generalized multi-dimensional adaptive ﬁltering for ... · patient data of spiral and sequential...

Generalized multi-dimensional adaptive filtering for conventionaland spiral single-slice, multi-slice, and cone-beam CT

Marc Kachelrieß,a) Oliver Watzke, and Willi A. KalenderInstitute of Medical Physics, University of Erlangen-Nurnberg, Germany

~Received 3 August 2000; accepted for publication 1 February 2001!

In modern computed tomography~CT! there is a strong desire to reduce patient dose and/or toimprove image quality by increasing spatial resolution and decreasing image noise. These areconflicting demands since increasing resolution at a constant noise level or decreasing noise at aconstant resolution level implies a higher demand on x-ray power and an increase of patient dose.X-ray tube power is limited due to technical reasons. We therefore developed a generalized multi-dimensional adaptive filtering approach that applies nonlinear filters in up to three dimensions in theraw data domain. This new method differs from approaches in the literature since our nonlinearfilters are applied not only in the detector row direction but also in the view and in thez-direction.This true three-dimensional filtering improves the quantum statistics of a measured projection valueproportional to the third power of the filter size. Resolution tradeoffs are shared among these threedimensions and thus are considerably smaller as compared to one-dimensional smoothing ap-proaches. Patient data of spiral and sequential single- and multi-slice CT scans as well as simulatedspiral cone-beam data were processed to evaluate these new approaches. Image quality was as-sessed by evaluation of difference images, by measuring the image noise and the noise reduction,and by calculating the image resolution using point spread functions. The use of generalizedadaptive filters helps to reduce image noise or, alternatively, patient dose. Image noise structures,typically along the direction of the highest attenuation, are effectively reduced. Noise reductionvalues of typically 30%–60% can be achieved in noncylindrical body regions like the shoulder. Theloss in image resolution remains below 5% for all cases. In addition, the new method has a greatpotential to reduce metal artifacts, e.g., in the hip region. ©2001 American Association of Physi-cists in Medicine. @DOI: 10.1118/1.1358303#

Key words: computed tomography~CT!, multi-slice spiral CT~MSCT!, cone-beam CT~CBCT!,artifact reduction, noise reduction, dose reduction, adaptive filters

I. INTRODUCTION

X-ray computed tomography~CT! has been an essential di-agnostic tool for three decades. The introduction of spiralCT, which made true three-dimensional~3D! data acquisi-tion available, further increased its importance.1 It becamepossible to scan complete organs with an isotropic spatialresolution during a single breathhold. The introduction ofmulti-row detector systems in CT in 1998 further reducedscan time and allowed to acquire data for larger volumes athigherz-resolution due to the simultaneous acquisition ofMslices. Future developments will probably lead to the so-called cone-beam CT scanners — CT systems that have ahigh number of simultaneously measured slices — whichwill improve the overall performance even more.

Nevertheless there is a drawback in CT. To measure theline integrals the patient must be exposed to x-rays. The dosevalues for typical examinations nowadays lie between 50%and 500% of the annual natural background radiationexposure.2 Future developments aim at increasing the~iso-tropic! spatial resolution while keeping the image noise leveland thus the contrast detectability constant. As a conse-quence, the patient dose must be increased. Increasing dosein turn requires an increase of x-ray power, even when con-sidering that the cone angles will increase in the future al-

lowing one to make more efficient use of the available x-rayflux.

It can easily be seen that dose will become an increasinglyimportant problem in the future, on one hand for patientreasons, and, on the other hand, regarding the limited x-raypower. Thus, there is a strong need for dose reduction whilekeeping the image quality constant.

A very effective way to do so is to adapt the tube currentto the anatomy of the patient: For projection angles with highattenuation~typically the lateral direction! the tube currenthas to be increased and for angles with lower attenuation~mainly anterior–posterior! it can be decreased with respectto its nominal value. Corresponding studies show that thisyields mAs reduction values of up to 50% and a very homo-geneous image noise distribution.3–5 Although this method isclose to the optimum it is limited due to~a! the tube current’sinertia ~because the current cannot be varied arbitrarily fast!,~b! the fact that the tube current influences complete projec-tions only ~which will become a more important fact whengoing to area detector systems!, and ~c! the inability to ret-rospectively adjust the noise level.

We therefore propose a method that allows one retrospec-tively to locally adjust the noise level in the projections andto drastically reduce image noise as well as anisotropic im-

475 475Med. Phys. 28 „4…, April 2001 0094-2405 Õ2001Õ28„4…Õ475Õ16Õ$18.00 © 2001 Am. Assoc. Phys. Med.

age noise structure. The method originally aimed at reducingmetal artifacts in the hip region,6 where it proved to be usefulfor dose reduction purposes as well. This adaptive, i.e., localfiltering, approach can be used as a ‘‘stand-alone’’ tool or itmay be used in combination with the online current controlwhich might become the most effective means to reduce pa-tient dose.

Since it is widely known that streak artifacts in computedtomography may result from quantum noise statistics inhighly attenuated projections there have been a number ofadaptive filtering approaches in the literature. Some of themoperate in the raw data domain, e.g., Refs. 7–9, others try toimprove the image quality in the spatial domain.10,11Operat-ing in the spatial domain is an image postprocessing tech-nique which cannot make use of the measured attenuationvalues and the photon statistics. Better results are obtainedwhen filtering is performed in the raw data domain. Sug-gested filter types are running mean filters, boxcar filters,trimmed mean filters, median filters, etc. All proposed raw-data-based methods have in common that the filtering is re-stricted to the detector row direction. In other words, filteringis done within a given projection only. Some methods arelocal in the sense that for each complete projection a specificconvolution kernel is chosen,12 others are local in the sensethat for each data value within a projection a local filterfunction, which uses neighboring channels, is used. This in-cludes the use of wavelet decompositions as well as win-dowed Fourier methods.9 The weakness of these approaches,besides the fact that filtering is performed in one dimensiononly, lies in the computational inefficiency: Fourier or wave-let transforms are rather complex procedures that might addsignificantly to the total reconstruction time.

We present a new pragmatic approach: For data pointsthat suffer from photon starvation a local filter~here an up tothree-dimensional triangle function! is applied to average be-tween neighboring data points. To balance between im-proved photon statistics and image resolution we do not onlyuse neighboring data points in detector row direction@one-dimensional~1D! filtering# but also incorporate neighbors inview direction@two-dimensional~2D! filtering# and, if avail-able, in thez-direction ~3D filtering!.6,13,14 There are tworeasons for this multi-dimensional approach.~a! Using threefiltering dimensions improves the photon statistics with thethird power of the applied filter width~instead of linear withthe filter width as in the 1D case!. ~b! CT is a true 3D imag-ing modality where one should aim at uniform resolutioncompromises in all three dimensions instead of only compro-mising the in-plane resolution and, consequently, it is advan-tageous to use smaller filters in the detector row directionwhile incorporating filters in the view and/orz-direction.

Reconstructions of measured as well as simulated 2D and3D data are presented in this paper using the new adaptivefiltering approach. Comparison is always performed relativeto the original reconstruction; difference images highlight themodifications introduced by the new method. Image noise~standard deviation of the pixel values! and spatial resolution@5% value of the modulation transfer function~MTF!# arequantified and, in addition, displayed using so-calleds andr

images, which give the distributions(x,y) of image noiseandr(x,y) of spatial resolution, respectively.

Additional examples show the potential to reduce metalartifacts ~especially in the hip region! and the capability ofprocessing true cone-beam data.

II. MATERIALS AND METHODS

Notations and definitions used throughout this paper aregiven in the following.

* convolution symbol•* n n-fold self-convolution of function•d~•! Dirac’s delta functionII ~•! rectangle function with support@21

2,12# and

height 1L~•! triangle functionL5II* IIII a* (•) rectangle function of widtha and heightuau21,

cf. the AppendixII a,b** (•) convolution of two rectangle functions, cf. the

AppendixII a,b,c*** (•) convolution of three rectangle functions, cf. the

AppendixII a,b,c,d**** (•) convolution of four rectangle functions, cf. the

Appendixb•c floor function, yields greatest integer lower or

equal•d•e ceiling function, yields smallest integer greater

or equal•x∨y maximum ofx andy, x∨y5max(x,y)x∧y minimum of x andy, x∧y5min(x,y)1M(•) indicator function, 1M(x)51 if xPM , other-

wise 0iff if and only ifAF index that stands for ‘‘adaptively filtered’’a projection angle,aP@0,2p) for a 360° scan,

aPR for a spiral scanb bP@2

12F, 1

2F# is the angle within the fan~fanangleF!

b bP@212MS, 1

2MS# is the z-position of the raymeasured at the center of rotation relative to thefocus. For a given slicem we havebm5mS2

12(M21)S

b sampling distance along the detector’sz-coordinate. Hereb[S

~C/W! window setting of the reconstructed images inHU; C is the window center, W is the windowwidth

x,x,Dx pseudovariablex which represents any of thefollowing a, b, b, j, q, ... . The parameterxdenotes the sampling distance of variablex andDx stands for the filter width of the correspond-ing variable.

d table increment per 360° rotationF fan angle, hereF552°f fraction of modified data points~modification

fraction! relative to the total number of datapoints contributing to the reconstruction, 0< f<1

476 Kachelrieß, Watzke, and Kalender: Generalized multi-dimensional adaptive filtering 476

Medical Physics, Vol. 28, No. 4, April 2001

f Dx(x) filter function for variablex ~normalized to unitarea, widthDx!

m, M slice index and number of simultaneously mea-sured slices, 1<m<M

MTF(u) modulation transfer function~frequencyu!

PSF(r) point spread function~radial distancer!p pitch, p5d/MSp(a,b,b) projection data~negative logarithm of relative

intensities! for multi-slice or cone-beam sys-tems

p(a,R,R) short form for the set$p(a,b,b)ubPR,bPR%which is a function of the view anglea

p(a,b,m) projection data for M-slice systems,p(a,b,m)[p(a,b,bm)

RM radius of the field of measurement~FOM!

rc c value of the MTF defined as MTF(rc)5cMTF~0!

r rel relative in-plane resolution defined asr rel

5r5%AF /r5%

Orig<1r rel

z relative z resolution defined as the ratio of thefull widths of half maximum of the original andadaptively filtered slice sensitivity profiles.r rel

z

5FWHMOrig/FWHM AF<1s rel relative noise~standard deviations! defined as

s rel5sAF/sOrig<1S nominal slice thicknessSeff effective slice thickness, i.e., the FWHM of the

reconstructed sliceSSP~z! slice sensitivity profile~positionz!q angle of the ray in parallel coordinatesT,T(a) filtering threshold, depending on viewat rot time for a 360° rotationj jP@2RM ,RM# distance of the ray to the rota-

tional center~parallel coordinates!z-axis axis of rotation

Spiral as well as sequence scans were performed withsubsecond rotation times on the CT scanners SOMATOMPLUS 4 and SOMATOM Volume Zoom~Siemens MedicalSystems, Erlangen, Germany!. The 2D and 3D adaptive fil-tering algorithms were implemented as a data preprocessingstep in the raw data pipeline followed byz-interpolation,rebinning to parallel coordinates, and filtered-backprojectionprocedure.

Cone-beam data were simulated using a virtual scannerwith in-plane geometry equivalent to our real scanners. Tableincrement and slice thickness wered564 mm and S51 mm, respectively. These settings require the number ofdetector rows to beM543. Cone-beam reconstruction wasdone with the ASSR~Advanced Single-Slice Rebinning!

algorithm15 that performs rebinning to parallel data corre-sponding to tilted reconstruction planes followed by a 2Dreconstruction and an interpolation step onto Cartesian coor-dinates. Phantom definitions as well as the method to addartificial quantum noise to the raw data were taken from theworldwide phantom database at http://www.imp.uni-erlangen.de/forbild.

All reconstruction algorithms are implemented on a stan-

dard PC with the dedicated reconstruction and image evalu-ation software ImpactIR~VAMP GmbH, Mohrendorf, Ger-many!; reconstruction time lies below 2 s per image on adual 700 MHz Pentium CPU with 512 Mbytes of memory.

III. ADAPTIVE FILTERS

To generalize the formulation of the filtering equations forvarious scanning geometries~and thus various ray variables!we use the pseudovariablex and formulate our equations asa 1D adaptive filtering.x can stand for any variable describ-ing the ray, e.g., it can denoteb, which is the ray’s anglewithin the fan, or it can stand for the fan geometry projectionanglea.

The following sections describe how the filtering is car-ried out, how the filter widths are set as a function of a given,view dependent thresholdT, how to replace the pseudopa-rameterx by physical variables, and how the threshold func-tion is determined as a function of the underlying object oranatomy.

A. Filtering equation

The adaptive filtering proposed here is a local smoothingwith a filter functionf Dx of characteristic widthDx such thatthis width is a function of the attenuation valuep(x) that iscurrently being smoothed:

Dx5Dx~p~x !!.

For the sake of simplicity we will neglect the discrete natureof the measured data since the sampling distancex will betaken into account in the following, when deriving explicitformulas for the various filter functions. We can now formu-late our approach as the filter integral

pAF~x !5E dx8 f Dx~x2x8!p~x8!uDx5Dx~p~x !!. ~1!

Various filter functionsf are possible but it has turned outthat our method is quite insensitive to the exact shape of thefilter function and thus we limit our description and analysisto a triangular filter function:

f Dx~x !5

1

DxLS x

DxD . ~2!

The functionp(x) is not known for allx but rather atdiscrete positionsx l5x01lx only. Assuming the nearest-neighbor interpolation

p~x !5(l

p~x l!II S x2x l

xD ~3!

continues the discrete data ontoR. Inserting Eqs.~2! and~3!into Eq. ~1! yields

pPF~x !5(l

p~x l!xIIDx,Dx,x*** ~x2x l!,

where we have used the identities II(x/x)5xII x* (x),

L(x/Dx)5DxIIDx,Dx** (x), and IIa,b** * II c* 5II a,b,c*** , which aregiven in the Appendix by recursive definition. A useful ex-



plicit expression of IIa,b,c*** can be found in the Appendix. Inthe limit Dx→0 the adaptive filtering will reduce to thenearest-neighbor interpolation~3!, which means that no fil-tering is done at all.

Since the adaptively filtered raw data are resampled againat the gridx l to be fed into the image reconstruction softwarethe kind of interpolation used in Eq.~3! does not play asignificant role. Nevertheless it may sometimes be useful toresample the adaptively filtered projection at a different grid.For example, when incorporating the adaptive filters into therebinning step from fan-beam geometry to parallel-beam ge-ometry it is necessary to sample the equidistant grid (b i ,a j)at nonequidistant steps to gain the parallel coordinates atequidistant (j i ,q j). Especially in this rebinning procedure alinear interpolation is used frequently. For completeness, wealso give the corresponding filtering equations assuming thelinear interpolation

p~x !5(l

p~x l!LS x2x l

xD

is used to continue the discrete data. The final adaptive filterequation then is

pPF~x !5(l

p~x l!xIIDx,Dx,xx**** ~x2x l!.

The corresponding explicit expression of IIa,b,c,d**** is given inthe Appendix to allow for an easy implementation.

B. Width settings

Since adaptive filtering aims at reducing the image noiseand since image noise is dominated by highly attenuatedprojection data the filter width setting will be performed us-ing a thresholdT in the following binary manner:

Dx5H Dxmax if p~x !.T

0 otherwise.

We have also investigated the possibility to varyDx continu-ously from zero to the maximum but no improvements ascompared to the above-proposed binary setting have beenfound.

If the projection value is too low, the filter width is set tozero ~no filtering and no modification of the data!. If theattenuation exceeds the threshold the filter width is set to amaximal value. Typical values lie in the rangex<Dxmax

<5x, wherex again represents the sampling distance alongthe coordinatex. The lower limit ensures that the adaptivefiltering has a visible impact on the image. It turns out thatexceeding the upper range introduces unacceptable artifactsand blurring in the images.

C. Switching to physical variables

In the following, two examples of how the adaptive filter-ing is carried out with realistic variables are given. In prin-ciple, this corresponds to substituting the pseudovariablexwith the physical beam coordinates of the respective scangeometry.

2D adaptive: Sequence raw datap(q,j) in the paralleldomain~q is the ray’s angle with respect to the coordinateaxes,j is the ray’s distance to the center of rotation!:

pAF~q,j !5E dq8 dj8 f Dq~q2q8!

3 f Dj~j2j8!p~q8,j8!uDq5Dq~p~q,j !!Dj5Dj~p~q,j !!

.

3D adaptive: Multi-slice or cone-beam raw datap(a,b,b) with cylindrical detector geometry~a is the viewangle,b is the ray’s angle within the fan,b parametrizes thedetector inz direction!:

pAF~a,b,b !5E da8 db8 db8 f Da~a2a8!

3 f Db~b2b8! f Db~b2b8!

3p~a8,b8,b8!u Da5Da~p~a,b,b !!Db5Db~p~a,b,b !!Db5Db~p~~a,b,b !!

.

Other generalizations, such as adaptive filtering of fan-beam sequence raw data or such as incorporating the adap-tive filtering into thez-interpolation procedure are straight-forward and are omitted for convenience.

Without loss of generality, we assume in the following tohave spiral cone-beam datap(a,b,b) available, which cor-responds to the typical situation nowadays~spiral multi-slicedata! and probably in the future~spiral cone-beam data!.

D. Automatic threshold determination

The threshold setting should preferably be performed au-tomatically. Since the attenuation properties of a completevolume change significantly as a function of the view anglea @and thus of thez position, sincez5z(a)# the thresholdvalue will be a function of the current view:T5T(a).

The user, however, should not be forced to select morethan one pseudoparameter, which we will call thefilterstrength. Let us denote this value bys and let us furtherdemand 0<s<1 wheres50 stands for no filtering ands51 means ‘‘full’’ filtering.

According to our experience the image quality remainsgood as long as not more than a small fraction 0, f max!1,e.g., f max53%, of the raw data are modified within690° ofthe current view. The resulting image quality will be sensi-tive to the choice off max, values above 5% should beavoided to prevent blurred image regions and new imageartifacts.

The adaptive filtering works well as long as the object’sattenuation properties vary significantly during a rotation.This ensures that only a few hot spots corresponding to thedirection of the highest attenuation will be modified by thealgorithm. Since the attenuation properties are strongly re-lated to the object’s geometrical properties we will alsospeak of the object’s eccentricity while being aware that thisrefers to the attenuation rather than the physical diameter ofthe object.



We want to avoid influencing views from many projectionangles in the case of noneccentric objects in order to avoidimage blurring. Consequently, the thresholding must alsotake into account the local eccentricitye, a value which shallbe 0 for circular objects and approach 1 for highly eccentricobjects. A suitable eccentricity measure will be given in thefollowing.

To be applicable during data acquisition, i.e., for real-timeimage reconstruction, the adaptive filtering and the thresholddetermination shall be computationally efficient and shoulduse local~in z-direction! information only. Thereby we un-derstand information within690° of the current view be-cause a data range of 180° contains all local informationabout the object of interest. Since the amount of photon star-vation within each projection is well characterized by themaximum attenuation value within each projection, i.e., bysupp(a,R,R) it suffices to regard this one-dimensionalfunction instead of the complete raw data set to determineT(a). To remove statistical fluctuations within the projec-tion maximum function we compute a running mean thereof.The angular range covered by the running mean must besignificantly smaller than 90° to preserve the eccentricityproperties of that function. We chose a value of 18° by ex-perience; the results, however, are insensitive to this specificchoice.

Therefore we will base the threshold computation uponthe smoothed projection maximum function, defined as fol-lows:

p~a !ªPp/10* ~a !* supp~a,R,R!.

Since the complete automated threshold determination willbe based on this function we will make no more use of singleprojection valuesp(a,b,b).

Two important functions to define are the local minimumand maximum ofp(a):

p2~a !5 inf p~a112@2p,p# !,

p1~a !5supp~a112@2p,p# !.

Here,local means in the range of690° of the current view.We can now define an appropriate eccentricity measure.

There are several possibilities. Either one regards the localstandard deviation ofp(a) normalized by the local meanp(a)* IIp* (a) or one uses the eccentricitye(a)P@0,1) de-fined as

e~a !ª12

p2~a !

p1~a !. ~4!

The latter approach has the advantage that the eccentricityvalues are normalized automatically whereas defining the ec-centricity using the local standard deviation would require aglobal normalization of the resulting values. This, however,would require reading the complete data set before adaptivefiltering can start, which would not allow for adaptive filter-ing during data acquisition. The quality of the adaptivelyfiltered images, however, is not influenced by this specificchoice of the eccentricity function.

Adaptive filtering should be avoided fore(a)'0 ~circu-lar object! and increased as the normalized local eccentricityapproaches 1. Typical eccentricity values are independent ofthe patient size but they depend on the anatomic level. Someare given in Table I.

Since values close to 0 or close to 1 do not occur fortypical objects usinge as defined previously to determine theamount of filtering would not allow to become specificenough for certain body regions since the lower and upperend of the interval@0, 1# would remain unused. Thus wedefine the truncated eccentricity which maps the eccentricityinterval @e0 ,e1# to @0, 1#:

e trunc~a !ª0~e~a !2e0

e12e0`1.

The valuese050.3 ande150.5 have turned out to be a goodchoice; thee window @0.3, 0.5# is mapped to@0, 1# in thatcase. This avoids filtering in the head and neck regionwhereas the effect of multi-dimensional adaptive filtering~MAF! in the other body regions is amplified.

The degree of filtering shall depend on the truncated localeccentricity as well as on the chosen filter strengths. We thusdefine a local modification fraction as

f ~a !ª f maxe trunc~a !s ~5!

and we determine the thresholdT(a) from

f ~a !5

1

pE

a2p/2

a1p/2

da8 1$p~a8!.T~a !%~a8!.

Since this expression is not an explicit one forT(a) theinversion is performed numerically using a binary searchtechnique to iteratively find the threshold resulting in thedesired modification fraction.

Figure 1 gives an expression of what the thresholding willlook like for a typical patient. The multiplanar reformation~MPR! is given for orientation purposes only. The plots, cor-rectly mapped to the MPR, depict the area between functionsp6(a) and a typical setting for the thresholdT(a). More-over, the local eccentricitye~a! is given as well as the trun-cation thereof. Since the modification fractionf (a) isasimple rescaling of the eccentricity function according to Eq.~5! it is not plotted explicitly.

TABLE I. Typical eccentricity values for some anatomic intervalsaPI,R. The numbers given are a rough estimate for orientation purposes.

I e(I)

Head @0.1, 0.3#Jaw @0.3, 0.6#Neck @0.1, 0.3#Shoulder @0.4, 0.7#Thorax @0.2, 0.5#Pelvis @0.3, 0.5#Femur @0.3, 0.5#



IV. RESULTS

Due to the huge parameter space available, the depen-dency of the adaptive filtering on the measured data, andconsidering the limited space in this article it is impossible topresent comprehensive results for all possible situations. Wewill therefore present results by means of an example for onepatient who has been scanned in the neck region. We willthoroughly analyze this patient, especially one reconstructedslice, and present the corresponding MTF and noise mea-surements. In addition to these quantitative results we willshow s andr images to get a qualitative impression of theinfluence of the adaptive filters upon image quality.

Nevertheless, with our specific choice we have takencare — and verified — that the data being presented here arerepresentative. The efficiency of the MAF approach usingthe specific parameters suggested here has been demon-strated in a clinical study for the shoulder region.16 Studiesof the thorax and pelvis region are under current investiga-tion. The underlying assumption, however, is that the origi-nal data are noisy and that removing the noise will improvethe image impression. For those scans which have been per-formed using a higher dose than necessary the adaptive fil-ters cannot significantly improve the image quality.

Examples of true cone-beam data as well as patient dataof a metallic implant are given in the last sections as well.

A. Images, difference images, and noise images

The image quality and the influence of the adaptive filtersas a function of the filter strength can be judged best byregarding the images themselves and the difference imagesof the adaptively filtered data minus the original image.

Therefore we have prepared Fig. 2, which shows recon-structed images (10243384 0.5 mm pixels! from a four-slicespiral scan withS51 mm and d54 mm ~z-interpolation:180° MFI with Seff51.25 mm!. The slice presented is repre-sentative for the shoulder region; the filter strengths varyfrom s50% ~original!, 25%, 50% to 100% using the filter

width settingDamax52a, Dbmax52b, andDbmax52b. Fromthose images it can be seen that the image noise and thenoise structure have been reduced significantly already fors525%. Although it appears that the 50% and 100% imagesintroduce new artifacts a thorough comparison to the originalimages shows that this is not the case. These apparently newartifacts may have been overlaid by noise and noise structurein the original image and thus have probably been hidden.Removing the noise makes those structures more apparent.

The difference images given in the middle of~a!–~d! inFig. 2 show that the adaptive filters have removed only noisebut not any structure from the image. And it can be seen thatthe adaptive filters only influence those parts of the imagewhich have been affected strongly by noise: The upper andlower regions of the difference images are zero since rays ofhighest attenuation do not contribute to these image parts.

The bottom images of Fig. 2 represent special noise im-ages. Such noise images can in principle be obtained byscanning the object twice and by calculating difference im-ages at the samez-position. However, the data used for thisstudy are standard patient data with only one scan availableand, moreover, dose considerations forbid scanning a patienttwice. Above all, a second patient scan would pose matchingerrors due to patient motion between the two scans and thusyield questionable results when regarding difference images.

FIG. 1. ~a! Overview MPR image.~b!The gray shaded area is the area en-closed byp6(a). The black curve is aplot of T(a). It must be noted that ifT(a) and p1(a) coincide no filteringand thus no modifications will be ap-plied. The vertical lines are placed atthe attenuation values~5 logarithm ofrelative intensity! 0, 2,..., 12. ~c! Ec-centricity functions e~a! ~gray! ande trunc(a) ~black!. The vertical lines areplaced at the eccentricity values 0,0.2,..., 1. The patient’s jaw and thedental fillings cannot be seen in theMPR but appear as high local maxi-mum attenuation and as high eccen-tricity values in the two plots in thehead region.~0/500!



Our way to produce noise images is to add artificial quantumnoise to the measured data~assuming the variance of a mea-sured value to be proportional to the expectation value, i.e.,to its intensity! and to subtract two of those reconstructed

images. The results are equivalent to measuring twice exceptfor the negligible fact that the underlying mean value istaken from a measurement and thus subjected to noise.Within these noise images we can then perform region of

FIG. 2. Adaptively filtered images~top!, difference images~middle!, and noise images~bottom! showing the original data~a! and AF images~b!–~d! with filterstrengths ofs525%, 50%, and 100%, respectively. The difference images are calculated by subtracting the original from the AF images. The noise imagesshow four ROIs~left, center, right, and upper!, which were used to calculate the image noise~standard deviation!. The open dots centered in those ROIs showthe location of the delta objects which were simulated to quantify the spatial resolution. The results are given in Table II.~0/500!



interest~ROI! evaluations to measure image noise. The loca-tions of the four ROIs used for this specific slice are depictedin the noise images as well. The left, center, and right ROIswere chosen to lie directly in the areas which are modified bythe adaptive filters whereas the upper ROI was chosen to belocated in a region which is not.

The results of the noise evaluation~standard deviationmeasurement of a noise image ROI! are given relative to theoriginal image@Fig. 2~a!# in Table II. It can be seen that evenfor the relatively low filter strength of 25% the image noisehas been reduced by approximately 20%, i.e., to 80% of itsoriginal value. There is no significant difference between theleft, the center, or the right ROI regarding the relative noiseand the relative resolution, respectively. The values of theupper ROI are not influenced at all, as has been expected;they remain at 100%. The noise reduction values and thusthe potential for dose reduction of course improve with in-creasing filter strength.

Multiplanar reformations give an even better impressionof the adaptive filtering technique than the axial images. Thisis demonstrated in Fig. 3, which shows a coronal MPR of thesame patient. The filter strength varies from 0% to 100% insteps of 25%. Again, the image noise is drastically reduced.The corresponding difference images show the anatomic lev-els which are modified~particularly the shoulder region! andthose which are left untouched~neck region!. Sagittal MPRsare not given since the adaptive filters mainly reduce noiseoriented in the lateral direction, visible in coronal MPRs.

B. Spatial resolution

As we have seen that the image noise and its structure canbe greatly reduced by this new approach the natural questionto ask is about the influence on image resolution. Typically,a reduction of noise by image smoothing goes hand in handwith a significant loss in resolution. Since the adaptive filtersonly modify a small fractionf within a 180° data interval itcan be supposed that the image resolution isnot reducedsignificantly.

To give quantitative results we have conducted the fol-lowing experiment. We have simulated four raw data setscorresponding to delta objects located at the center of theROIs given in Fig. 2. Each of these delta phantoms under-went adaptive filtering while the filter operation~filter

strength, threshold setting, etc.! was driven by the originalpatient raw data. Thus, the same operation that had beenapplied to the patient data was performed on the delta phan-toms. For illustration purposes we have superimposed thetraces of those four delta peaks with the sinograms corre-sponding to the slices shown in Fig. 2. These sinograms,given in Fig. 4, also show the hot spots of the adaptive filters,i.e., those points that have been modified since they exceedT(a). Of course, these hot spots become larger as the filterstrength increases. The only delta peak trajectory which isnot influenced by the AF operation is the one correspondingto the upper ROI~cf. Fig. 2!. The size of the hot spots isgiven as the relative modification fractionf in Table II.

After these virtual data have undergone adaptive filteringwe have measured the PSF and SSP and performed a MTFcalculation — this is the Hankel transform of the in-planePSF — to qualify and to quantify the changes. Figure 5shows the result for the left and central ROIs for the typical

setting s550% and Damax52a, Dbmax52b, and Dbmax

52b. The dotted graphs show the original PSF as a functionof the distancer to the center of the delta object and the MTFas a function of spatial frequencyu, respectively, the solidlines correspond to the AF data. The units are arbitrary, sincecomparisons are meant to be done relative to the originalfunction; absolute resolution values shall not play a rolehere. There are two important points to be concluded fromFig. 5. First, the point spread function hardly changes ascompared to the original. In particular, the FWHM remainsalmost constant. Second, the first zero of the MTF — whichis a measure of maximally achievable resolution — remainsunchanged as well. The MTF itself is slightly reduced by theadaptive filters but, since this does not apply to the first zerovalue, this reduction can be compensated by choosing differ-ent convolution kernels.

To describe the influence of the parameterss, Damax,Dbmax, andDbmax we have prepared Fig. 6 where each ofthe parameters is varied while keeping the others constant.Results for the left and for the central ROI are given. Thedefault values have been chosen quite extremely ass

5100%, Damax52a, Dbmax52b, andDbmax52b52S. Thisextreme setting helps to emphasize the differences intro-duced by the various settings. The first case of increasingfilter strength ~upper row! obviously decreases the MTF.

TABLE II. Relative noise, relative resolution, and modification fractionf for various filter strengths usingDamax52a, Dbmax52b, and Dbmax52b. The values correspond to the left, center, and right ROIs as depictedin Fig. 2. Values for the upper ROI are omitted since they are 100% always. The data correspond to the slicedepicted in Fig. 2. Low values for the relative noise and values close to 100% for the relative resolution aredesirable.

s

s rel in % r rel in % r relz in %

fLeft Center Right Left Center Right Left Center Right

0% 100 100 100 100 100 100 100 100 100 0.0%25% 74 73 76 98 98 99 99 99 99 1.0%50% 57 54 51 97 97 97 99 98 97 1.7%75% 40 38 40 96 95 96 98 97 96 2.6%

100% 33 33 34 95 94 96 98 96 95 3.4%



Varying the filter width~second row! in view direction doesnot change the MTF in the center but for the off-center ROI.Theb filtering reduces the MTF for both the left and centralROIs ~third row!. The first zero of the MTF, defining theresolution limit, remains unchanged for all cases. We haveomitted the cases for varyingDbmax since thez-filtering does

not influence the in-plane resolution at all. We do not showthe MTFs for the right ROI which are similar to those of theleft ROI and the results for the upper ROI which lies in aregion which is not influenced by the adaptive filters. Forcompleteness, the bottom row of Fig. 6 shows images of thePSF for the original unfiltered case and for the adaptively

FIG. 3. Coronal MPRs including difference MPRs for varying filter strengths.~0/500!



filtered case~using the default values!. The artifacts in theclose vicinity of the delta object are of the order of 1% of thedelta peak maximum for the unfiltered original image and ofthe order of 5%–10% for the adaptively filtered image~AF!.

Similar to Fig. 6 which shows the in-plane resolution interms of MTF for the center and left ROI we have preparedFig. 7, which shows the resolution in thez-direction as afunction of varyingDb. The effects of theb-filtering on theSSP are very similar to the effects ofa- and b-filtering onthe in-plane PSF~cf. Fig. 5!; a slight broadening of the SSPin the lower regions is observed.

Further on, we have quantified the image noise and thespatial resolution as a function of varying filter strength~Table II! and of varying filter widths~Table III!. Using allthree dimensions for adaptive filtering adds to the potentialof noise reduction: As the last line of Table III shows, thelowest noise values can only be achieved when filtering isperformed in all three dimensions. From line 5 of Table III itcan be seen that filtering in view direction does not reducethe noise in the image center but only for the off-centerROIs.

C. Visualization of image noise and spatial resolution

The image noise values, the noise reduction, and the mea-surements of spatial resolution have, up to now, only beenpresented for four distinct locations~left, center, right, andupper ROIs!. Although we had ensured the values given tobe representative we additionally show the spatial distribu-tion of image noise and of image resolution usings and r

images to give a qualitative impression of the overall behav-ior.

The distributions(x,y) of image noise has been calcu-lated using the laws of error propagation which, in principle,corresponds to a filtered backprojection of the variances ofthe measured data using a squared reconstruction kernel. Thedistributionr(x,y) of spatial resolution has been calculatedby performing simulations of delta objects for each position~x, y! and by computing ther5% value of the MTF. Thus,here we use the same method to quantify spatial resolution aswe used in previous sections.

The results, corresponding to Fig. 2, are given in Fig.8 — the original and the adaptively filtered cases and the

FIG. 4. Sinograms showing the hotspots of the adaptive filters~homoge-neous light gray areas! and the tracesof the four delta objects correspondingto the left, center, right, and upper lo-cations ~see Fig. 2!. As the filterstrength increases, the hot spots be-come larger. The trajectory of the up-per delta object does not cross the hotspots. Consequently, no loss in resolu-tion is expected there.

FIG. 5. Point spread function andmodulation transfer function of theoriginal, unfiltered delta phantom~dot-ted! and for the adaptively filtereddelta object using the typical settings

s550% andDamax52a, Dbmax52b,

and Dbmax52b corresponding to~a!the left and~b! the center ROI.



corresponding quotient imagess rel(x,y) and r rel(x,y). Re-garding the image noise@Fig. 8~a!# it becomes clear that thedistribution of the image noise has become more homoge-neous after adaptive filtering. The quotient image showsthose regions where image noise has not been changed~white! and those with a significant noise reduction~gray!;the window settings were chosen such that ranges froms rel

50 are black~noise completely removed!, s rel51 white~noise at its original value!, and the minimum values occur-ring arounds rel'0.3 are displayed as gray.

Spatial resolution@Fig. 8~b!# is constant for the originalimage ~top! and slightly reduced for some regions of theadaptively filtered image~middle!. The quotient image quan-tifies the loss of resolution. Please note the window settings:

FIG. 6. In-plane MTF and PSF images for~a! the left and~b! the central ROI of Fig. 2. The units are arbitrary since comparisons should be performed relative

to the original, unfiltered MTF and not to absolute values. The default values have been set tos5100% andDamax/a5Dbmax/b5Dbmax/b52 to show theinfluence of the adaptive filters in a pronounced way. The dotted graphs represent the original data~i.e., s50!. No influence on in-plane resolution is observedwhen varyingDb. MTFs remain unchanged in the upper ROI.

FIG. 7. SSP for the original data~dotted! and for the adaptively filtered case

with the settingss5100% andDamax52a, Dbmax52b, andDbmax52b.



values range fromr rel50.9 ~black, slight loss of resolution!to r rel51 ~white, no loss in resolution!, the minimum valuein the quotient image isr rel'0.92.

Evidently, thes andr images confirm the results obtainedfrom evaluating the four ROIs used in the previous sectionsquite well. Thus, the above-given quantitative values can betaken as representative. Image noise is greatly reduced inthose regions corresponding to high attenuation. Spatial reso-lution is only slightly impaired.

D. Cone-beam CT

We have also used 3D adaptive filtering with medicalcone-beam data to demonstrate its efficiency and its applica-bility for this case also. Figure 9 shows a simulated thoraxphantom that has been sampled using a virtual 43-slice scan-ner with a table increment ofd564 mm per rotation~pitch1.5!. The algorithm used to reconstruct the adaptively filtereddata is the advanced single-slice rebinning algorithmASSR, apromising approach for medical cone-beam CT.15 Significantimprovements can be seen in the axial images and in thecoronal MPRs. Noise has been reduced by about 38% for acentral 50 cm3 VOI ~volume of interest!; the overall imageimpression has been greatly improved. The sagittal MPR,however, shows no visible improvement although the same

noise reduction values apply there as well. The filter strengthhas been set tos550% to produce these images.

V. DISCUSSION AND CONCLUSIONS

The 3D adaptive filtering allows one to reduce the imagenoise and the image noise structure without significantly im-pairing spatial resolution. Noise can be typically reduced to50% of its original level. The method is applicable to single-slice, multi-slice, and cone-beam CT in conventional or spi-ral mode and can be easily adapted to other scan geometriesas well. MAF and the automatic threshold and filter widthsetting can be implemented as part of the raw data pipeline.Since only local data~within 690°! are used for the auto-matic adjustment of the parameters the adaptive filtering canbe applied during the data acquisition to allow for real-timeimage reconstruction. The operations required are computa-tionally inexpensive and will therefore not increase the im-age reconstruction time significantly.

Although the automatic threshold setting requires a num-ber of experimentally determined constants we have demon-strated the usefulness of the parameters suggested here. Themost critical choice is the selection of the truncated eccen-tricity window which directly influences the number of ana-tomic regions that can be improved by MAF. The setting for

TABLE III. Relative noise and relative resolution for varying filter widths at 50% filter strength (f 51.7%). The values correspond to the left, center, and rightROIs as depicted in Fig. 2. Values for the upper ROI are omitted since they are 100% always.

Damax

a

Dbmax

b

Dbmax

b

s rel in % r rel in % r relz in %

Left Center Right Left Center Right Left Center Right

0 0 0 100 100 100 100 100 100 100 100 1000 0 1 80 76 79 100 100 100 99 98 970 1 0 75 67 75 97 97 97 100 100 1000 1 1 65 64 68 97 97 97 99 98 971 0 0 71 99 81 97 99 97 100 100 1001 0 1 66 79 64 97 99 97 99 98 971 1 0 63 67 69 96 97 96 100 100 1001 1 1 59 62 60 96 97 96 99 98 97

FIG. 8. Spatial distribution of imagenoise~a! and of image resolution~b!.The upper images correspond to theoriginal, the middle images are adap-tively filtered with s5100%, and thelower images are the correspondingquotient, i.e.,s rel(x,y) and r rel(x,y).The window settings for the originalimages are arbitrary, those for theadaptively filtered images are the sameas for the corresponding original im-age. The window settings for the quo-tient images are given in the images.



the maximum modification fraction is critical as well but canbe easily determined once for a given imaging system. Avalue of a few percent seems to be plausible: The impact onresolution remains negligible whereas the image noise is sig-nificantly reduced. Other steps involved in the computationof T(a) such as the definition of the eccentricity or the defi-nition of the projection maximum are rather uncritical andmay be modified to better meet specific~performance orhardware! requirements. Above all, MAF shall not be char-acterized by the automatic threshold determination presentedin this paper. It is rather characterized by the multi-dimensional filtering procedure and may be likewise efficientusing other more or less manual methods to determine thefraction of data that suffers from photon starvation.

Adaptive filtering will only improve the image quality incase of a photon starvation situation as it frequently occurs inthe shoulder, pelvis, or hip region. Especially highly eccen-tric cross sections can be drastically improved, as long as theoriginal data contain enough image noise and noise structure.

The adaptive filters have the potential to improve imagequality in case of metal artifacts assuming image noise to bethe predominant cause. The common case of titanium hipprostheses can be improved significantly; a fact which is ofhigh importance for diagnostic purposes and for revisionplanning. The case shown in Fig. 10 demonstrates this quiteimpressively. The left, uncorrected image shows streakingartifacts which are greatly reduced in the right image. Auto-

matic contour finders, as frequently used in the hip region tomeasure bone densities, would require more user interactionthan in the right, corrected image.

No advantages, however, have been found by trying toreduce metal artifacts from dental fillings using adaptivefilters.6 In those cases, noise is not the dominant problem.We presently further evaluate the dedicated application ofMAF for metal artifact reduction in general and for bonedensity measurements in the vicinity of implants.17

The MAF approach has great potential for both dose andnoise reduction. The new method may be used in futuremedical CT applications as an easy and efficient standardtool to improve image quality by means of data preprocess-

FIG. 9. The cone-beam reconstruction by theASSR algorithm using simulated data of a thorax phantom shows significant improvements in image noise andnoise structure when applying the 3D adaptive filtering technique~middle row!. Noise, measured as the standard deviation of a spherical 50 cm3 VOI centeredabout the cross hairs, was reduced from about 210 HU from the original volume~upper row! to 130 HU, i.e., to 62% of its original level. The lower row showsthe corresponding difference images. Collimation: 4331 mm, d564 mm. ~0/500!

FIG. 10. Axial scan of a titanium hip endoprosthesis at 140 kV,S52 mm.Left: original, right: 2D adaptively filtered.~500/2000!



ing. The advantages are both the improved image quality andthe alternatively reduced patient dose. In addition, MAF al-lows one to increase the total scan length by reducing thetube current prior to the scan and it can be used to reduce thedemands on x-ray tube power ratings.

ACKNOWLEDGMENTS

This work was supported by ‘‘FORBILD’’ Grant No. AZ286/98 ‘‘Bayerische Forschungsstiftung, Mu¨nchen, Ger-many.’’ We thank Dr. Ulrich Baum, Institute of DiagnosticRadiology, University of Erlangen, for a very efficient andpleasant cooperation.

APPENDIX: RECTANGLE FUNCTIONS

Here we will state some helpful expressions concerningthe convolution of rectangle functions. Starting from

II a* ~x !51

uauII S x

a D ,

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

II a,b** 5II a* * II b* ,

II a,b,c*** 5II a* * II b* * II c* ,

II a,b,c,d**** 5II a* * II b* * II c* II d* .

The n-fold convolution is recursively defined as

II a1 ,...,an* n

5II a1 ,...,an21* ~n21!

* II an* .

The functions IIa1 ,...,an* n are invariant under permutation as

well as under a change of sign of the parametersa1 ,...,an .For scale transformations we have

II a1 ,...,an* n

5S •

aD5uauIIaa1 ,...,aan

* n~• !.

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

following, but rather the functions ofdoubled widthII2a1 ,...,2an

* n . Moreover, we assume the parameters to be

sorted to bedescending, i.e., we assume 2a1>¯>2an>0.

1. Convolution of two rectangle functions

2a>2b>0:

II2a,2b* ~x !51

4ab H 0 if a1b,uxu

a1b2uxu if a2b,uxu<a1b

2b if uxu<a2b.

2. Convolution of three rectangle functions

2a>2b>2c>0:

II2a,2b,2c*** ~x !

51

8abc

350 if a1b1c,uxu12~a1b1c2uxu!2 if a1b2c,uxu<a1b1c

2c~a1b2uxu! if a2b1c,uxu<a1b2c

4bc212~a2b2c2uxu!2

if ua2b2cu,uxu<a2b1c

4bc2~a2b2c !22x2 if uxu<2~a2b2c !

4bc if uxu<a2b2c.

3. Convolution of four rectangle functions

Due to the lack of space the explicit expression forII2a,2b,2c,2d**** has been placed in Table IV.

4. Integration of convolved rectangle functions

Integrating the given rectangle functions can easily bedone, since for arbitrary integrable functionsf we have

Ea21/2b

a11/2b

dx f ~x !5E2`

`

dx sgn~b !II S a2x

b D f ~x !

5sgn~b !II S a

b D * f ~a !

5bII b* ~a !* f ~a !.

Obviously, we can formulate the integral functionFk of anarbitrary function f as a convolution off and a rectanglefunction

Fk~x !5Ek

x

dt f ~ t !5~x2k !II x2k* ~ t !* f ~ t !u t5~x1k !/2 . ~A1!

For symmetric functionsf with f (k1x)5 f (k2x);x wehave

Fk~x !5Ek

x

dt f ~ t !51

2 Ek2~x2k !

k1~x2k !

dt f ~ t !

5~x2k !II2~x2k !* ~ t !* f ~ t !u t5k . ~A2!

Using the symmetry of the rectangle functions and their con-volutions (k50) and Eqs.~A1! and ~A2! we can find twoexplicit expressions of their integrals:



E dx II a* ~x !5xII a,x** S 1

2x D5xII a,2x** ~0!,

E dx II a,b** ~x !5xII a,b,x*** S 1

2x D5xII a,b,2x*** ~0!, ~A3!

E dx II a,b,c*** ~x !5xII a,b,c,x**** S 1

2x D5xII a,b,c,2x**** ~0!.

5. Limits

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

lima→0

II a* 5d,

limak→0

II a1 ,...,an* n

5II a1 ,...,ak21 ,ak11 ,...,an* ~n21! ,

limak→6`

akII a1 ,...,an* n

561

limn→`

II aAn

,...,aAn

* n~x !5

1

A2pse21/2~x/s !2

with

s25

112a

2.

6. Second moment

The second moments, which characterizes the width ofthe function, is given by

s25E dx x 2II a1 ,...,an

* n~x !5

1

12 (k51

n

ak2.

a!Author to whom correspondence should be addressed; electronic mail:[email protected]. A. Kalender,Computed Tomography ~Wiley, New York, 2000!.2W. A. Kalender, B. Schmidt, M. Zankl, and M. Schmidt, ‘‘A PC programfor estimating organ dose and effective dose values in computed tomog-raphy,’’ Eur. J. Radiol.9, 555–562~1999!.

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TABLE IV. Explicit expression of II2a,2b,2c,3d**** for 2a>2b>2c>2d>0.

II2a,2b,2c,2d**** ~x !5

1

16abcd ¦0 if a1b1c1d,uxu

16 ~a1b1c1d2uxu!3 if a1b1c2d,uxu<a1b1c1d

d~a1b1c2uxu!21

13 d3 if a1b2c1d,uxu<a1b1c2d

4~a1b2uxu!cd216 ~a1b2c2d2uxu!3 if a1ub2c2du,uxu<a1b2c1d

4bcd2~a2uxu!~ 13 ~a2uxu!2

1~b2c1d !224bd! if b1ua2c2du,uxu<a2~b2c2d !

4bc~a1d2uxu!216 ~a1b1c2d2uxu!3

2~a2b2c !2~d2uxu!213 ~d2uxu!3 if a1b2c2d,uxu<2a1b1c1d

4cd~a1b2uxu! if a2b1c1d,uxu<a1b2c2d

4cd~a1b2uxu!216 ~a2b1c1d2uxu!3 if c1ua2b2du,uxu<a2ub2c2du

4cd~a1b2uxu!213 ~c1d2uxu!3

2~a2b !2~c1d2uxu! if a2b1c2d,uxu<b2ua2c2du

8bcd2d~a2b2c2uxu!22

13 d3 if ua2b2cu1d,uxu<a2b1c2d

8bcd116 ~a2b2c2d2uxu!3

2~a2b2c !2~d2uxu!213 ~d2uxu!3 if ua2b2c1du,uxu<c2ua2b2du

8bcd116 ~a2b2c2d2uxu!3 if ua2b2c2du,uxu<a2b2c1d

8bcd113 ~a2b2c2d !3

1~a2b2c2d !x2 if uxu<2ua2b2cu1d

8bcd22d~a2b2c !22

23 d3

22dx2 if uxu<2~a2b2c !2d

8bcd if uxu<a2b2c2d



12K. L. Lauro, D. J. Heuscher, and H. Kesavan, ‘‘Bandwidth filtering of CTscans of the spine,’’ Radiology177, 307 ~1990!.

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generalized multi-dimensional adaptive ﬁltering for ... · patient data of spiral and sequential...

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