presampling, algorithm factors, and noise: considerations ... · presampling, algorithm factors,...

Presampling, algorithm factors, and noise: Considerations for CTin particular and for medical imaging in general

Marc Kachelrießa! and Willi A. KalenderInstitute of Medical Physics, University of Erlangen–Nürnberg, D-91052 Erlangen, Germany

sReceived 24 November 2004; revised 1 March 2005; accepted for publication 1 March 2005;published 18 April 2005d

CT scanners acquire noisy data at discrete sample positions. Typically, a convention of how tocontinue these data from discrete integer positions to the continuous domain must be applied duringprocessing. We study the properties of three typical one-dimensional spatial domain interpolationalgorithms in terms of a cost or quality factorQ. This figure of meritQ is a function of spatialresolution, data noise, and dose and is used to optimize detector design. Spatial resolutionR isdefined as either mean square widthD or as the full width at half maximumW of the point spreadfunction sPSFd. Our results show that a trapezoidal interpolation algorithm is optimal for the highresolution domainsrelative to the detector aperture sizegd and should be replaced by a triangular orGaussian interpolation function for spatial resolutions of about 1.3g or larger; these result in bell-shaped PSFs. Assuming such a hybrid algorithm we find a 1.5-fold increase ofQ2—this is equiva-lent to 50% improved dose usage—when smoothing the data to a spatial resolution of 3g or morecompared to a highest resolution reconstruction. Therefore it is advisable to use detectors of one-third of the size of the desired spatial resolutionW and to compensate for the 1.5-fold increase inQ2 by reducing dose by 33%. Under the presence of moderately sized septase.g., 10% of the spatialresolution element sized the benefit of optimizing still lies in the order of 30% improved dose usage;in that case the detector sizeg should be on the order ofW/2 and a dose reduction of 23% can beachieved. Again, bell-shaped PSFs show a better tradeoff between noise and resolution for a givendose than rectangular-shaped PSFs. The general interpretation of our results is that the degree offreedom of choosing the weighting or interpolation function for a given resolution is large for smalldetectors and small for large detectors. Thus systems with smallg have a higher potential ofoptimization compared to systems with largeg. Similarly, detector binning, which corresponds toreplacing g by 2g, should be avoided. Note that the figures reported correspond to a one-dimensional interpolation. Two-dimensional detectors typically separate and resulting quality fac-tors can be easily obtained by multiplication. Then,Q2 is expected to improve by a factor of 1.52

without septa and by a factor of 1.32 with septa. This indicates that dose can be reduced by about56% and about 41%, respectively. Our findings are general and not restricted to CT. They can bereadily applied to medical or nonmedical imaging devices and digital detectors and they may alsoturn out to be useful in other fields. ©2005 American Association of Physicists in Medicine.fDOI: 10.1118/1.1897083g

Key words: computed tomography, noise, spatial resolution, dose

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I. INTRODUCTION

The introduction of multislice CT scanners in 1998 induced a race for more and more slices that become thand thinner. It is a well-known fact that image noisecreases with increased spatial resolution and that theretrade-off among image noise, dose, and spatial resolu1

The question is: Will the race for more slices continue?what will it converge to?

Here, we want to give a partial answer by showing thmakes sense to further decrease the detector elemeneven if one is already content with the spatial resoluachieved so far. Our primary focus lies in multislicesMSCTd and cone-beam CT imaging. To emphasize thaconsiderations are of general nature a rather generamathematical notation is used and we will switch bactypical CT terminology only when presenting specific
amples.
1321 Med. Phys. 32 „5…, May 2005 0094-2405/2005/32„5…/1

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To find the optimal detector element size for a givential resolution the influence of data presampling and spling as well as the influence of the algorithm onto spresolutionR and noiseN is studied. We will restrict ourselvto linear systems which are easily traceable using stansignal theory tools. To relate spatial resolutionR, noiseN,and doseD to obtain a single figure of merit a quality facis defined as follows:

Q2 =1

RN2D. s1d

This definition accounts for the fact that noise variashould be inversely proportional to the quanta used fomeasurement and that it should be inversely proportionthe integration lengthR since the number of samples cont
uting should be proportional toR. Although we will concen-
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1322 M. Kachelrieß and W. A. Kalender: Presampling, algorithm factors, and noise in CT 1322

trate here on CT imaging the results may apply to other fias well. For exampleD can be thought of as the time avaable to carry out the measurements or, equivalently, fonumber of repeated measurements. Of course there msituations where our cost functionQ is inadequate. In thcase the reader can easily adapt the figure of merit to hisneeds, define a new quality factorQ8, and redo the finaperformance comparisons with respect toQ8. For two-dimensional detectors,Q2=1/sR2N2Dd would be the appropriate generalization.

It may appear at this point that a contrast detail dsCDDd analysis could be an appropriate tool to quantifyformance, too. However, CDD means quantifying perction, it would require one to define two additional paremters: the contrast and the observer and results wouldonly to these specific settings. We provide a far more brelation among noise, spatial resolution, and dose. Ousults can be used as input to a CDD analysis, if desired

Two different spatial resolution measures are used foinvestigations. The second momentD and the full width ahalf maximumW of the point spread functionsPSFd. There-fore R will be replaced by eitherD or W in the definitions1dof the quality factor. NoiseN and doseD is of interest to uonly on a relative scale and the figuresQ we derive will begiven at relative scales, too. Noise will often be givenstating the noisesreduction or amplificationd factor F that isintroduced by the algorithm applied and we will haveN=Fin these cases.

We will further restrict our considerations to ondimensional functions. The results can easily be adapttwo-dimensional or higher systems. This adaptation is trif the functions involved separate inx and y and in othedimensions.

A literature survey shows that many authors have alrtreated noise in radiological imaging. Basic relationstween noise and spatial resolution and basic know-hoerror propagation from CT raw data into CT images isvided in the textbooks.2–4 Other important papers to mentiare Refs. 5–15. Almost all of them deal with the propertiefiltered backprojection that is known to amplify noise vance as the thirdstwo-dimensionald or fourth sthree-dimensionald power of spatial resolution. Absolute staments were derived regarding relations between resoland image noise which had often been related to spescanner setups, specific reconstruction algorithms, anspecific reconstruction kernels. New algorithms to imprfiltered backprojection, rebinning, and reconstruction kerhave been proposed and validated. None of the saidences treats the relationship between detector elemenand desired spatial resolution. Our paper, in contrast, ilated to the more basic issue of interpolating odimensional functions by smoothing them down to a spfied spatial resolution. It is therefore directly relatedmultislice spiral CT where the effective slice thicknessdefined during reconstruction by specifying the desirez

1
filter.
Medical Physics, Vol. 32, No. 5, May 2005

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II. GENERAL CONSIDERATIONS

A. System model

To model the system and to quantify its performanceuse the well-known tools of linear system theory.16–18 Letfsxd be some function that we would like to assess by msuring it. Every detector performs some kind of presampwhich is typically some smoothing. This can be describea presampling functionssxd which also comprises smoothieffects introduced by a finite sized focal spot. The FoutransformSsud of ssxd is the presampling MTF. The outputthe detector is of the formfsxdpssxd. This output is availablehowever, only at discrete sampling positions, say at spaDx=1, and all we can assess is

sfsxd p ssxddIII sxd,

with III sxd=ondsx−nd being the shah function.Signal processing introduces an algorithm factorasxd

such that the final result is given by

f̂sxd = ssfsxd p ssxddIII sxdd p asxd. s2d

The main purpose of the algorithm factor is to providcontinuation of the discrete data samples ontoR. In this con-text we can think ofasxd being an interpolation functioNonlinear algorithms, such as spline interpolation, thatnot be treated as a convolution with an algorithm factorasxdyield data-dependent resultsse.g., splines use infinite impulse response filtersd that are not traceable. Furthermore,MTF or PSF concepts do not apply and noise predictionot possible for those nonlinear approaches. However,imaging cases linear and shift-invariant results are desirleast approximately. Hence one can find a linear algorasxd that sufficiently approximates a given nonlinear arithm and make use of the results obtained in this pape

A very simple algorithm factor is the rectangle functII sxd. It corresponds to a nearest neighbor interpolatioadjacent samples. Often, the algorithm factor is implicapplied. For example when a digital image sampledsquare grid is displayed on a computer screen. Each sapoint will control the brightness of a display pixel andshape of that pixel then determinesasx,yd which is, in thacase, a two-dimensional function. The choice of the arithm factor mainly depends on the application. In our imdisplay example an algorithm factor that leaves gapstween neighboring sample points is rather undesiredzooming into the image would then show a black gridraster in-between the measured samples. If, however, ointerested in determining the point-spread function ofdetector that had acquired the image data one might desalgorithm factor that can even approximate a delta funcUnfortunately, the real world is noisy and we will see tsharply peaked algorithm factors tend to significantlycrease noise.

To characterizef̂sxd we assume that the absolute sampposition is irrelevant to the results. Our intent is to repthe shah function of Eq.s2d by III sx− td and to average ov
all t. This is justified by the fact that the relative orientation

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of our primary signalfsxd with respect to the detectorrandomsthe details that we want to visualize do not knwhere the sampling positions ared. Mathematically, this requirement is necessary since our system, which shoulinear and translation-invariant to be traceable, is only linand translation-invariant on a scale much larger thansampling distance. We can obtain the desired propertieaveraging overt fconvolution with IIstdg:

f̄sxd = ssfsxd p ssxddIII sx − tdd p asxd p II std

= fsxd p ssxd p asxd s3d

and now are ready to characterizef̄ in place of f̂.An example where this averaging is often enforced is

measurement of the response function, which means thais either interested in the combined knowledge of the pspread function PSFsxd=ssxdpasxd or one desires to extrassxd or asxd. In radiography, for example, the said averagis performed by taking care to continuously vary the spling positions by imaging an edge that is tilted with respto the detector axes. To determine the in-plane point spfunction in CT one measures a delta objectse.g., a thin wiredthat is not located in the rotational center of the scanner.ensures that the projection of the wire continuously swover a number of detector elements during one rotation.CT slice sensitivity profilesSSPd, which is the longitudinapoint spread function, can be determined by either usinfact that spiral CT provides these varying sampling positanyway or one can use a stepping motor to move someobject in small increments through the slice. Summarizall these efforts average out the undesiredsand randomd ef-fect of absolute sampling positions.

We now assume proper normalization of the presampfunction and of the algorithm:

E dx ssxd = 1, E dx asxd = 1.

These demands ensure the conservation of mass

E dx fsxd =E dx fsxd p ssxd =E dx fsxd p ssxd p asxd.

Violating this normalization in CT, for example, would mereconstructing the wrong density values and thereforeplaying a wrong HU scale.

B. Spatial resolution measures

Given that the measurement and the algorithm aremalized properly we can obtain the system respPSFsxd=ssxdpasxd. This point spread function can be useddefine a spatial resolution measure. We define two spresolution measures, the full width at half maximsFWHMd W and the second momentD of the point spreafunction.W is best defined implicitly by

PSFs0d/2 = PSFsW/2d,

whereasD is given by the integral


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D2 =E dx x2PSFsxd.

The third possibility to quantify spatial resolution using,example, 10% of the modulation transfer functionsMTFd,which is the Fourier transform of the PSF, will not be csidered here.

C. Noise factor

Assuming that the detector samples are independendom variables we can determine the noise introducedsorremovedd by the algorithm by error propagation. Basicawe are dealing with the filtering of noisy signals anderror propagation characteristics are well understood.3 Lets2sxd be the variance corresponding to the valuefsxdpssxdmeasured at positionx. Error propagation yieldss2sxdpa2sxdfor the variance after signal processing. Integrating overxgives the mean varianceS2F2 with S2=edx s2sxd being themean variance of the raw data and

F2 =E dx a2sxd

being the noise factor of the algorithm.As an example assume the algorithm either to perfo

nearest-neighbor interpolation or to perform a linear intelation. In the first caseasxd=IIsxd and the noise varianwould be multiplied by the factor 1. For the linear interlation we haveasxd=Lsxd and the factor introduced by talgorithm isF2=2/3.

D. Noise performance at specified MTF

A rather general result can be obtained by regardingnoise factor as a function of presamplingSsud for a specifiedsystem modulation transfer function MTFsud. To do so, leMTFsud be fixed. From PSFsxd=ssxdpasxd we findMTFsud=SsudAsud where Asud is the Fourier transform oasxd. Using Rayleigh’s theorem allows us to computenoise factor:

F2 =E dx a2sxd =E du A2sud =E duMTF2sud

S2sud.

We here assume that the specified MTF has its cut-off bthe first zero ofSsud occurs. In spatial domain this meathat the specified PSF must not be “sharper” than thesampling functionssxd. Regarding two systems, systemwith small detectors and system B with large detectorsfind that for well-behaved presampling functions the ineqity SAsud.SBsud will hold. ConsequentlyFA

2 ,FB2 and thus

QA2 ,QB

2, which implies that smaller detectors are to beferred over larger detectors. Note that case B can alsachieved by binning detector elements of type A. Hebinning should be avoided!

To become quantitative we will now continue and ana
if these findings apply to more practical algorithms that op-

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III. SYSTEM AND ALGORITHMS

Let us consider a system as shown in Fig. 1. It hasform response detectors whose integrating size is giveng.We further assume a point sourcesinfinitely small focal spotdand we assume that cross-talk effects between the deelements can be neglected. The presampling function odevice is then

ssxd = IIg*sxd,

where the active widthg is also known as the geometricefficiency of the detectorssee the Appendix for definitionand explicit expressions of rectangle functions and convtions of rectangle functionsd. In many casesg,Dx sincedead space, a gap or some septum is in-between adchannels. For CT, a typical value isg<0.9Dx. Howeverthere are situations whereDx,g can be achievedsin termsof the Nyquist theoremDx=g/2 is desiredd. For exampleperforming a second acquisition with the detector shiftehalf of its original sampling distance can be used to decrDx. In CT, quarter detector shift and flying focal spot teniques are used to double the sampling density. Onealso use nonstructured scintillators to obtaing.Dx. How-ever, in this case modelings as a rectangle function is inaequate since a bell-shaped presampling function, e.Gaussian, results. We will, however, restrict ourselverectangular sensitivity functions.

A considerable simplification of the following mathemacal expressions can be achieved by scaling the wholelem to achieveg=1 sotherwise dozens of fractions withgbeing the denominator would resultd. Sinceg determines spatial resolution it is the adequate parameter to put other levariables in relation to. It must be emphasized that ousults are independent of samplingfthe sampling distanceDxhad been removed by averaging in Eq.s3dg and one musresist the temptation to identify the sampling distanceDxwith g sor with g+d with d being the septa thicknessd. Theconsequence of our agreement is

ssxd = II1*sxd,

which will be used in the following. Later we can convertresults to other values ofg by simple scaling.

Now let us define some sample algorithms. Besides b

FIG. 1. Uniform response detector system, three detector elemenshown. The sampling distance for that specific detector is given asDx=g+d.

properly normalized we seek for the algorithm factors that


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allow one to adjust the spatial resolution from a reasonminimum to infinity. We define the reasonable minimumbe an algorithm that achieves a point spread function wFWHM matches the detector apertureg. Note that the use oenhancing techniquessalgorithms with negative valuesd hasthe potential to go even below that limit. Even more,second moment measureD2 may even become negativechoosing filters with negative components. This papernot consider using enhancing techniques because thesto increase noise disproportional.

The three basic algorithmssand their point spread funtionsd used for our investigation are

a1sxd = IIw,w** sxd, PSF1sxd = II1,w,w

*** sxd, 12 ø w,

a2sxd = II1,w** sxd, PSF2sxd = II1,1,w

*** sxd, 0 ø w, s4ad

a3sxd =1

Îp/3we−3x2/w2

, PSF3sxd = II1*sxd p a3sxd, 0 ø w.

Algorithm a1 does a convolution with a narrow triangle fution of full width at half maximumw. The lower limit of thawidth has been chosen such that the full width at half mmum of the corresponding PSF just equals the detecto

FIG. 2. Point spread functionssad for w=2 andsbd for w=3. The algorithm1, 2, and 3 are coded with solid, dotted and dashed style, respective

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Algorithm a2 consists of a trapezoidal filtering. The tpeze has a plateau of widthuw−gu and a base width ofw+g. We chose the sizeg=1 of the detector aperture as oparameter since then the PSF becomes a combinationlinear filter with a base width of 2g and a box-car filter owidth w.

With algorithma3 smoothing to lower spatial resolutionachieved using a Gaussian. Its PSF can be evaluated inof the error function, but the result is analytically not furttraceable:

PSF3sxd =1

2o±

erfSÎ31 ± 2x

2wD .

As one can see in the following the FWHM of the pospread function of algorithm 3 is fairly close tog for w beingon the order of 1/2. It would not make sense to decreawbelow 1/2 for that algorithm since spatial resolution wohardly improve while noise will significantly increase.

The full width at half maximum values are given as

W1swd =51 if w ,

1

2

2w + 1 −Î4w − 1 else ifw ,5

4

w +1

4else,

6W2swd =51 + w/4 if w ,

4

5

2 + w − Î4w − w2 else if w , 2

w else,6 s4bd

W3swd → 51 for w ! 1

1

3Î12 ln 2w < 0.96w for w @ 1.6

The second moment spatial resolution valuesscf. Sec. 4 othe Appendixd are given as

12D12 = 12 + w2 + w2,

12D22 = 12 + 12 + w2, s4cd

12D32 = 12 + 2w2

and noise evaluates to

F12swd = 2/3w,

F22swd = Hs3 − wd/3 if w , 1

s3w − 1d/s3w2d else,J s4dd

F32swd = Î3/s2pd/w.

To compare system performance in terms of theQ factor
defined in Eq.s1d we must ensure equal spatial resolution for

a

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all approaches. This can be easily done by inverting thelution Eqs.s4bd and/ors4cd to solve forw. We find

w1sDd = w3sDd =1

2Î24D2 − 2 for D ù Î1/12,

w2sDd = Î12D2 − 2 for D ù Î1/6

and forWù1 we obtain

w1sWd =1

25W+ Î2W− 2 if W,3

2

2W−1

2else, 6

w2sWd =1

258W− 8 if W,6

5

W+ ÎWs8 − Wd − 8 else ifW, 2

2W else.6

The inversion ofW3swd cannot be given analytically and w

FIG. 3. Point spread functions for given FWHM:sad W=2, sbd W=3.

use a numerical approximation thereto for our plots.

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IV. RESULTS

In the following, algorithms 1, 2, and 3 will be plottedred, green, and blue, respectively. To allow for reproducon gray scale and even black and white printers the plotis chosen to be solid, dotted, and dashed, respectively, fothree algorithms.

A. Algorithm analysis

Samples of point spread functions are given in Fig. 2interpolation widthsw=2 andw=3. Similarly, Figs. 3 andshow point spread functions for constantWi and for constanDi, respectively. The relation of both spatial resolution msuresW andD is shown in Fig. 5sad. Figure 5sbd shows a ploof the spatial resolution for all three algorithms and bspatial resolution measures. We have added a vertical linw=1 which is a widely used value wheneverDx=1 is givenWe find

W1s1d = W2s1d = 3 −Î3 < 1.27, W3s1d < 1.22

for the FWHM at that position. This means that the “natuspatial resolution lies about 25% above the detector sizmultislice spiral CT this is reflected by the fact that mmanufacturers offer an effective slice thickness aboutabove the collimated slice thickness as the default re

FIG. 4. Point spread functions for givenD: sad D=1, sbd D=3/2.

struction setting.sRegarding theD values we findD=1/2 for


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all three algorithms atw=1 which compared to the valuew=0 is an increase of about 73% for algorithms 1 and 3of 22% for algorithm 2.d

Let us continue and regard the quality figures of mer

Qi2sDd =

1

DFi2swisDdd

, Qi2sWd =

1

WFi2swisWdd

.

For algorithms 1, 2, and 3 one finds

0 ø Q12sDd = 3

4Î24 − 2/D2,

limD→`

Q12sDd = 3

2Î6 < 3.67,

Î6 ø Q22sDd =5

3/D2

3/D − Î12 − 2/D2for

1Î6

ø D ,1

2

36 − 6/D2

3Î12 − 2/D2 − 1/Dfor

1

2ø D, 6

limD→`

Q22sDd = 2Î3 < 3.46,

0 ø Q32sDd =Îp

3Î12 − 1/D2,

limD→`

Q32sDd = 2Îp < 3.54,

whereDù1/Î12 for algorithms 1 and 3 andDù1/Î6 foralgorithm 2. Interestingly, the limits obtained for heasmoothing are very similar for all algorithms.

We will not state the corresponding figures forFWHM-based spatial resolution measure. The expresturn out to become somewhat inconvenient due to theplicated analytical form ofWiswd and its inversions. The limits, however, can be derived easily and yield

Q12s1d = 3

4, limW→`

Q12sWd = 3

2 ,

Q22s1d = 1, lim

W→`Q2

2sWd = 1, s5d

Q32s1d = 0, lim

W→`Q3

2sWd =Î p

2 ln 2< 1.51

for large W and show that algorithm 2 significantly diffefrom the other two approaches.

Figure 6 plots theQ2 as a function of spatial resolutioUsing theD measurefFig. 6sadg shows no significant diffeences between the respective algorithms, apart from ththat algorithm 2 cannot go to very high spatial resolutiWe also find that the quality factor goes down to zero forvery high resolutions. Clearly, for algorithms 1 and 3w→0 the number of contributing quanta approachesand noise increases to infinity and it is obvious that restruction at the highestD-based resolution level should nbe attempted.

The comparison at equal FWHM shows an interesbehavior of algorithm 2. From Fig. 6sbd we see that th
corresponding curve exhibits a maximum atW<1.5 and

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thenQ2 decreases back to its original value. The reasonthat behavior is that algorithm 2’s PSF is close to rectangfor very small W and for very largeW. In between PSF2becomes bell-shaped, which is of advantage in termnoise.

The deviation of the ratio of the quality factors at mamum spatial resolutionW=1 and at “natural” resolutionw=1 from unity

Bi =Qi

2sWis1ddQi

2s1d− 1

gives us a first idea of the achievable benefit of a dete
size optimization. We obtain

rr

f

r

B1 =1Î3

< 58 % , B2 =1

4sÎ3 − 1d < 18 % , B3 = `

for algorithms 1, 2, and 3, respectively. These values cadirectly converted into the necessary dose increase wheforcing the maximum spatial resolution compared tonatural interpolation.

Figure 6sbd further indicates that a hybrid algorithm cosisting of a combination of algorithm 2 in the high resoludomain and algorithm 3 in the low resolution domain wo

FIG. 5. Illustration of the relationshamongw, W, andD.

be advantageous. The turn-over point lies atW<1.1 and the

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hybrid algorithm is denoted asa23sxd. In the following, wemake use of this hybrid approach when quantifying thetimization benefit.

No significant algorithm differences are found withspect to theD resolution measure. We therefore considerFWHM descriptorW to better distinguish between the vaous algorithms.

B. Septa

We now design a sampling system that achieves a g
spatial resolution at minimum cost, i.e., at optimal quality. In

n

contrast to the previous considerations we want to apenalty that accounts for detector septa or other kind ofspace. Without such a penalty one would have to recommusing infinitely small detectors, as we saw earlier, and cbine that detector with an efficient smoothing algoriswhich is the Gaussian smoothing imposed by algorithmd.

Let the thickness of the septa bed and assume the apture to beg. The septa decrease efficiency by a factog/ sg+dd because doseD must be increased bysg+dd /g toobtain the same statistics. The equations are scaledaperture sizeg swhich is now released from its previo

FIG. 6. Quality Qi2 as a function o

spatial resolutionR. All algorithms behave similarly with respect to the seond moment spatial resolution mesureD. With respect to the full widtat half maximum spatial resolutiomeasureW there are significant diffeences between the algorithms.

value 1d as

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Q̂i2sR,g,dd = Qi

2SR

gD g

g + ds6d

with R standing forD or W and where the new penalty facaccounting for the septa has been multiplied.

To optimize the detector for a given spatial resolutionnow fix R and regard quality as a function ofg. Figure 7gives an example that has been computed usingd=1/10.Clearly, the curves now exhibit a well-pronouncmaximum—if we had setd to zero the curves would increatoward the limitss5d for g decreasing to 0. From the ordnates we find that using the optimum valueg<0.5 increase
quality by almost one-third compared tog=1.

Figure 8 gives an idea of where the optimumgopt

is located as a function of septa widthd and it shows

the Q̂i2 values corresponding to these maxima. The

result here is that increasing septa thickness sugincreasing detector aperture. One should, however, not fthat sampling conditions can become an issue forthick septa and, even more, dose efficiency is gettinglow.

The benefit of optimizing can be found by comparing=gopt to a system withg=W. The plot of Fig. 9 indicateimprovements on the order of 30% or more for mode

FIG. 7. Plots ofQ̂i2sR,g,dd. As refer-

ence values we choseD=Î1/6 andW=1, which is the maximal resolutiothat can be obtained for all three algrithms. The results can easily be scato other values by simple multipliction. Obviously, about 30% qualitysordosed can be gained for this septhickness by using detectors thathalf of the size of the desireresolution.

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V. EXAMPLES

A. Detector binning

Many detector systems allow for a pairwise electrocombination of neighboring detector elements. This binis used to reduce data transfer rates or to speed up datcessing and it is justified as long as spatial resolution islimited by the detector element size.

In our terminology, binning of two detectors correspoto replacingg by 2g swhere we neglect septa, for convnienced. Regarding Fig. 6sbd, where binning correspondsjumping fromW/g to W/2g, we see that binning reducesquality factor as long as the desired spatial resolutionW is onthe order of 2g. For example with the Gaussian algorithmcan obtainQ3

2sWd<1.41 for W=2 if no binning is use
whereas the best we can do with binned data to obtainW

o-t

=2 is to use algorithm 2 and obtainQ22sWd=1 which is sig

nificantly less. Similar results are obtained from Fig.sadregarding theD resolution measure.

B. Single-slice spiral 180°LI vs 360°LI

As a simple example we want to use the formalismevaluate the image quality in single-slice spiral CT. Wspiral CT became available two different algorithms woffered. The 360°LI algorithm is an algorithm that perfora linear interpolation between the two closest data padjacent to the desired reconstruction plane. The 180°Lgorithm makes additional use of redundant data obtaineconsidering that each ray is measured twice during one

FIG. 8. Optimum g as a function ogiven d scurves from lower left to up

per rightd and the correspondingQ̂i2

valuesscurves from upper left to lowerightd.

tion. There, one of the two data points corresponds to the

ob

tion,ling

thee-

r

thfrom

t forsw

fthew

lityual-

ms:

-age

atedduce

e tot

fzetimal

efit

t

toved.

m


actually measured line integral and the other value istained from a spiral offset by approximately 180°.1

For both algorithms, the slim and the wide reconstruca linear interpolation is used and differs only by the sampdistance and the interpolation width which ared/2 for the180°LI algorithm andd for the 360°LI algorithm withd being the table increment per rotation. Thereforesingle-slice algorithm is of typea1sxd where we replace thgeometrical efficiencyg by the collimated slice thicknessS and the interpolation weightw by d/2 and d, forconvenience. We further introduce the spiral pitch factop=d/S.

The algorithm factors are given as

a180szd = IId/2,d/2** szd, PSF180szd = IIS,d/2,d/2

*** szd,

a360szd = IId,d** szd, PSF360szd = IIS,d,d

*** szd.

The spatial resolution figures become

W180spd = SW1sp/2d

= S51 if p , 1

p + 1 −Î2p − 1 else ifp , 5/2

p/2 + 1/4 else,6

W360spd = SW1spd

= S51 if p , 1/2

2p + 1 −Î4p − 1 else ifp , 5/4

p + 1/4 else6

and

12D1802 = S2 + s 1

2dd2 + s 12dd2 = S2s1 + 1

2p2d ,

12D3600 = S2 + d2 + d2 = S2s1 + 2p2d. s7d

FIG. 9. Benefits obtained when optimizing the detector sizeg as a functionof septa thicknessd for a given spatial resolution. Top curve: ben

Q̂32sW,gopt,dd /Q̂2

2sW,1 ,dd−1 for W=1. Bottom curve: benefi

Q̂32sD ,gopt,dd /Q̂2

2sD ,1 ,dd−1 for D=Î1/6. For moderate septa sizes up30% benefit and therefore a 30% increase in dose efficiency is achie

Note that


-

W360spd = W180s2pd, D360spd = D180s2pd,

which means that the system’sz resolution scales for boresolution measures in the same way when switching180° interpolation to 360° interpolation.

When computing the noise factors one has to accounthe sampling distance that isd/2 and d rather than 1 aassumed in Eq.s4dd. This is done by scaling to the nesampling distance by multiplication withd/2 or d and oneobtains

F1802 sdd = 4

3, F3602 sdd = 2

3 .

We find that image noise is increased by a factor oÎ2with 180°LI compared to 360°LI. However regardinggain in spatial resolutionsFig. 10d we find that it stays belo2 for all pitch values that are used in single-slice CT.

To find out whether the slim algorithm is of lower quathan the wide reconstruction we will now compute the qity factors. DoseD is proportional top−1 and for a givenspatial resolutionD we use Eq.s7d to solve forp. We thenobtain—surprisingly?—the same result for both algorith

Q1802 = Q360

2 =1

DF2D=

3

4Î24/S2 − 2/D2.

Apparently, both single-slice spiralz-interpolation algorithms are of the same quality and the increase in imnoise with 180°LI reconstruction is correctly compensfor by the possibility to increase the pitch and thereby repatient dose compared to the 360°LI algorithm.

C. MSCT

1. Achieving a specified resolution

Let us consider a typical MSCT scanner. It allows onsimultaneously acquireM slices of thicknessS smeasured athe rotation centerd. Assume one desires az resolution oW=0.5 mm. And assume that there are septa of sid=0.1 mm between the detector elements. What is the op

FIG. 10. GainW360spd /W180spd in spatial resolution when switching fro360°LI to 180°LI.

detector element sizeg to use?

10or

n

tylgo-al-

us

mmofsagnoot

oidarisely-

en-

havin th

solu

Asis ano-

ssedtial

zedntialasescondmea-werelyzerth

nearbi-

s de-

nottiontwoffer.tical

ereTF.

t thattion

in-ical.rec-

y also

d asersusas a

tiontalk,in-

pre-

thisivenllowsuisttionre isnguently


We cannot use Fig. 7 since it assumes the septa to beof the desired spatial resolution. However from Fig. 8from Eq. s6d, we can find that the optimal aperture sizeg isgiven by

gopt,i = arg maxg

Q̂i2s0.5 mm,g,0.1 mmd

= 50.34 mm for i = 1

0.36 mm for i = 2

0.30 mm for i = 36

and achieves a quality factor of

Q̂opt,i2 = max

gQ̂i

2s0.5 mm,g,0.1 mmd = 50.96 for i = 1

0.95 for i = 2

1.02 for i = 3.6

Regarding the typical situation withg=W sspatial resolutiopushed to its extreme limitd the quality factors are lower:

Q̂i2s0.5 mm,0.5 mm, 0.1 mmd = 50.63 for i = 1

0.83 for i = 2

0.00 for i = 3.6

The best algorithm to use for detector aperturesg being aslarge as the spatial resolutionW is algorithm a2szd, which

achieves a quality index ofQ̂2=0.83. The highest qualiindex with optimized aperture can be achieved with arithm a3szd. This requiresg=0.3 mm and one obtains a qu

ity index of Q̂2=1.02. The benefit of optimization is thgiven by 1.02/0.83−1=22%salso see Fig. 9d.

Summarizing, to achieve a spatial resolution of 0.5sFWHM of the slice sensitivity profiled assuming septathickness 0.1 mm we find a 22% increase in dose uwhich is equivalent to a 19%s=1−0.83/1.02d dose reductiowhen using 0.3 mm detector elements and Gaussian sming compared to 0.5 mm detector elements with trapezsmoothing. In the case of standard samplingsarea detectowithout flying focal spotd the sampling distance itselfgiven byg+d and evaluates to 0.4 mm. Due to the relativlarge septa thickness doublez sampling is indicated to improve sampling characteristics.

Note that the spatial resolutionW is also known as theffective slice thicknessSeff in MSCT and that we may idetify g with the nominal slice thicknessS of the scanner.

2. Preferred spatial resolution for a givendetector

In contrast to the previous section we here assume toa MSCT scanner available whose detector element sizez direction is given byg=0.5 mm. We still used=0.1 mmfor the septa thickness. What is the preferred spatial re
tion W to head for during reconstruction?

%

e

h-l

ee

-

Since bothd and g are fixed we find from Eq.s6d that

maximizing Q̂isW,g,dd with respect toW is equivalent tomaximizingQisW/gd. Consequently, the results of Sec. Vapply and we find from Fig. 6sbd that W must become alarge as possible for algorithms 1 and 3 whereas thereoptimum at W<1.5g=0.75 mm for the trapezoidal algrithm.

VI. DISCUSSION AND CONCLUSIONS

Given a rectangular presampling function we discuthe behavior of three typical algorithms to perform spadomain interpolation. All three algorithms are of finite sisupport given that the Gaussian, which exhibits exponedecay sin x2d, is truncated as soon as its value decrebelow a certain threshold. We derived expressions for semoment-based and for FWHM-based spatial resolutionsures. Algorithm noise, spatial resolution, and dosetaken into relation to define a quality index and to anathe algorithms with respect to that figure of merit. It is woto mention that our results easily generalize to any licombination of the algorithms as long as this linear comnation is taken at equal spatial resolution.

The reader should be aware that spatial resolution ifined in this paper using a figure of meritseither D or Wd.Achieving a given spatial resolution therefore doesspecify the complete point spread function or modulatransfer function and even if the spatial resolution ofsystems is identical their PSFs and MTFs are likely to diIt depends on the specific application whether this is crior not.

We did not construct algorithms in Fourier domain whone could theoretically achieve a specified PSF and MThis seeming advantage is counterbalanced by the facFourier-based algorithm design is likely to yield cancellaproblems close to the zeroes of the presampling MTFSsud.Further, such interpolation algorithms are likely to be offinite size in spatial domain and thus may not be practNevertheless, it was shown that our general finding ofommending smaller detectors than absolutely necessarapplies to this group of algorithmssc.f. section II Dd.

Our detector and algorithm model is to be understooan example to demonstrate the effects of detector size vspatial resolution. The detection process was modeledrectangular profile which is a valid first-order approximato most realistic CT systems. Effects of detector crossdetector after glow, or finite sized focal spot should becluded into the model if higher accuracy of performancediction is desired.

Sampling conditions are handled only marginally inpaper. Our predictions of optimal detector size for a gresolution assume that the detector or system design aone to additionally achieve sufficient sampling. The Nyqcriterion, which is especially important in the high resoludomain, should be fulfilled as usual. Consequently, theno difference with respect toQ between standard samplischemes and double sampling schemes as they are freq
used. For example our evaluations already compriseQ fac-

the

sam

evedingctorterpon.

s isd-50%imierenedreable.andse

asenen

uctior, wed

terbe-

Speex

ion.s anasema-so-will

ning

y

as

orted


tors of typical CT double sampling techniques such asdetector quarter offset or the flying focal spot.19 This is, ofcourse, not true regarding the artifact behavior wherepling becomes an important issue.

The optimal resolution and noise tradeoff can be achiby a hybrid algorithm that performs trapezoidal smoothfor spatial resolution values below 1.1 times that deteaperture size and that switches to a Gaussian-shaped inlation function to optimally achieve lower spatial resoluti

We have further shown that the ideal behavior ofQ2 beingconstant over a wide range of spatial resolution valuevalid only for the low resolution domain. Therefore it is avisable to use detector elements that are on the order ofor more smaller than the desired resolution element. Slarly, detector binning should be avoided, if possible. This less freedom to design an optimized algorithm on bindata than on unbinned data and we have shown an incof Q2 on the order of 40% if unbinned data are availaFurthermore, binning reduces sampling by a factor of 2aliasing is more likely to occur. And binning will increanonlinear partial volume artifacts in CT.

On the downside of using smaller detectors are increrequirements regarding data bandwidth, electrical compocosts, and increased data processing and reconstrtimes. In view of increased dose awareness, howevebelieve that these disadvantages are negligible comparthe potential gain in dose efficiency.

What about the slice war in MSCT? Today, submillimeisotropic resolution on the order of 0.5 to 0.75 mm hascome clinical routine and is widely accepted and used.cial applications, such as studies of the inner ear, forample, could benefit from even higher resolutManufacturers are driven by market share considerationit is likely that the slice thicknesses will further decrewhile the number of slices will increase. Given that thejority of CT scans will be reconstructed with the spatial relution one uses today patient dose and image quality
certainly profit from this decrease in slice thicknesses.

-

o-

-

se

dtn

eto

--

d

APPENDIX: RECTANGLE FUNCTIONS

We will now state some helpful expressions concerthe convolution of rectangle functions. Starting from

IIa*sxd =

1

uauIIS x

aD ,

a rectangle function of widtha and area 1, we will explicitlgive the following functions:

IIa,b** = IIa

* p IIb* ,

IIa,b,c*** = IIa

* p IIb* p II c

* .

The n-fold convolution is recursively defined as

IIa1,. . .,an

*n = IIa1,. . .,an−1

* sn−1d p IIan

* .

The functions IIa1,. . .,an

*n are invariant under permutationwell as under a change of sign of the parametersa1, . . . ,an.For scale transformations we have

IIaa1,. . .,aan

*n s·d =1

uauIIa1,. . .,an

*n S ·

aD .

For the sake of simplicity we will not state IIa1,. . .,an

*n in thefollowing, but rather the functions ofdoubled widthII 2a1,. . .,2an

*n . Moreover, we assume the parameters to be sto bedescending, i.e., we assume 2a1ù ¯ ù2anù0. Sincethe functions are symmetric with respect to they axis wefurther assumexù0 without loss of generality.

1. Convolution of two rectangle functions

aùbù0,x.0:

II 2a,2b** sxd =

1

4b50 if a + b , x

a + b − x if a − b , x ø a + b

2b if x ø a − b.6

2. Convolution of three rectangle functions

aùbùcù0,x.0:

II 2a,2b,2c*** sxd =

1

8abc50 if a + b + c , x

1

2sa + b + c − xd2 if a + b − c , x ø a + b + c

2csa + b − xd if a − b + c , x ø a + b − c

4bc−1

2s1 − b − c − xd2 if ua − b − cu , x ø a − b + c

4bc− sa − b − cd2 − x2 if x ø − sa − b − cd4bc if x ø a − b − c.

6

sefu

of

bol

er

ter

or

nd

,

cf.

cf.

ure,

on

re,

e

en-Math-

m-

uc-

ount-ogr.

gra-

om-

go-

phytion,”

e in

sys-

eam

s ofe-slice

r in

tedts of

l,

al-g-


3. Limits

For the sake of completeness, we will state some ulimits:

lima→0

IIa* = d,

limak→0

IIa1,. . .an

*n = IIa1,. . .,ak−1,ak+1,. . .,an−1

* sn−1d ,

limak→±`

akIIa1,. . .,an

*n = ± 1,

limn→`

IIa/În,. . .,a/În*n sxd =

1Î2ps

e−1/2sx/sd2 with s2 =1

12a2.

4. Second moment

The second moments, which characterizes the widththe function, is given by

s2 =E dx x2IIa1,. . .,an

*n sxd =1

12ok=1

n

ak2.

NOMENCLATURE

This section lists some of the symbols used. The symrelated to signal processing are adapted from Ref. 16.

b·c 5 floor function, yields greatest integer lowor equal

d·e 5 ceil function, yields smallest integer greaor equal

ds·d 5 Dirac’s delta functionsgns·d 5 sign function such thatx= uxusgnxIII s·d 5 shah function, also known as sampling

comb function, IIIsxd=ondsx−ndII s·d 5 rectangle function of width and area 1 a

of centroid 0IIa

*s·d 5 rectangle function of widtha and area 1IIa

*sxd=IIsx/ad / uauIIa,b

** s·d 5 convolution of two rectangle functions,the Appendix, IIa,b

** sxd=IIa*sxdp IIb

*sxdIIa,b,c

*** s·d 5 convolution of three rectangle functions,the Appendix, IIa,b,c

*** sxd=IIa,b** sxdp II c

*sxdLs·d 5 triangle functionLsxd=IIsxdp II sxd

erfs·d 5 error function, erfsxd=s2/Îpde0xdt e−t2

f*n 5 n−1 fold self-convolution of functionf snfactors in the convolution productd

sincs·d 5 sinc function, sincsxd=sinspxd /px is theFourier transform of IIsxd

asxd 5 interpolation algorithm used,edx asxd=1ssxd 5 presampling function,edx ssxd=1

D 5 second moment spatial resolution measD2=edx x2PSFsxd

Dx 5 sampling distance


l

s

D 5 doseF2 5 algorithm noise factor,F2=edx a2sxdg 5 detector apertureN 5 noiseQ 5 quality measure,Q2=1/sRN2Dd with R be-

ing D or Ww 5 width parameter of the interpolati

algorithmsW 5 FWHM spatial resolution measu

PSFs0d /2=PSFsW/2dFWHM 5 full width at half maximum

MTF 5 modulation transfer functionPSF 5 point spread functionSSP 5 slice sensitivity profilesequivalent to th

PSF inz directiond

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