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’etre 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 imagingtask . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 distributionf(x, y). If the projection pl is defined as pl =

∫

l f(x, y)ds for the straight linel, the set {pl} for all l ∈ R2 is the Radon transform of f(x, y), henceforthdenoted Rf(x, y).

In computed tomography, the goal is to reconstruct the interior distri-bution of linear attenuation coefficients from projection data. This requiressolving 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, forinstance [2, 3].

Conventional, or energy integrating CT, is based on indirect detectorswhere a scintillator material (for example GOS) first converts the energy ofthe x-ray quantum to visible light. This light is then converted to electriccharge 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 essentialphysics of the imaging chain (neglecting electronic noise), and describesmany of the imperfections associated with energy integrating systems:

It,θ = n0

∫

Φ(E)D(E)E exp

(

−∫

lµ(x, y;E)ds

)

dE (1)

The parameters and variables of Eq. (1) are described in table 1 and depictedin 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 atangle θ in one integration pe-riod

- -

I energy integrated signal keV -Φ(E) x-ray energy distribution

functionkeV−1 Normalized such that

∫

Φ(E)dE = 1D(E) Detection efficiency - Convertion efficiency of

the detector times thegeometric efficiency

t detector coordinate/element - see fig 1θ rotation angle of the projec-

tionrad. see fig 1

l line connecting the x-raysource and detector element tat 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 = − logIt,θI0

= − logn0

∫

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 =

∫

lµ(x, y;E′)ds (3)

where E′ is an interior energy of the spectrum Φ(E). E′ will depend onthe particular effective spectrum for the projection line l. If the effectivespectrum Φ is different in different views, for instance due to passing aparticulary radio-dense region such as bone, the linear attenuation coefficientwill be reconstructed at a higher energy. Since the linear attenuation tendsto decrease with energy, reconstruction at a higher effective energy resultsin darker/lower values, which results in characteristic dark streaks in theimage. Even for homogenous objects this will be a problem due to thedifferent attenuation lengths which result in different effective spectra andthereby different reconstruction energies. The result of this is the cuppingartifact, where the interior of the image appears darker. Both these problemsare referred to as beamhardening. This is one of the problem associated withenergy integrating CT systems, artifactual inhomogeneities within the imagedue to beamhardening.

A related problem arises when one wants to compare CT-numbers (inHounsfield units) between different systems. In CT, it is customary to dis-play the effective linear attenuation coefficient normalized to water:

HU(x, y) = 1000µeff (x, y)− µH2O

eff

µH2Oeff − 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, ifany forward model parameter such as x-ray spectrum or detection efficiencyD(E) is different for different systems, the line integrals will correspond todifferent energies E′ in Eq. (3) and thus CT-numbers will differ betweensystems and depend on exposure conditions. Normalization with water ispartly used to compensate for this, but systematic differences still remainbetween 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 linearattenuation coefficients with energy, the contrast between tissues tend to

4

decrease with increasing energy. A system that, as in the case of energyintegrating systems, places more weight at high energy events is thus notachieving ALARA1 since it is not utilizing the contrast information availablein the most optimal fashion.

Fourthly, if the linear attenuation coefficients of a particular target andbackground when weighted over the spectrum Φ according to Eq. (1) aresimilar, their contrast will cancel.[6] This is a clinically relevant issue that ex-plains the observed loss of contrast for certain iodine concentrations2 againstplaque and bone.

A fifth problem with energy integrating detectors is that electronic noiseduring the integration time is added to the signal. This makes real low-dose acquisitions difficult as the performance of the system relative the dosedecreases with decreases dose as electronic noise becomes more and moreprominent.[7]

In summary, the problems with standard CT can be grouped in thefollowing categories:

1. Beamhardening

2. CT-numbers depend on parameters of the forward equation and thusdiffer between systems, this makes quantitative CT difficult

3. The contrast and the contrast to noise ratio between two tissues in thebody is energy dependent, and low energy photons usually carry moreinformation. This information is lost in energy integrating detectorsand 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

1As Low As Reasonably Achievable; a doctrine stating that dose should be kept as lowas possible while still maintaining diagnostic quality.

2Iodine contrast agents are very common in CT exams.

5

2 Raison d’etre for multibin CT

All five limitations with energy integrating CT can be removed with photoncounting multibin CT. The solution to limitations 1 and 2 are based on themethod of basis decomposition and the solution to the third and fourth canbe based on energy weighting as well as well as on basis decomposition. Theproblem with electronic noise integrated into the signal is removed by thepossibility of thresholding the signal. In this chapter, basis decompositionand 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 signalcorresponds to an energy between the thresholds Tj and Tj+1 (TN+1 =∞). Further assume a detection efficiency D(E) and an energy responsefunction R(E,E′). The response function denotes the probability that anx-ray quantum of energy E gives rise to a signal corresponding to E′ and isnormalized 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 elementt at a projection angle θ where x rays have traversed a path l with linearattenuation µ(x, y;E) is given by

λ(t, θ;Bj) = λj = I0(t, θ)

∫ ∞

0Φ(E)D(E)Sj(E)e−

∫lµ(x,y;E)dsdE. (6)

The actual number of registered events in each bin, mj, will be Poissondistributed 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 oneassumes that the unknown attenuation coefficient can be decomposed intoM bases with known energy dependencies:

µ(x, y;E) ≈M∑

i=1

ai(x, y)fi(E). (7)

6

With

Ai(t, θ) =

∫

lai(x, y)ds, (8)

the exponent∫

l µ(x, y;E)ds in (6) can be approximated by∑

iAi(t, θ)fi(E)allowing us to express the expected number of counts in each energy bin asa function of the M parameters Ai:

λj(t, θ) = I0(t, θ)

∫ ∞

0Φ(E)D(E)Sj(E)e−

∑Mi=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 Nbins are independently Poisson distributed, the likelihood function can bewritten

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 termsnot 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 thatminimize (11) for the observed data m = (m1, . . . ,mN ):

A∗1, . . . , A

∗M = argminL(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 ai(x, y) which then constitutethe 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 functionsM in Eq. (7) should be used and which the bases f(E) should be. It isinstructive to first consider the dependencies of the cross section of photoninteraction, 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 Eare 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 attenuationcoefficients for bodily constituents (being mixtures of low Z elements) is alsospanned by the three energy basis functions fα(E):

µ(E) = ρNA

∑

i

wi

massiσ(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 theatomic mass of element i and wi the fraction by weight for element i.

Although the above theoretical considerations conclude that the rankof the linear attenuation space for human tissues at clinically relevant x-ray energies should be at least three, possibly higher if the assumption onseparability must be waived, basis decomposition methods[10, 11, 12] as-sume that the linear attenuation space, disregarding k-edges, is spannedby only two basis functions. This is a reasonable assumption, since thephotoelectric effect and Compton scattering dominate over Rayleigh scat-tering, i.e. for clinically relevant energies and atomic numbers it holds thatfcoh(E)gcoh(Z) � fph(E)gph(Z) and fcoh(E)gcoh(Z) � finc(E)ginc(Z) inEq. (14). Furthermore, any part of a third basis not being orthogonal to thefirst two will be captured by the first two bases. Indeed such two functiondecompositions work well; good basis decomposition results have been ob-tained on real data by for instance Schlomka et al.[13]. Although theoreticalwork on the dimensionality of the linear attenuation coefficient space has re-sulting in a statistically significant dimension of four [14], the last two basesare very weak and negligible for (most?) practical purposes – for naturallyoccurring human tissues. When contrast agents with k-edge discontinuitiesin the x-ray spectrum are used, the decomposition must be expanded toaccount for this, using one additional basis for a each contrast compoundwith 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 shouldbe the linear attenuation coefficient of the corresponding k-edge element.

2.1.3 Benefit of basis decomposition

When we have the set of ai(x, y)’s for each image coordinate x, y, (seeEq. (7)) the corresponding linear attenuation is derived as

µ(x, y;E) =∑

i

ai(x, y)fi(E). (16)

By construction, due to the ML-solution taking the exponential nature ofphoton attenuation into consideration, the linear attenuation coefficients arereconstructed beam hardening free.

If the forward model is accurately specified (see also Sec. 5.3), the basisdecomposition results in unbiased estimates of the true linear attenuation co-efficient at all energies. This allows quantitative CT, i.e. the use of absoluteimage values for diagnostic purposes. Translation of radiologists’ rules-of-thumb, such as “more than 2% higher attenuation in a liver cyst comparedto the background indicates. . . ”, between different systems will not be aproblem as long as the forward models of each system are known.

Basis decomposition also allows for generating a map of a k-edge contrastagent concentration. This can be used for quantifying the contrast agent inspace and time and also for post-acquisition removal of the contrast agent,thereby generating an artificial pre-contrast image. In diagnostic proceduresthat 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 becauseany problem with missregistration due to patent movement between expo-sures is eliminated.

Finally, if the linear attenuation coefficients of target and backgroundcross each other for certain energy they might cancel in energy integratingimaging. However, as the entire energy dependence of the linear attenuationcoefficients is reconstructed, this method allows for the generation of syn-thetic monoenergetic images that present the linear attenuation coefficientat any energy. This ensures that cancellation does not occur as the linearattenuation coefficients will not be equal for all energies.

2.2 Energy weighting

A more straight forward alternative to use the energy information is tosimply linearly weight the counts in each bin. While simple and intuitive,

9

such an approach does not eliminate beam hardening and does not resultin an image quantity which is easily interpreted physically. The weights arenormally selected to achieve one of two tasks: to maximize the contrast-to-noise ratio (i.e. detectability) of a certain imaging task, for instance acyst against a homogenous liver background, or to approximate the basisdecomposition by determining the linear transform that most accuratelyapproximates the Ai’s of Eq. (12).

The weights can be applied either prior or after reconstruction, denotedprojection based weighting and image-based weighting respectively. Image-based weighting was proposed by Gilat-Schmidt[15] and has some desirabletheoretical properties, but the difference to projection based weighting inclinical applications appears to be small.[16] The mathematical treatmentis similar for the two methods. The only difference is whether the weightsoperate on transmission intensities (I, see Eq. (1)) or on the images re-constructed by each bin. We will only present a rigorous derivation of theweights in the projection domain. Due to the linearity of the inverse Radontransform, the image based weights are applied to the projections {pt,θ} foreach 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 ineach bin for rays 1,2 and b is given by:

n1j = I0

∫ ∞

0Φ(E)D(E)Sj(E)e−(L−d)µb(E)−dµ1(E)dE (17)

n2j = I0

∫ ∞

0Φ(E)D(E)Sj(E)e−(L−d)µb(E)−dµ2(E)dE (18)

nbj = I0

∫ ∞

0Φ(E)D(E)Sj(E)e−Lµb(E)dE (19)

In Appendix A we derive the weights with which each bin count shouldbe weighted in order to optimize the detectability of objects 1 and 2 againstthe background. If nb is the vector with individual elements nbj and similarfor 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/escapephotons 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 inthe 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 whichis why the more general form Eq. (21) must be used when weighting isapplied 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 andthat weights that optimize detectability of any known material can easilybe determined; either by use of forward model or by measurements overregions-of-interest in images. Optimal weighting has been shown to increasethe signal-difference-to-noise ratio by 15-60% [15, 19]. Another benefit isthat it can be applied in both the image domain and projection domain; thesame general formula Eq. (21) holds.

A problem is that determination of the optimal weights require priorknowledge of what one is searching for, i.e. knowledge of n1 and nb. This willnot 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 forenhancing a lung nodule are certainly not optimal when searching for smallcracks 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 − 1dimensions as the entire surface of a hypersphere in R

N must be covered.This is not feasible and means that the displayed image has to be re-weightedaccording 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 listis indeed exhaustive. Furthermore, beam hardening can not be removed byenergy weighting.[41]

Despite these practical drawbacks, energy weighting plays an importantrole when different system configurations are compared and it is reasonableto 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 decompositionbased on the phantom shown in Fig. 3. The example is taken from a 2014SPIE contribution.[42]

Figure 3: Spectral CT phantom made of PMMA with inserts of (starting at11 o’clock) water, plaster (high in calcium), gadolinium solution, iodine andoil.

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 rightpanel. Clearly, there is a distinct profile for each material.

1

2

34

5

6

HU−1000 0 1000 2000 3000

1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Energy bin number

Ave

rage

pix

el v

alue

in R

OI,

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 PMMAphantom with water, calcium, gadolinium, iodine and oil. Also shown arethe ROIs used for measuring the image values for the different materials.Right panel: Reconstructed image value in each energy bin, as measured inthe 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 energybin and assume that a number of basis materials, with known linear attenu-ation coefficient vectors µ1, . . . , µM have been chosen. Furthermore let a bethe [M × 1]-vector of densities of the different basis materials, normalizedto a unitless quantity in such a way that a component ai being equal to 1corresponds 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-rayattenuation, 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 basismaterial attenuation profiles.

13

Calcium image Gadolinium image

Iodine image Color overlay

Figure 5: Basis images for imaged based decomposition according toEq. (23). The lower right panel shows the photon counting image witha 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 thiscase, Eq. (22) must be solved in the least squares sense, which gives anestimate a of a as

a =(

MTM)−1

MTm (23)

where T denotes transpose. The different noise level in different bins couldbe 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 basiscoefficients. In practice, the basis m vectors were measured as the averagesover regions of interest (ROIs) in the reconstructed images. The method isillustrated 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 withthat of water is displayed, see Eq. (4). The choice of what to display inspectral CT is much more complex. At least five different options existwhich 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 onEq. (6), this would constitute forming the set of projections

p(t, θ;Bj) =I0(t, θ)

∫∞0 Φ(E)D(E)Sj(E)e−

∫lµ(x,y;E)dsdE

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 anenergy 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 samplesof the linear attenuation coefficient at an effective energy inside the corre-sponding bin. They could be normalized with the attenuation of water andthus yield N different CT-number images in Hounsfield units.

It is very difficult to find any circumstance under which this approachwould be optimal. From the preceding chapters we know that a weightfactor w consisting of all but one zeros will not be optimal for realisticbackgrounds and targets if the goal is to maximize detectability (and a binimage correspond to bin weight vector of the form [0, . . . , 0, 1, 0]).Such binimages will thus be quite noisy and also qualitatively difficult to interpretas rules of thumb such as fat exhibiting a HU value of −100 to −50 will bedifferent for different energy bins.

3.2 Gray scale images from weighting

We have shown how to determine an optimal weight vector w with which toweight 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 whichin turn depends on the imaging task at hand. Thus, while intuitive andgood for detecting abnormalities, displaying optimally weighted gray scaleimages does not allow quantitative CT.

3.3 Basis images

The reconstructed basis images ai(x, y) are scalar and can easily be displayedfor views. This can be highly beneficial if one particular base is chosen tobe clinically relevant, for instance being a contrast agent, in which case theconcentration can be readily determined, or calcium.

Displaying pure “photoelectric” and Compton cross sectional images addlittle value. However, as shown by Alvarez[18] these bases can be linearlycombined to obtain the same signal to noise ratio as would optimal energyweighting - but free of beam hardening. (In our opinion, this is the strongestcase 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 areselected to be physiologically relevant, is not recommended.

3.4 Synthetic mono-energetic images

Once the ai(x, y)’s are reconstructed, the scalar entity µ(x, y;E) =∑

i ai(x, y)fi(E)can be displayed as a cross sectional image for any single energy E. Theseimages are beam hardening free and allow quantitative use of the imagedata.

An added benefit is that for most imaging cases, i.e. a target volumewith µt(E) against a background µb(E)) as in Fig. 2, there exists an energyE′ such that the signal-to-noise ratio in the mono-energetic image is equalto the optimal achievable with energy weighting, i.e.

µt(E′)− µb(E

′)√

varµb(E′)=

wtT (λt − λb)

√

wtTCov(λb)wt

(25)

with wt according to Eq. (21). λt and λb being the vectors of expectedcounts corresponding to a particular target and the background, see Eq. (6).

The nice thing is that Eq. (25) holds for most imaging cases. This meansthat by just a one-dimensional search across the reconstruction energy onecan 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 thecomplexity of the search problem associated with finding the optimal weightsin R

N since it can be implemented with a scroll wheel. This is illustrated inFig. 6 below and is analogous to the finding by Alvarez and Seppi[51] thatnoise in monoenergetic images exhibit a clear minimum at a reconstructionenergy 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 thecurve is given by the expression on the left hand side, or, more explicitly, inthe 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 giventypical 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 homogenousmaterials, a1 and a2 are determined for target and background fromA1 and A2 for the target and background by division with the pathlength

5. From a1 and a2, the linear attenuation coefficients are estimated:µj(E) = a1jf1(E) + a2jf2(E) where j ∈ {t, b}.

6. The left hand side of Eq. (25) is determined as

(a1t − a1b)f1(E) + (a2t − a2b)f2(E)√

f21 (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 themethods described by Alvarez[18].

17

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Reconstruction energy E(keV)

SdN

R2 (

arbi

trar

y un

its)

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 correctselection is very important. This section is intended to bring some order tothe 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 ai(x, y) is reconstructed fromthe ML-estimates of Ai(t, θ) and displayed. This yields a beamhardening free image of the selected material.

18

(b) Bin images can be reconstructed and then linearly weighted toenhance specific materials, Eq. (23). This is in essence a linearapproximation of the above and will therefore not yield beamhardening 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 theattenuating properties of target and background materials

(b) Optimal weighting of image data; requires same pre-reconstructionknowledge as above.

(c) Basis decomposition and linear weighting of basis images; requirespre-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 allenergies E; requires scrolling the reconstruction energy E.

4. To quantify a concentration or a density there is only one feasiblemethod:

(a) Basis decomposition of projection data where one base is thematerial for which the concentration should be determined.

5. For discrimination between materialsAll combinations of domain and method work for discriminating be-tween materials.

Note that for cases 3 and 4, basis decomposition in the projection domainis 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 canresult in lost counts and distorted energy response functions. Due to theexponentially distributed times between x-ray interactions pile-up to someextent is always present.

No agreement exists within the community regarding how high x-rayfluxes a photon counting detector must be designed for, but rates of 100million 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 noagreement exists on how to present and compare the count rate capabilitiesof different detectors. Several valid but not completely satisfactory measuresexist. Consider Fig. 7 showing two systems with roughly equal count rateperformance; for an input rate of 10 Mcps per channel3 (denoted segmentin the left panel) approximately 7.5 million event per second and channelare counted. 25% of the counts are thus lost due to pile-up at an incidentconversion rate of 10 Mcps.

There is no agreement on how the performance depicted in Fig. 7 shouldbe reduced to a single scalar; is it the maximum count rate before 1% of thecounts are lost, 25% or 50%? From the figure it is also clear that the actualcount 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 thisis valid up to 38 Mcps input rate). Statistical techniques[25, 26, 27] havebeen derived for this conversion, but although the correct expected value

310 MCps might sound like a far cry compared to the required/desired flux rate capacityof 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 diodesystem, 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 willincrease (and this translates to image noise). An issue of immense practicalimportance is how the detector behaves in the time period following a peakoutput count rate; does the detector break down or does it give reasonablyreliable readings in subsequent projections?

As noted above, pile-up not only results in lost counts but also decreasedenergy resolution for the events that are actually detected. By scanning theenergy thresholds in a beam of synchrotron radiation the energy resolutioncan 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

2erfc

(

x− µn√2σ

)

(A(x− µn) +B). (26)

x is the detector threshold value (in mV) and µn and σ the parameters ofthe underlying normal probability distribution function in the same units.A captures charge sharing; for lower thresholds, more counts triggered bycharges that leak in from neighboring pixels are registered. The results inthe 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 thisreason it is important to state the count rate at which a reported energyresolution has been obtained.

21

30 35 40 45 50 55 600

500

1000

1500

2000

2500

3000

x (mV)

counts

Figure 8: Registered counts vs. threshold value x and complementary errorfunction 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 andcharge sharing is that in photon counting mode, where all events are weightedequally, a scatter-induced double count to a first approximation is just asdetrimental as a lost primary count. The same holds for the effect of objectscatter on image quality. To see this, consider Fig. 10 where it assumedthat 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 = Cpre1

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 costsas much as 1% missed counts. For this reason it is paramount to keepscatter, both from the object and from within the detector, to a minimumfor 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 skewedenergy distributions towards lower energies. This skewness needs to be in-corporated into the forward equation Eq. (6) if one intends to perform basismaterial decomposition.

5.3 Calibration issues

Different detection efficiencies or energy response functions among individualdetector elements result in ring artifacts in the final image if not properly

23

compensated for. There are two distinct ways to approach the problem ofring artifacts in multibin systems, depending on whether energy weightingis applied or basis decomposition.

If weighting is applied, the total counts in each bin need to be identicalacross detector elements in homogenous areas of the image, or rings willoccur. Since there is always some threshold dispersion between detectorchannels this must be compensated for. For multibin systems, such com-pensation schemes tend to be more complex than for energy integratingsystems; and will depend on imaging parameters. Nevertheless, a suggested“reshuffling” of bin-counts by means of an affine transformation have shownsatisfactory results in simulations under realistic conditions.[29] For eachdetector element j a N × N matrix Aj and an N × 1 vector bj is appliedto the bin counts mj to get the adjusted counts:

madjj = 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 worksfor the full range of different tissue combinations one can expect to encounterin practice.

A second approach to eliminate the negative effects of threshold spreadis to perform material basis decomposition with separate forward models foreach detector element. The calibration process then focuses on determiningthe forward model “accurately enough”. A reasonable question is just howaccurate such a calibration has to be. The question of how accurate theparameters of the forward model must be known in order to performmaterialbasis decomposition in practice has been addressed by the authors[30] andrecapitulated here.

The figure of merit is based on the observation that when the param-eters of the forward model are incorrect the estimates a1(r) and a2(r) (asreconstructed from the ML-estimates of Eq. 12) will be biased. This biaswill add to the variance and result in a mean square error that is larger thanthe variance of the unbiased estimate since:

MSE(a) = E[

(a− a)2]

= E[a2]− 2E[a]a+ a2 =(

E[a2]− E[a]2)

+ (E[a]− a)2 = var(a) + bias2(a).(30)

If MSE(a) is determined for different deviations of the forward parametersfrom the true values and related to the Cramer-Rao lower bound (CRLB)of the variance in the case of a correctly specified forward model, one candetermine the allowable misspecification for each parameter of the forward

24

model. Below this is shown for the most important parameter, the set ofthresholds {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 methodologyapplies equally well to other detector materials. Recall that the goal is tofind how large misspecifications of the forward model that can be toleratedbefore the bias component of the mean square error of Eq. (30) is larger thanthe variance part. This appears straight forward: the CRLB of the varianceof the A1-estimate (see Eq. (8)) is given by element 1,1 of the inverse of theFisher information matrix[46]

σ2A1

≥(

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 giventhe observed bin counts {mi}:

A∗1, A

∗2 = arg max

A1,A2

P ({mi};A1, A2) = arg minA1,A2

N∏

i=1

(λi −mi log λi) (33)

where λi = λi(A1, A2) according to Eq. (6).[46] For the bias calculation theexpected value of counts in each bin (i.e. noiseless) was used, i.e. we applythe approximation EA({mi}) ≈ A({λi}).

The results will unfortunately depend on dose and on the size of theregion of interest (ROI) that is examined; for larger dose or when averagedover a larger area, the random variance component of Eq. (30) will decreasewhereas the contribution from the bias will not (at least in the case ofa homogenous cylindrical object where the central volume is considered).Thus we have to select typical x-ray fluence for which the comparison iscarried out, and also determine how the size of the region of interest affectsthe result. First however, we show how variance and bias in the projectiondomain is translated to the reconstructed image domain. Note that we inthis model are assuming that all thresholds move in parallel, i.e. to have thesame errors. This is the expected behavior of a temperature increase of oneparticular photon counting detector.[34]

Hanson[40] has shown how the single-pixel variance in a reconstructedimage depends on the variance of a projection measurement:

σ2a1 =

σ2A1

(λ)

Nθa2k2. (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 dependingon filter kernel for the filtered backprojection and determined to k = 0.62for 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 thereconstructed image of a homogenous cylinder with diameter L is given bybias(a) = bias(A)/L since by definition, Eq. (8), A(t, θ) =

∫

l a(r)ds = aL fora central ray l. When estimates are biased, as in the case of a misspecifiedforward equation, one cannot however be sure that A(t, θ) = a

∫

l ds for allpossible 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 inthe 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 thefiltered back projection due to the filtering step and thus, if relative biasesdiffer across projections for instance due to path length, this will propagateunpredictably to the image domain bias.

A sufficient condition for bias(a) = bias(A)/L to hold in the centralregion of reconstructed image of a homogenous cylinder with diameter L is

A = c a

∫

lds (35)

with constant c for different path lengths∫

l ds. If Eq. (35) holds, we have

bias(A) = A − A = (c − 1)a∫

l ds since A = a∫

l ds by definition. In thereconstructed domain, we have bias(a) = a − a where a is obtained viathe inverse radon transform of A of Eq. (35). Due to the linearity of thetransform, a = 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 A ≈ c a∫

l dsfor

∫

l ds ∈ [0.85, 1]L . Under these feasible circumstances Eq. (35) holdsapproximately and thus bias(a) ≈ bias(A)/L.

The correlation structure in the reconstructed image makes the variancein an ROI consisting of M pixels decaying faster than M−1. In the limit of acontinuous image, with perfect bandlimited interpolation of the projectiondata before backprojection, the variance depends on M as M−3/2.[38] Inprevious work[48], we have shown (with a discrete image and typical inter-polation) that the variance decreases asM−1.33. This derivation is reiterated

26

in App. B. For an ROI consisting of M pixels, the standard deviation willthus 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 willmost likely be carried out over ROIs much larger than a single pixel, wechose to compare the bias with the variance over a 100 pixel large ROI.

A homogenous cylindrical object with L = 20 cm diameter consisting of50% 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. Theunattenuated x ray fluence, N0, must be selected as the total number ofx rays per reconstructed pixel area for an entire gantry revolution. (Notethat 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

A1

(λ(N0/Nθ))/Nθ = σ2A1

(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 angleand 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 Cramer-Rao lower bound of thevariance of the A1-estimate is given by Eqs. (31) and (32) in conjunctionwith Eq. (6). Combined with Eq. (34) this yields σ(a1) = 0.68 (unitless)(which should be related to E(a1) = 0.5 (also unitless)). For an ROI withM=100, σ(a1) is 0.033. This is the value the bias should be compared to.

Fig. 11 indicates that the misspecified thresholds are very detrimentalto quantitative CT. For parallel shifts in the thresholds of 0.1 keV or more(as compared to calibration values), the bias introduced will dominate theMSE. This indicates that thresholds uncertainties of more than some 0.1 keVcannot be tolerated for quantitative CT. We have recently devised a broadspectrum calibration scheme that fulfills this requirement and is also feasiblein clinical practice.[17]

27

−1 −0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

CRLB σ(a1) (100 pix)

|bias(a1)|

CR

LB

σ(a

1),|b

ias(

a1)|

∆T (keV)

Figure 11: Effect of uncertainty in thresholds.

6 How to evaluate

Linear systems theory[50] is normally applied x-ray imaging modalities toexpress the DQE, detective quantum efficiency, the spatial frequency equiva-lent of signal-to-noise ratio. The metrics derived using linear systems theoryhave been shown to correlate very well with the results of human observerstudies[31], and has the benefit of allowing system optimization while stillon the drawing desk of the engineer.

The method has been adapted to computed tomography[32, 33] but issomewhat elusive to apply to multibin systems; the noise and signal char-acteristics of of multibin systems depend heavily on the method for imagegeneration; for instance the weight factor applied to the bin will have aneffect on the DQE and thus the intended system figure of merit dependson what on intends to image. For the non-linear basis decomposition tech-niques MTF and NPS (constituents of DQE) cannot be defined, so neitherfor energy weighting multibin systems nor for basis decomposition systemsis there such a thing as one system DQE, but for different reasons.

To make matters even worse, DQE is intended as a system measure andthus does not take the contrast of a particular object into consideration. Yetone of the benefits of using the spectral information was to avoid contrastcancellation and to selective enhance the contrast of different objects; if thiscapability is not included in the figure of merit at all, certainly it cannotbe used for high-level system comparison. Instead the related detectabilityindex[50] has to be used as it combines spatial and contrast resolution. InRef.[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 iswith iterative reconstruction techniques). Indeed, one cannot even give astraight answer to the following (very reasonable) question: “What is thespectral resolution of your system?” because the answer would depend onhow the spectral resolution is interpreted and could mean, at least, any ofthe below:

• The underlying standard deviation σ (in keV) of the photo peak ofEq. (26) at low count rates (i.e. composed mainly of the the intrinsicenergy resolution of the direct conversion material and the electronicnoise).

• The σ (in keV) of the photo peak of Eq. (26) at various realistic countrates as in Fig. 9, mainly capturing pile-up effects.

• The width of an energy bin, Ti+1 − Ti (also in keV). This correspondsto determining the energy of a single photon by a multibin system.

• A very technical measure, and quite difficult to translate to imagequality, would be the capability to use the detector as a spectrograph,determining the unknown energy of the photon of a monochromaticbeam (this would also be stated in keV and is a measure where duallayer detectors come out fairly well).

• The ability to selectively enhance energy dependent contrast (how thisproperty should be captured by a figure or merit single scalar is veryhard to see though. . . ).

• The ability to quantify different tissues? The unit of this measurewould be unitless since in essence it is the mean square error of abasis coefficient estimate, Eq. (30) (or linear combinations of the basisestimate coefficients ai).

29

A Derivation of optimal weights

This derivation is the courtesy of Mats Persson and first appeared in hisMaster’s thesis.[44] A similar but not as detailed derivation can be foundin the the seminal paper of Tapiovaara and Wagner[45] where it was firstpointed out to the community how the contrast carrying information differsfor 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 thatexactly one of them is true. We want to use the image data to determinewhich one is true and which one is false. In the cases which we will beconcerned with here, H1 represents the presence of some feature, e.g. atumor or a bone, which is different from the background tissue, while H0

represents the absence of the feature. Assume we count the interactions inin each bin, mi, each of which can be seen as the outcome of a randomvariable gi. Let

g = (g1, g2, . . . , gN )T (36)

be a vector containing all these random variables. Note that if we wantto use g to determine whether H0 or H1 is true, the probability densitiesprg|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 thecovariance 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-valuedfunction of the data vector, possibly nonlinear. Then, t is compared tosome 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 tvalues). The so-called discriminant function T and the decision thresholdtc should of course be chosen so that, if possible, this test identifies thecorrect hypothesis for most outcomes g. However, it is usually impossibleto find a discriminant function that takes entirely separate values underH0 and H1, and so there will always be a risk that the test makes someincorrect decisions, so-called false positives and false negatives. There areseveral ways of comparing the performance of different tests. One approachis to assign costs to the different outcomes (true positive, false positive, truenegative and false negative) and then say that the best test is that whichminimizes the expected value of the cost. This is called the Bayes criterion.Another approach, which does not require that one assigns costs to thedifferent 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 fixedfalse positive rate. It can be shown that [49] there is an ideal discriminantfunction which is optimal under both these criteria, namely the likelihoodratio

Λ(g) =prg|H1

(g|H1)

prg|H0(g|H0)

(37)

The observer that decides between H0 and H1 according to (37) is called theideal observer. In practice, the complete probability distribution function ofg is seldom known, meaning that one has to use other, nonideal, observermodels to assess image quality. It also worth noting that medical imagesare normally intended to be used by human observers, who might performsignificantly worse than the ideal observer. In the present study we shallmeasure image quality by the performance of the linear observer, i.e. theobserver with discriminant function given by

T (g) = wTg (38)

where w = (w1, . . . , wN )T is a vector of weights. Furthermore, we willoptimize this with respect to the squared signal-difference-to-noise ratio

SDNR2 =(〈T (g)|H1〉 − 〈T (g)|H0〉)2V [T (g)|H0] + V [T (g)|H1]

=(wTg1 −wTg0)2

wTK0w +wTK1w=

=(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] = 〈(wTg −wTg)2|Hm〉 = 〈(

wT (g − g))2 |Hm〉 =

= 〈wT (g − g)(g − g)Tw|Hm〉 = wT 〈(g − g)(g − g)T |Hm〉w = wTKmw

has been used. Note that an additional factor of two is included in thedefinition (39) by some authors, since this gives a neater formula in the casewhen K0 = K1. We shall show that whenever K0 +K1 is invertible, whichis 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). Notethat K0+K1 has a square root matrix (K0+K1)1/2 that is symmetric andinvertible (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)1/2w)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)1/2w‖2 · ‖(K0 +K1)−1/2∆g‖2 ≤

≤ wT (K0 +K1)1/2(K0 +K1)1/2w∆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 linearlydependent, i.e. for some real-valued constant k,

(K0 +K1)1/2w = 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 isactually equivalent to the ideal observer for the case when g is multivariatenormal with equal covariance but different mean under the two hypotheses.We shall use (44) and the corresponding optimal weight factors (40) in orderto measure and optimize detectability in images.

It should be pointed out that the above discussion based on SDNR isrelevant only in the limit where the structures to be detected in the imageare uniform over very large areas. When investigating the detectablity ofsmaller structures, one has to compare the signal strength with the noiselevel for each of the spatial frequences present in the image. In other words,it becomes necessary to use linear systems theory to investigate how signaland noise propagates through the imaging system in Fourier space. Whenstudying x rays, the linear systems theory approach is complicated furtherby the fact that the number of photons in each measurement is few enoughthat the flux distribution cannot be treated as a continuous function butmust be represented by a point process, which is a certain kind of stochasticprocess whose realizations are sums of delta functions. These ideas formthe basis for a useful framework which is, however, beyond the scope ofthis thesis. The interested reader is referred to [49]. Even though SDNRis a crude measure of imaging system performance, it is useful since it isso easy to calculate while still giving some important information about theimaging performance, at least in large homogeneous regions of the images. Itis therefore useful during the process of designing an imaging system, whenit is desirable to have a figure of merit in order to make design decisionsquickly, 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 semidefinitematrices and assume that K0 + K1 is invertible, like in section A. ThenK0+K1 is also symmetric and positive semidefinite. The symmetry impliesthat K0 +K1 = PDP−1 for some matrix P such that P T = P−1, where Dis the diagonal vector of eigenvalues of K0 +K1:

D =

λ1 0 · · · 00 λ2 · · · 0...

.... . .

...0 0 · · · λN

(45)

(See [52], theorem 7.3.1.) It also follows that all λi are real and strictlypositive. (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

λ1/2i 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

λ1/2i > 0. The symmetry of (K0 +K1)1/2 follows from the equation (K0 +

K1)1/2 = PD1/2P T . To conclude, (K0+K1)1/2 exists and is symmetric andinvertible.

B Variance as a function of ROI size

Due to correlations and the effect of the filter, the lower limit variance overan ROI of M pixels can be expressed as h(M) times the variance over onepixel:

CRLBσ2a(M) = CRLBσ2

a(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 theimage 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(a3;M) = varPo(a3)h(M) + Eε

[

bias2(a3)]

. (47)

To determine h(M), a large sinogram (Nθ = 300 angles and Nt,θ = 1000detector positions) is filled with gaussian noise with zero mean and unitvariance. In Eq. (34) this corresponds to σ2

A/a2 = 1, i.e. the variance in each

a2 large detector element is unity. The backprojection algorithm is appliedand the resulting image is smoothed with square filters with increasing sidelengths of

√M pixels. The resulting standard deviation is measured in the

reconstructed image. Since σ2A/a2 = 1, a combination of Eqs. (34) and (46)

indicate that it is the normalized entity

σa(M)√

Nθ = k√

h(M) (48)

that is of interest. k√

h(M) is plotted in Fig. 12. Since h(M = 1) = 1, k is0.62. The dashed line has a slope corresponding to h(M) = M−1.33, but itis the simulated functional form of h(M) that is used in the derivations ofSec. B.

C Constancy of c for different path lengths

In this section we show that c of Eq. (35) is close to constant for somedifferent forward model misspecifications. For values of the path length

34

100

101

102

103

104

10−3

10−2

10−1

100

M (pixels in the square ROI))k√

h(M

)

Figure 12: k√

h(M) of Eq. (48) as a function of ROI size M .

∫

l ds ranging from 0.85L to L A1 is estimated by Eq. (33) (by insertionof the erroneous forward parameter in Eq. (6)). c is then determined byc = A1/

∫

l a1ds and plotted against path length expressed as a ratio tothe diameter L. Relative changes of c seem to be confined to around or lessthan 1% over the interval, for which reason we can model it as approximatelyconstant.

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0.85 0.9 0.95 10.9875

0.988

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