normalisation of histogrammed list mode data

IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 55, NO. 1, FEBRUARY 2008 543

Normalisation of Histogrammed List Mode DataKris Thielemans, Member, IEEE, Christian Morel, Member, IEEE, Matthew W. Jacobson, Jerome H. Kaempf, and

Sanida Mustafovic

Abstract—Many PET scanners nowadays have the possibility torecord event-by-event information, known as list mode data. Thishas the advantage of keeping the data in the highest possible reso-lution (both temporal and spatial). In most cases, list mode dataare then binned into sinogram format before reconstruction. Inthis paper, we discuss at which stage normalisation factors shouldbe introduced. It is shown that noise is greatly reduced by per-forming the normalisation after the binning. We illustrate this withacquired and simulated data for the quad HiDAC camera.

Index Terms—Data normalisation, list mode, PET.

I. INTRODUCTION

L IST mode data contains both temporal and spatial infor-mation on the detection of a coincidence. If one bins the

events into sinograms, the temporal information is lost. Never-theless, in the case of rotating cameras, this time informationis necessary to determine correct detection efficiency and geo-metrical acceptance normalisation factors. The question then isif one should apply corrections before binning into sinograms,or use a time-averaged normalisation factor after binning. Simi-larly, list mode data generally contains higher resolution spatialinformation than is practical to use in sinogram format. So, evenfor static cameras the question is: should one normalise beforebinning or after? We will call the first option pre-normalisation,and the second post-normalisation.

We will show with a simple statistical analysis that post-nor-malisation has better noise properties. It will follow that thedifference between pre- and post-normalisation increases whenevents with different efficiencies are combined.

A system where this is very much the case is the HiDACcamera. This is a small animal 3D PET scanner [1] with ap-proximately 1 mm resolution. In the configuration installed atHammersmith, the camera consists of four planar, rectangulardetector banks each consisting of 8 HiDAC modules, see Fig. 1.In normal operation, the camera rotates backwards and forwardsevery 6 sec over 180 degrees. We found that the detection ef-ficiencies for this camera are very much dependent on where

Manuscript received February 13, 2007; revised June 24, 2007. This workwas supported in part by the Medical Research Council (U.K.) and in part bythe Swiss National Foundation for Research under Grant No. 2153-063870.00.Paper no. TNS-00079-2007.

K. Thielemans is from Hammersmith Imanet Ltd., part of GE Healthcare,London W12 0NN, U.K. (e-mail: [email protected]).

During the course of this work, C. Morel and J. H. Kaempf were from theInstitute for High Energy Physics, University of Lausanne, CH-1015 Lausanne,Switzerland (e-mail: [email protected]).

During the course of this work, M. W. Jacobson was from the Departmentof Electrical Engineering and Computer Science, University of Michigan,Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]).

During the course of this work, S. Mustafovic was with Hammersmith Imanet,part of GE Healthcare, London W12 0NN, U.K.

Digital Object Identifier 10.1109/TNS.2007.914207

Fig. 1. Drawing of the HiDAC design (image courtesy Oxford PositronSystems).

the gamma photon interacts with the detector. Due to the dif-ferent layers of detectors and the rotation, this means that eventswith large differences in detection efficiencies will end up in thesame sinogram bin. We illustrate the difference between pre-and post-normalisation on Monte Carlo data, and use those datato find post-normalisation factors which we apply on uniformcylinder data acquired on the camera. Both 3DRP [4] and OSEM[5] followed by postfiltering reconstructions are used to showthe difference in noise properties of the two normalisation pro-cedures. Resolution measurements are also given. Some of theresults in this paper were previously presented in [6].

II. THEORY

We first consider the static case where the radioactivity dis-tribution and scanner position do not change. The total countsdetected in each detector pair will be denoted by (with meanvalue and finite variance ). The normalized values ofthese counts are obtained by applying normalization factors ,meant to correct for varying detection efficiencies. The values of

are obtained, as a standard practice in PET, by pre-scanninga known test object. Each is approximately inversely pro-portional to the product of the detector efficiencies in the -thdetector pair but will also include geometrical and dead-timefactors.

To construct a sinogram, the detector pairs are grouped intosubsets (bins) denoted where all LORs in each subsetare geometrically similar to each other. In this section, we shalldiscuss how best to consolidate the counts from the detector

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544 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 55, NO. 1, FEBRUARY 2008

pairs of each bin to obtain binned sinogram data , wherewe restrict ourselves to taking linear combinations

(1)

where is a vector of weights to be selected.The mean values of the normalised counts are propor-

tional to the total activity along the Line Of Response (LOR) foreach detector pair, with a fixed proportionality constant acrossall detector pairs . Consequently, for all detector pairs of eachbin , we have in very good approximation that their meannormalised counts will be equal to each other. We denote thisbin-dependent value as

(2)

The accuracy of the approximation is better the more uniformthe activity distribution and the more geometrically similar thedifferent LORs in the bin are.

To analyze the noise properties of the linear combination in(1), we can look at its Squared Coefficient of Variation (SCV),defined as the ratio of variance to the mean squared. The SCVof is given by

(3)

and is defined for all for which the denominator of (3) isnon-zero.

The SCV obviously does not depend on a scale factor in thevector . In the Appendix, we find that the SCV is minimizedonly when the weights are proportional to the mean and in-versely proportional to the variance and normalization factors:

(4)

A well-known case of (4) is when the are normally dis-tributed with identical means (and ).

The proportionality constant in can be fixed by imposingsome desired condition on the mean of . For example,choosing

(5)

implies with (1) and (2) that the mean of the normalized binneddata is consistent with that of the normalized non-binned data,

i.e., independently of the . The constraint in (5)determines the proportionality constant in (4) leading to

(6)

In addition, (2) and (5) imply that (3) reduces approximately to

That is, minimizing SCV of is approximately the same asminimizing the variance.

Uncorrected PET data are in very good approximationPoisson distributed.1 This means that the variances of theunnormalised counts are equal to their means, .The optimal weights, as given in (6), therefore satisfy

(7)

leading to the following formula for the binned counts (1):

(8)

Hence, optimal binning is obtained by summing the raw countsand post-weighting this sum. We therefore call this post-nor-malisation. The formula (8) for post-normalisation can be un-derstood easily in terms of detection efficiencies, which are thereciprocal of the normalisation coefficients. Post-normalisationsimply divides the sum of the measured counts by the sum ofthe detection efficiencies. This procedure has been suggestedfor example in [2].

It is natural to compare post-normalisation to the case wherethe normalized counts are uniformly averaged, yielding binneddata

(9)

This corresponds to with the number of detec-tors pairs combined into bin . We call this case pre-normalisa-tion. Because of the unique optimality of , we see that, in thePoisson case, pre-normalisation always has strictly higher SCVthan post-normalisation, except when the normalisation coeffi-cients for the combined LORs are identical (in which case pre-and post-normalisation are the same).

Another interesting result for the Poisson case is obtainedwhen we compare the SCV of both pre- and post-normalisation.

1 This disregards radioactive decay. The actual distribution is binomial, but formeasurement times smaller than the half-life, it can be shown that the Poissonapproximation is very good.

THIELEMANS et al.: NORMALISATION OF HISTOGRAMMED LIST MODE DATA 545

Fig. 2. Histograms of the LOR coordinate perpendicular to the anode wire di-rection for events detected at directly opposing location on the 2 dectector heads.Horizontal axis is in mm. The LOR statistics observed demonstrates a Gaussianefficiency-like pattern with a periodicity corresponding to the pitch �1:5 mm

– of the anode wire plane.

Substituting the corresponding weights in (3) and using (2), weget after some algebraic manipulation

(10)

which shows that the reduction in between pre- and post-normalisation is object-independent.

This proof can easily be extended to the case where thescanner moves with respect to the object in discrete steps, e.g.,by rotation. We now have to consider the counts in detectorpairs for each position of the scanner. We can do this by ap-plying the previous proof for a virtual scanner for which eachvirtual detector pair corresponds to a physical detector pair in aparticular scanner position. The normalisation factors willthen include the dwell-time, i.e., how long each virtual detectorpair is active.

The current proof works only for discrete detectors and dis-crete motion. The case of list mode data with continuous de-tectors and/or motion can be considered as a limiting case withvery fine discretisation. We leave a more rigorous analysis ofthe continuous case to the interested reader.

III. HIDAC LIST MODE NORMALISATION FACTORS

Each HiDAC module comprises 2 converters (consisting ofseveral lead layers with drilled holes and insulating material)each 50 thick), each with 1 plane of cathode tracks spacedby 1 mm. In between the converters, there is 1 anode plane withwires spaced 1.5 mm apart. The list mode data contains for eachphoton ( , ) position information with a sampling of 125 ,and the converter number. The main feature of interest for thispaper is that we observed [3] a detection efficiency pattern witha periodicity of 1.5 mm for the cathode tracks which are parallelto the anode wires (Fig. 2).

Thus, the probability to detect an event at a given positionis clearly dependent on the location of the interaction of thephoton within the detector. Consequently, events from differentconverters will be binned in the same sinogram bin, sometimeswith very different efficiencies.

Fig. 3. Sinogram acquired at a fixed detector positions for coincidences ob-served between converter 0 in a detector (the closest converter to the object) andconverter 10 in the opposite detector for a uniform Ge-68 cylindrical phantom.The vertical axis represents the azimuthal angle in radians. The horizontal axisrepresents the radial coordinate in mm.

Furthermore, there is a dependence on the incidence angle ofthe photon on the converter. Monte-Carlo simulations [3] haveshown that this dependence is small, and can be parametrisedusing a simple absorption model, for which the absorption prob-ability is given by

(11)

where is the impinging angle on the converter, the corre-sponding ’effective’ linear attenuation coefficient of the con-verter and is the converter thickness. The HiDAC convertersare made from lead (attenuation coefficient at 511 keV of 0.1766

) with 0.4 mm diameter holes with a pitch of 0.5 mm. TheMonte Carlo simulation used hexagonal spacing. The fit to (11)obtained a value for the ’effective’ attenuation coefficient 1.98times smaller than lead, which is roughly consistent with theratio of the surface of the holes to the total area.

However, the probability of detection in the outer modulesis smaller, as there is a chance of absorption into the lead con-verters of the previous modules. The probability to detect a co-incidence between any pair of converters is given by

(12)

where is the total number of converters where the photonshave not interacted.

Finally, because the detectors are planar and do not surroundthe object completely, there is a large geometrical acceptanceeffect in the sensitivity. Actually, the sinograms are completelyfilled with the rotation of the detectors, but some regions of thesinograms are more sampled than others [3], [7].

As shown in Fig. 3, a sheered diamond shaped portion of thesinogram is sampled for every detector position, and special caremust be taken in the calculation of the normalisation factors forgeometrical acceptance, especially for azimuthal angles wherethe detectors switch their rotation direction [3].

For the rotating acquisition mode, the sensitivity dropsroughly triangular in radial direction (Fig. 3). Basically, the


Fig. 4. Profile through the reciprocal of the post-normalisation factors for 2Dsinograms, i.e. with average polar angle 0. The profile is taken in radial direction,i.e., orthogonal to the scanner axis. Bin size is 0.5 mm. Horizontal axis is in mm.Dots show result when all detection efficiencies are taken into account, solid lineshow result when only geometric effects are modelled. Results were scaled tomatch near the tails.

angular range covered by the rotating detectors for a givenpoint in the sinogram can be defined by

(13)

where is the detector angle position, and defines amask of the sinogram areas covered for detectors at angle posi-tion . Then for a given LOR , the geometrical acceptancenormalisation is given by the reciprocal of .

IV. EXPERIMENTS

A. Simulations

We have developed a simple – not considering any scatter –Monte Carlo simulator for the HiDAC. The main effects simu-lated are the geometrical acceptance of each converter, absorp-tion by lead in the converters and the detection efficiency patternof the anode wire as discussed in the previous section. Briefly,the detection probability of a photon is determined by checkingfirst in which converter it is absorbed (if at all) using (12). Thedetection position is then simply determined as the intersectionof the LOR with the converter. Finally, this detection position isused to find the detection efficiency due to the anode wire pat-tern, for which we used a sum of 3 Gaussians centred on theclosest anode wires, each with FWHM given by half the anodewire pitch.

For the simulated data, we have complete control over allefficiencies, so determining the pre-normalisation factors istrivial, aside from the geometric factors. Instead of tediouslycomputing the post-normalisation factors, we used a highcount Monte Carlo simulation of a (scatter free) plane sourceacquisition, and binned the data without pre-normalisation. Thepost-normalisation factors follow then by dividing these databy line integrals through the plane source, using only viewswhich are approximately orthogonal to the plane source.

Fig. 5. Profile along the axis of the scanner through normalised sinogramdata of the simulated cylinder. Horizontal axis is in mm, vertical axes isin arbitrary units. Solid curves are using post-normalisation, dotted curvespre-normalisation.

For the data presented here, we used bins of 0.5 mm (in bothtangential and axial direction), using 80 azimuthal angles and 9polar angles with a maximum acceptance of 18 .

A profile through the reciprocal of the obtained post-normal-isation factors along the tangential coordinate is given in Fig. 4.Note that the profile approximates the geometric result of a tri-angle very closely, except for the central LORs. This is becauseexcept at the centre, all efficiency effects nearly average out dueto the rotation of the scanner and the fact that non-orthogonalLORs intersect the converters in different places. Note that thepost-normalisation factors show a pattern of groups of 3, corre-sponding to the 1.5 mm spacing between the anode wires.

We then simulated the acquisition of a uniform cylinder(100 mm diameter) and binned it using both pre- and post-nor-malisation (bin size was 3 mm in both tangential and axialdirection). Profiles in axial direction are given in Fig. 5. Thedifference in noise is obvious. Also, quantification does notseem to be affected.

We reconstructed these data using 3DRP and 3D OSEM(8 subsets, 80 subiterations, post-filtered with a Gaussian withFWHM of 4 mm), implemented in the STIR reconstructionlibrary [8], [9]. We used cubic voxels of size 3 mm.

We considered 3 cases for the OSEM reconstructions. Forpre-normalisation we simply precorrected the data and henceused unit normalisation factors in the probability model (OSEM-pre), while for post-normalisation we either precorrected thedata with the post-normalisation factors (OSEM-post), or in-cluded the post-normalisation factors into the probability matrix(OSEM-prob).

Horizontal profiles through the 3DRP images are given inFig. 7, while slices through the reconstruction are shown inFig. 6. We also computed ROI standard deviations for the dif-ferent reconstructions, see Table I. From this, we see that CV


Fig. 6. Transaxial slices through images of the simulated cylinder re-constructed with 3DRP. Left is with pre-normalised data, right is withpost-normalisation. The gray scale is such that black corresponds to 0 andwhite to 1.2 times the value in the simulated cylinder.

Fig. 7. Horizontal profiles in a slice perpendicular to the scanner axis through3DRP reconstruction of the simulated cylinder. Horizontal axis is in mm, ver-tical axes is in arbitrary units. Solid curves are using post-normalisation, dottedcurves from pre-normalisation.

TABLE ICOEFFICIENT OF VARIATION IN ROIS FOR THE SIMULATED DATA AND THE

ACQUIRED DATA. OSEM-PRE AND OSEM-POST USE PRECORRECTED DATA,WHILE OSEM-PROB IS THE CASE WHERE THE (POST-)NORMALISATION

FACTORS ARE APPLIED IN THE PROBABILITY MATRIX

reduces with nearly a factor of when using post-normal-isation in this simulation.

B. Uniform Cylinder Measurement

We also applied the above normalisation factors to acquireddata of a Ge-68 cylinder (75 mm length, 30 mm diameter),

located slightly off-centre. List mode data was histogrammedinto 3D sinograms with tangential and axial bin size of 0.5 mm(116 azimuthal angles and 15 polar angles with a maximum ac-ceptance of 34 ) and reconstructed using cubic voxel sizes of

with 3DRP (with Hanning filter in both tangentialand axial direction) and MLEM (80 iterations post-filtered witha 3D Gaussian with ). Attenuation, scatterand random coincidences were ignored in the reconstruction. Acylindrical ROI (42.5 mm length, 10 mm diameter) centred onthe cylinder was chosen and a global scale factor was appliedto all images such that the ROI mean value was 1. Transverseslices through the reconstructed images are shown in Fig. 8.

Clearly, post-normalisation gives much better noise proper-ties. As in the simulation, we see that CV reduces with a factorof when using post-normalisation. Note that the differ-ence in CV in the ROIs between OSEM-post and OSEM-probis negligible. However, near the edge of the simulated cylinderand also the edge of the FOV in the real data, spikes occurred inthe pre-corrected OSEM images. This is because the normalisa-tion factors there are very large, so any noise will be amplifieddramatically. OSEM-prob did not have these artefacts.

Averages of transverse slices (bottom row images of Fig. 8)show that both pre- and post-normalisation provide already verygood uniformity of the reconstructed images, even with the cur-rent simplistic measurement model.

To investigate the effect of ignoring the detection efficien-cies, we also reconstructed data using only the geometric nor-malisation factors.2 This gives already good results, except forring artefacts centred on the axis of the scanner (Fig. 9). Thisis because the detection efficiency variations in effect averageout (see also Fig. 4). Only near the centre there is reinforce-ment of the anode wire detection efficiency patterns. To quantifythis effect Fig. 9 shows average radial profiles through the im-ages. These were constructed as follows: images were re-inter-polated (using linear interpolation) on a polar grid (with originon the axis of the scanner) with 80 angles and 0.5 mm radialspacing. Radial profiles were then averaged over all angles and85 transaxial planes. The profiles confirm that uniformity is im-proved by taking the detection efficiency pattern into account.We hypothesize that the differences at 0 and 0.5 mm betweenpre- and post-normalisation are due to noise (these points havethe highest standard deviation in the average radial profile).

C. Line Source Measurements

To test the resolution properties of the two normalisationmethods, we used experimental data from line source scans.The line sources were fibres of 0.11 mm diameter radiated toproduce F-18. The list mode data were binned with a bin-sizeof 0.5 mm and reconstructed with FBP. To test radial andtangential resolution, the line source was placed off-centre andoriented along the axis of the scanner and hence parallel withthe anode wires causing the large efficiency variations on thedetectors. Profiles through the line source averaged over 30planes are shown in Fig. 10. They clearly illustrate that radial

2Results shown used pre-normalisation but are nearly identical to post-nor-malisation. This is because for this detection model, all LORs contributing toone sinogram bin have effectively the same normalisation factor.


Fig. 8. Transaxial slices through images of the measured cylinder reconstructed with 3DRP (0.5 mm voxel sizes in all directions). Left column is with pre-normalised data, right is with post-normalisation. Top row: single plane. Bottom row: average over 85 planes. The gray scale is such that black corresponds to 0and white to 1.2 times the value in a cylindrical ROI (see text). Axes are in mm.

resolution for the two methods is virtually identical. Similarresults were obtained in tangential and axial direction.

V. DISCUSSION

The theoretical analysis shows that post-normalisation givesbetter noise properties than pre-normalisation when binning listmode data. This has been confirmed in experiments with theHiDAC PET scanner, both on the sinogram data and in recon-structed images by 2 different algorithms. Experiments showthat resolution is not affected.

Similar experimental results were obtained in [12] on data ac-quired on an EXACT 3D PET scanner [13]. On uniform cylinderdata ROI standard deviation of pre-normalisation images recon-structed with 3DRP was about 1.04 higher than for post-nor-malisation. A similar ratio was found when comparing pixel-

level standard deviation using a bootstrap approach. This ratiois smaller than for the HiDAC as the differences in detection ef-ficiencies are not as large.

Using post-normalisation also has practical advantages. Post-normalisation factors can be measured in the standard way, forinstance using a moving line source [10], with much less noisethan pre-normalisation factors, as there are much more lines ofresponse to be normalised in the latter case. Also, for a systemlike the HiDAC with such a complicated detection set-up, the ef-ficiency variations that depend on the detection position are dif-ficult to determine accurately. Luckily, for post-normalisationalmost all these variations average out such that the post-nor-malisation factors turn out to be quite smooth, especially in therotating acquisition mode (see Fig. 4).


Fig. 9. Left: average over 85 transaxial planes of the measured cylinder data using pre-normalisation with geometric effects only (compare with Fig. 8). Right:average radial profiles from the axis of the scanner (see text); squares: pre-normalisation, stars: post-normalisation, triangles: pre-normalisation using geometriceffects only.

Fig. 10. Profiles through FBP reconstruction of an acquired line source alignedaxially. Stars: pre-normalised data, diamonds: post-normalised. Horizontal axisshows distance from the centre of the FOV.

For iterative algorithms on the binned data, such as MLEM,an important advantage of post-normalisation is that it allowsthe incorporation of the normalisation factors in the probabilitymodel. It is perhaps no surprise that optimal binning of coinci-dence events in list mode data preserves the Poisson nature. Ithas been shown in [11], and also observed here, that taking at-tenuating factors and normalisation factors into account in theprobability model results in better MLEM reconstructions.

In principle, pre- and post-normalisation lead to differentmean values for the counts, leading one to consider possiblebias and/or resolution effects. In practice (2) holds in verygood approximation such that any differences will be small.We have confirmed this in this paper for HiDAC data and in[12] for EXACT 3D data. If however (2) does not hold, it

would mean that detector pairs are combined that correspondto (very) different LORs, resulting in misplacement during thereconstruction in both pre- and post-normalisation. It remainsto be seen which normalisation procedure would lead to higherresolution/lower bias in this case.

Recently, a number of papers have implemented an LORrepositioning method [14] to correct for movement of the heador other cases where affine transformations are sufficient. Inthis method, the LOR for each list mode event is repositionedaccording to known motion and then binned into a sinogramwhich is then processed with a standard (although possiblyadapted) reconstruction method. Normalisation factors are gen-erally taken into account before repositioning, see for instanceBloomfield et al. [15], which corresponds to a pre-normalisa-tion scheme. The theory of Section II can also be applied to thismethod, at least when nearest neighbour interpolation is usedfor the repositioning to keep the counts in the bin statisticallyuncorrelated. Thus, we expect better noise properties for apost-normalisation scheme where a motion-averaged normali-sation factor is used [16]. The relevance of this will depend onthe degree in which the normalisation factors differ.

The analysis in this paper obviously holds for attenuationfactors as well. It can also be used when non-list mode dataare binned into coarser sinograms, or indeed other cases wheremultiple Poisson measurements are combined to produce ‘nor-malised’ average counts.

The theory of Section II can be applied to non-Poisson cases.In general, the application of (6) is difficult, as it requires priorknowledge of the weights , which might be object depen-dent. It is a special feature of the Poisson case that these weights


turn out to be 1 and that this optimum corresponds to post-nor-malisation. Note that for PET data that are precorrected for ran-doms by online subtraction of delayed events, the weights willbe proportional to

(14)

where is the randoms to trues ratio for that detector pair.We leave the study of the consequences of (14) for the future,but note that its incorporation into an OSEM-type algorithm isrelated to NEC scaling as suggested by Nuyts et al. [17].

In this paper, we have looked at combining counts from sev-eral detector pairs into one bin. A different, but related, case hasbeen studied in [18] where counts from different energy win-dows (for the same detector pair) are combined. Worstell et al.derive optimal weights to maximise the Signal to Noise Ratio(SNR) from the theory of “diversity combining” and use themaximal ratio combination, defined as “the gain of each channelis proportional to the rms signal and inversely proportional to themean square noise in that channel”. It is possible to combine theresults of [18] and the current paper, but we leave this for futurework.

APPENDIX

In this appendix, we find the weights that minimize theSquared CV (SCV) of an arbitrary linear combination of inde-pendent random variables. Results similar to those presentedhere have long been observed in the specific context of commu-nication channel signal processing [19]. For convenience, wewill use similar notation as in Section II, except that we dropthe -index for brevity and absorb the normalisation factorsinto the weights. The derivation holds for independent randomvariables with almost arbitrary statistics, as we only requirefinite variances.

Let

be a finite linear combination of statistically independentrandom variables with means and variances .For a given vector of weights (with components ), theSCV of is denoted by

(15)

We will show that the choice of weights (which corresponds to(4))

(16)

yields the optimal SCV, given by

(17)

We do so by comparing this value with the SCV for generalweights (see (15)). The ratio of the two may be expressed as

(18)

By the Cauchy-Schwarz inequality, we know that for any 2 non-orthogonal vectors , :

(19)

with equality iff one of the vectors is a scalar multiple of theother. Applying (19) with and thenshows that the ratio (18) is always at least 1 with equality iffall for some proportionality constant . Thisproportionality constant can be fixed by imposing a conditionon the mean value of the linear combination.

ACKNOWLEDGMENT

We thank P. Bloomfield and G. Houston for providing theHiDAC data and A. Ranicar for the production of the linesources. We wish to thank the reviewers for their comments andsuggestions that allowed us to improve the text considerably.

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normalisation of histogrammed list mode data

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