 + (E[â] − a)2
= var(â) + bias2
(â).
(30)
If MSE(â) is determined for different deviations of the forward parameters
from the true values and related to the Cramér-Rao lower bound (CRLB)
of the variance in the case of a correctly specified forward model, one can
determine the allowable misspecification for each parameter of the forward
24

model. Below this is shown for the most important parameter, the set of
thresholds {Ti} for each detector element.
The method is developed and illustrated for a photon counting multi-
bin system with silicon detector diodes[35, 36, 37, 34] but the methodology
applies equally well to other detector materials. Recall that the goal is to
find how large misspecifications of the forward model that can be tolerated
before the bias component of the mean square error of Eq. (30) is larger than
the variance part. This appears straight forward: the CRLB of the variance
of the A1-estimate (see Eq. (8)) is given by element 1,1 of the inverse of the
Fisher information matrix[46]
σ2
Â1
≥ F−1
11
(31)
where
Fjk =
N
i=1
1
λi
∂λi
∂Aj
∂λi
∂Ak
. (32)
The bias is obtained by solution of the maximum likelihood problem given
the observed bin counts {mi}:
A∗
1, A∗
2 = arg max
A1,A2
P({mi}; A1, A2) = arg min
A1,A2
N
i=1
(λi − mi log λi) (33)
where λi = λi(A1, A2) according to Eq. (6).[46] For the bias calculation the
expected value of counts in each bin (i.e. noiseless) was used, i.e. we apply
the approximation E Â({mi}) ≈ Â({λi}).
The results will unfortunately depend on dose and on the size of the
region of interest (ROI) that is examined; for larger dose or when averaged
over a larger area, the random variance component of Eq. (30) will decrease
whereas the contribution from the bias will not (at least in the case of
a homogenous cylindrical object where the central volume is considered).
Thus we have to select typical x-ray fluence for which the comparison is
carried out, and also determine how the size of the region of interest affects
the result. First however, we show how variance and bias in the projection
domain is translated to the reconstructed image domain. Note that we in
this model are assuming that all thresholds move in parallel, i.e. to have the
same errors. This is the expected behavior of a temperature increase of one
particular photon counting detector.[34]
Hanson[40] has shown how the single-pixel variance in a reconstructed
image depends on the variance of a projection measurement:
σ2
â1
=
σ2
Â1
(λ)
Nθa2
k2
. (34)
25

a = 1 mm is the size of the detector elements (and distance between samples)
and Nθ the number of projection angles. k is a unitless factor depending
on filter kernel for the filtered backprojection and determined to k = 0.62
for Matlab’s iradon with cropped Ram-Lak ramp filter.[48] Please see Ap-
pendix B for details.
One would easily be lead to believe that the bias in the middle of the
reconstructed image of a homogenous cylinder with diameter L is given by
bias(a) = bias(A)/L since by definition, Eq. (8), A(t, θ) = l a(r)ds = aL for
a central ray l. When estimates are biased, as in the case of a misspecified
forward equation, one cannot however be sure that Â(t, θ) = â l ds for all
possible path lengths { l ds} in the sinogram/projection domain. If the rel-
ative bias does depend on the path length it is not at all clear how a bias in
the projection domain (A-space) translate to the image domain (a-space).
The basic problem is similar to a characteristic of the Fourier transform:
a change at one point in the Fourier domain of a function alters the spa-
tial representation of the function at all points. The same holds for the
filtered back projection due to the filtering step and thus, if relative biases
differ across projections for instance due to path length, this will propagate
unpredictably to the image domain bias.
A sufficient condition for bias(a) = bias(A)/L to hold in the central
region of reconstructed image of a homogenous cylinder with diameter L is
Â = c a
l
ds (35)
with constant c for different path lengths l ds. If Eq. (35) holds, we have
bias(A) = Â − A = (c − 1)a l ds since A = a l ds by definition. In the
reconstructed domain, we have bias(a) = â − a where â is obtained via
the inverse radon transform of Â of Eq. (35). Due to the linearity of the
transform, â = ca and bias(a) = (c − 1)a and it follows that bias(A) =
L bias(a).
Eq. (35) unfortunately does not hold for all path lengths l ds and differ-
ent misspecifications. However, in Appendix C we show, that Â ≈ c a l ds
for l ds ∈ [0.85, 1]L . Under these feasible circumstances Eq. (35) holds
approximately and thus bias(a) ≈ bias(A)/L.
The correlation structure in the reconstructed image makes the variance
in an ROI consisting of M pixels decaying faster than M−1. In the limit of a
continuous image, with perfect bandlimited interpolation of the projection
data before backprojection, the variance depends on M as M−3/2.[38] In
previous work[48], we have shown (with a discrete image and typical inter-
polation) that the variance decreases as M−1.33. This derivation is reiterated
26

in App. B. For an ROI consisting of M pixels, the standard deviation will
thus have decrease by a factor of
√
M−1.33 = M−0.66. For a 10 × 10 pixel
large ROI this evaluates to 100−0.66 = 0.0479. Since quantitative CT will
most likely be carried out over ROIs much larger than a single pixel, we
chose to compare the bias with the variance over a 100 pixel large ROI.
A homogenous cylindrical object with L = 20 cm diameter consisting of
50% soft tissue and 50% adipose tissue is assumed for the evaluation (at-
tenuation data from ICRU 44,[39]). The bases f1(E) and f2(E) of Eq. (7)
are taken as the linear attenuation coefficients for the same materials. The
unattenuated x ray fluence, N0, must be selected as the total number of
x rays per reconstructed pixel area for an entire gantry revolution. (Note
that the lower bound of the variance in the reconstructed image is indepen-
dent of the number of angles and depends only on N0 since λ = λ(N0/Nθ)
in Eq. (31) and σ2
Â1
(λ(N0/Nθ))/Nθ = σ2
Â1
(λ(N0)).) Assuming a typical
current-time product of 120 mAs the x-ray tube model of Cranely et al.[47]
with a 120 kVp tungsten anode spectrum, 6 mm Al filtration, 7◦ anode angle
and 1000 mm source-to-isocenter distance gives 8.1 · 108 photons mm−2s−1
on the detector. With 3 revolutions per second as a typical rotation speed,
N0 = 2.7 · 108 are directed towards a central pixel area of 1 mm×1 mm
(which is the area henceforth assumed).
For the default parameter values, the Cramér-Rao lower bound of the
variance of the A1-estimate is given by Eqs. (31) and (32) in conjunction
with Eq. (6). Combined with Eq. (34) this yields σ(â1) = 0.68 (unitless)
(which should be related to E(â1) = 0.5 (also unitless)). For an ROI with
M=100, σ(â1) is 0.033. This is the value the bias should be compared to.
Fig. 11 indicates that the misspecified thresholds are very detrimental
to quantitative CT. For parallel shifts in the thresholds of 0.1 keV or more
(as compared to calibration values), the bias introduced will dominate the
MSE. This indicates that thresholds uncertainties of more than some 0.1 keV
cannot be tolerated for quantitative CT. We have recently devised a broad
spectrum calibration scheme that fulfills this requirement and is also feasible
in clinical practice.[17]
27

−1 −0.5 0 0.5 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
CRLB σ(â1) (100 pix)
|bias(â1)|
CRLBσ(â1),|bias(â1)|
∆T (keV)
Figure 11: Effect of uncertainty in thresholds.
6 How to evaluate
Linear systems theory[50] is normally applied x-ray imaging modalities to
express the DQE, detective quantum efficiency, the spatial frequency equiva-
lent of signal-to-noise ratio. The metrics derived using linear systems theory
have been shown to correlate very well with the results of human observer
studies[31], and has the benefit of allowing system optimization while still
on the drawing desk of the engineer.
The method has been adapted to computed tomography[32, 33] but is
somewhat elusive to apply to multibin systems; the noise and signal char-
acteristics of of multibin systems depend heavily on the method for image
generation; for instance the weight factor applied to the bin will have an
effect on the DQE and thus the intended system figure of merit depends
on what on intends to image. For the non-linear basis decomposition tech-
niques MTF and NPS (constituents of DQE) cannot be defined, so neither
for energy weighting multibin systems nor for basis decomposition systems
is there such a thing as one system DQE, but for different reasons.
To make matters even worse, DQE is intended as a system measure and
thus does not take the contrast of a particular object into consideration. Yet
one of the benefits of using the spectral information was to avoid contrast
cancellation and to selective enhance the contrast of different objects; if this
capability is not included in the figure of merit at all, certainly it cannot
be used for high-level system comparison. Instead the related detectability
index[50] has to be used as it combines spatial and contrast resolution. In
Ref.[43] it is shown how both the spatial an contrast resolution are affected
28

by different weights and how they need to be optimized simultaneously.
Let us conclude by noting that it is still an open question how imaging
performance should be compared across spectral CT systems (just like it is
with iterative reconstruction techniques). Indeed, one cannot even give a
straight answer to the following (very reasonable) question: “What is the
spectral resolution of your system?” because the answer would depend on
how the spectral resolution is interpreted and could mean, at least, any of
the below:
• The underlying standard deviation σ (in keV) of the photo peak of
Eq. (26) at low count rates (i.e. composed mainly of the the intrinsic
energy resolution of the direct conversion material and the electronic
noise).
• The σ (in keV) of the photo peak of Eq. (26) at various realistic count
rates as in Fig. 9, mainly capturing pile-up effects.
• The width of an energy bin, Ti+1 − Ti (also in keV). This corresponds
to determining the energy of a single photon by a multibin system.
• A very technical measure, and quite difficult to translate to image
quality, would be the capability to use the detector as a spectrograph,
determining the unknown energy of the photon of a monochromatic
beam (this would also be stated in keV and is a measure where dual
layer detectors come out fairly well).
• The ability to selectively enhance energy dependent contrast (how this
property should be captured by a figure or merit single scalar is very
hard to see though. . . ).
• The ability to quantify different tissues? The unit of this measure
would be unitless since in essence it is the mean square error of a
basis coefficient estimate, Eq. (30) (or linear combinations of the basis
estimate coefficients âi).
29

A Derivation of optimal weights
This derivation is the courtesy of Mats Persson and first appeared in his
Master’s thesis.[44] A similar but not as detailed derivation can be found
in the the seminal paper of Tapiovaara and Wagner[45] where it was first
pointed out to the community how the contrast carrying information differs
for photons of different energies and how this should be taken into consid-
eration.
Assume that we have two hypotheses H0 and H1 and that we know that
exactly one of them is true. We want to use the image data to determine
which one is true and which one is false. In the cases which we will be
concerned with here, H1 represents the presence of some feature, e.g. a
tumor or a bone, which is different from the background tissue, while H0
represents the absence of the feature. Assume we count the interactions in
in each bin, mi, each of which can be seen as the outcome of a random
variable gi. Let
g = (g1, g2, . . . , gN )T
(36)
be a vector containing all these random variables. Note that if we want
to use g to determine whether H0 or H1 is true, the probability densities
prg|H0
(g|H0) and prg|H1
(g|H1) must be different. Let gm = g|Hm be the
expectation value of g under hypothesis m for m = 0, 1 and let Km be the
covariance matrix of g, with entries Km
ij = (gi − gi)(gj − gj)|Hm .
In order to use the measured data g to decide whether it is H0 or H1
that is true, one forms a test statistic t = T(g) where T is some real-valued
function of the data vector, possibly nonlinear. Then, t is compared to
some threshold value tc, and the outcome of the test will be H0 if t <
tc and H1 if t > tc (or vice versa if it is H1 that corresponds to lower t
values). The so-called discriminant function T and the decision threshold
tc should of course be chosen so that, if possible, this test identifies the
correct hypothesis for most outcomes g. However, it is usually impossible
to find a discriminant function that takes entirely separate values under
H0 and H1, and so there will always be a risk that the test makes some
incorrect decisions, so-called false positives and false negatives. There are
several ways of comparing the performance of different tests. One approach
is to assign costs to the different outcomes (true positive, false positive, true
negative and false negative) and then say that the best test is that which
minimizes the expected value of the cost. This is called the Bayes criterion.
Another approach, which does not require that one assigns costs to the
different outcomes, is the so-called Neyman-Pearson criterion which states
30

that the best test is that which gives maximal true positive rate for a fixed
false positive rate. It can be shown that [49] there is an ideal discriminant
function which is optimal under both these criteria, namely the likelihood
ratio
Λ(g) =
prg|H1
(g|H1)
prg|H0
(g|H0)
(37)
The observer that decides between H0 and H1 according to (37) is called the
ideal observer. In practice, the complete probability distribution function of
g is seldom known, meaning that one has to use other, nonideal, observer
models to assess image quality. It also worth noting that medical images
are normally intended to be used by human observers, who might perform
significantly worse than the ideal observer. In the present study we shall
measure image quality by the performance of the linear observer, i.e. the
observer with discriminant function given by
T(g) = wT
g (38)
where w = (w1, . . . , wN )T
is a vector of weights. Furthermore, we will
optimize this with respect to the squared signal-difference-to-noise ratio
SDNR2
=
( T(g)|H1 − T(g)|H0 )2
V [T(g)|H0] + V [T(g)|H1]
=
(wT g1 − wT g0)2
wT K0w + wT K1w
=
=
(wT (g1 − g0))2
wT (K0 + K1)w
=
(wT ∆g)2
wT (K0 + K1)w
(39)
where ∆g = g1 − g0, V [ · ] denotes variance and the identity
V [T(g)|Hm] = (wT
g − wT
g)2
|Hm = wT
(g − g)
2
|Hm =
= wT
(g − g)(g − g)T
w|Hm = wT
(g − g)(g − g)T
|Hm w = wT
Km
w
has been used. Note that an additional factor of two is included in the
definition (39) by some authors, since this gives a neater formula in the case
when K0 = K1. We shall show that whenever K0 + K1 is invertible, which
is the case in most practical situations, (39) is maximized by choosing
w = K0
+ K1 −1
∆g (40)
or some scalar multiple thereof. In order to show this, we note that (K0 + K1)−1 T
=
(K0 + K1)−1, since Km is symmetric and the inverse of a symmetric matrix
31

is symmetric. Now, equation (39) with w given by (40) yields
SDNR2
=
(∆gT (K0 + K1)−1∆g)2
∆gT (K0 + K1)−1(K0 + K1)(K0 + K1)−1∆g
=
=
(∆gT (K0 + K1)−1∆g)2
∆gT (K0 + K1)−1∆g
= ∆gT
(K0
+ K1
)−1
∆g (41)
We want to show that the above expression is an upper bound for (39). Note
that K0 + K1 has a square root matrix (K0 + K1)1/2 that is symmetric and
invertible (See appendix A), meaning that wT ∆g can be rewritten as
wT
∆g = wT
(K0
+ K1
)1/2
(K0
+ K1
)−1/2
∆g =
= (K0
+ K1
)1/2
w
T
(K0
+ K1
)−1/2
∆g
Interpreting the above expression as a scalar product and using the Cauchy-
Schwarz inequality (see [52], theorem 6.2.1) then gives
wT
∆g
2
≤ (K0
+ K1
)1/2
w 2
· (K0
+ K1
)−1/2
∆g 2
≤
≤ wT
(K0
+ K1
)1/2
(K0
+ K1
)1/2
w∆gT
(K0
+ K1
)−1/2
(K0
+ K1
)−1/2
∆g
wT
∆g
2
≤ wT
(K0
+ K1
)w∆gT
(K0
+ K1
)−1
∆g
Dividing both sides by wT (K0 + K1)w gives
wT ∆g
2
wT (K0 + K1)w
≤ ∆gT
(K0
+ K1
)−1
∆g
SDNR2
≤ ∆gT
(K0
+ K1
)−1
∆g (42)
with equality if and only if (K0+K1)1/2w and (K0+K1)−1/2∆g are linearly
dependent, i.e. for some real-valued constant k,
(K0
+ K1
)1/2
w = k · (K0
+ K1
)−1/2
∆g
w = k · (K0
+ K1
)−1
∆g (43)
which is the required formula. The conclusion is that the discriminant func-
tion of the linear observer which gives optimal SDNR is obtained by substi-
tuting (40) into (38):
T(g) = ∆gT
K0
+ K1 −1
g (44)
32

The above formula is called the Hotelling observer for the problem of dis-
criminating between H0 and H1. It can be shown [49] that this observer is
actually equivalent to the ideal observer for the case when g is multivariate
normal with equal covariance but different mean under the two hypotheses.
We shall use (44) and the corresponding optimal weight factors (40) in order
to measure and optimize detectability in images.
It should be pointed out that the above discussion based on SDNR is
relevant only in the limit where the structures to be detected in the image
are uniform over very large areas. When investigating the detectablity of
smaller structures, one has to compare the signal strength with the noise
level for each of the spatial frequences present in the image. In other words,
it becomes necessary to use linear systems theory to investigate how signal
and noise propagates through the imaging system in Fourier space. When
studying x rays, the linear systems theory approach is complicated further
by the fact that the number of photons in each measurement is few enough
that the flux distribution cannot be treated as a continuous function but
must be represented by a point process, which is a certain kind of stochastic
process whose realizations are sums of delta functions. These ideas form
the basis for a useful framework which is, however, beyond the scope of
this thesis. The interested reader is referred to [49]. Even though SDNR
is a crude measure of imaging system performance, it is useful since it is
so easy to calculate while still giving some important information about the
imaging performance, at least in large homogeneous regions of the images. It
is therefore useful during the process of designing an imaging system, when
it is desirable to have a figure of merit in order to make design decisions
quickly, without having to analyze and model the complete system.
In this section we will fill in some details justifying the calculations lead-
ing up to equation (43). Let K0 and K1 be symmetric, positive semidefinite
matrices and assume that K0 + K1 is invertible, like in section A. Then
K0 +K1 is also symmetric and positive semidefinite. The symmetry implies
that K0 + K1 = PDP−1 for some matrix P such that PT = P−1, where D
is the diagonal vector of eigenvalues of K0 + K1:
D =





λ1 0 · · · 0
0 λ2 · · · 0
...
...
...
...
0 0 · · · λN





(45)
(See [52], theorem 7.3.1.) It also follows that all λi are real and strictly
positive. (The possibility of some eigenvalue being zero is ruled out by the
33

invertibility of K0 + K1.) Now D1/2 is the diagonal matrix with entries
λ
1/2
i on the diagonal, and then (K0 + K1)1/2 is defined as PD1/2P−1. The
invertibility of (K0 + K1)1/2 follows from this equation and the fact that all
λ
1/2
i > 0. The symmetry of (K0 + K1)1/2 follows from the equation (K0 +
K1)1/2 = PD1/2PT . To conclude, (K0 +K1)1/2 exists and is symmetric and
invertible.
B Variance as a function of ROI size
Due to correlations and the effect of the filter, the lower limit variance over
an ROI of M pixels can be expressed as h(M) times the variance over one
pixel:
CRLB σ2
â(M) = CRLB σ2
â(1)h(M). (46)
This section also appears in Ref. [48] and determines k of Eq. (34) and h(M)
of Eqs. (46) and (47), i.e. how the variance and means square error in the
image is affected when averaging is performed over an ROI with M pixels.
Since the bias is not affected by ROI size, we can write:
MSE(â3; M) = varPo(â3)h(M) + E bias2
(â3) . (47)
To determine h(M), a large sinogram (Nθ = 300 angles and Nt,θ = 1000
detector positions) is filled with gaussian noise with zero mean and unit
variance. In Eq. (34) this corresponds to σ2
Â
/a2 = 1, i.e. the variance in each
a2 large detector element is unity. The backprojection algorithm is applied
and the resulting image is smoothed with square filters with increasing side
lengths of
√
M pixels. The resulting standard deviation is measured in the
reconstructed image. Since σ2
Â
/a2 = 1, a combination of Eqs. (34) and (46)
indicate that it is the normalized entity
σâ(M) Nθ = k h(M) (48)
that is of interest. k h(M) is plotted in Fig. 12. Since h(M = 1) = 1, k is
0.62. The dashed line has a slope corresponding to h(M) = M−1.33, but it
is the simulated functional form of h(M) that is used in the derivations of
Sec. B.
C Constancy of c for different path lengths
In this section we show that c of Eq. (35) is close to constant for some
different forward model misspecifications. For values of the path length
34

10
0
10
1
10
2
10
3
10
4
10
−3
10
−2
10
−1
10
0
M (pixels in the square ROI))
kh(M)
Figure 12: k h(M) of Eq. (48) as a function of ROI size M.
l ds ranging from 0.85L to L Â1 is estimated by Eq. (33) (by insertion
of the erroneous forward parameter in Eq. (6)). c is then determined by
c = Â1/ l a1ds and plotted against path length expressed as a ratio to
the diameter L. Relative changes of c seem to be confined to around or less
than 1% over the interval, for which reason we can model it as approximately
constant.
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0.85 0.9 0.95 1
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