ProcIEEE_Kak_computerized_tomography_with_xray_emission_ultr(5)

‘Sometimes,because of its speed, the FFT interpolation method is used (see, for example, reference[ 5 7, p. 1901). Thismethodconsistsof takingthe FFT of aprojection,paddingthetransformwith a large number of zeros and taking the IFFT. It has recently been shown[ 82 1 that this method can lead to large errors especially near the beginning and the end of a data sequence.

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1250

PROCEEDINGS OF THE IEEE,

VOL. 67, NO. 9, SEPTEMBER 1979

where Qs(7)is the filtered projection correspondingto Rb(y). we need to introduce two more parameters U and t'. ParamIts digital implementation is given by (exactforthe band- eter U is for each pixel ( x,y ) (or equivalently ( r, 9)) the ratio of (Fig. 2(b))tothesourceto origin distance.Note that limited case and when a satisfies the Nyquist condition): E is theprojection of thesource to pixel distanceonthe Qg(na)= a~; ( m g((n - m) a). a) (1 7) central ray.m

The discrete impulse response g ( n a ) is given by the samples of (1 5: )n=O

-D+rsin(o-$) D

(21)

t' The other parameter is the value of t that correspondsto the iay passing through the pixel (r,$). In Fig. 2(b) t' is equal to the distance OF The convolution in (1 7) possesses a frequency-domain implementation very similar to (9). The comment we made there With all the parameters defied as above, theimage f ( r, 6) and regarding the desirability of incorporatingsomesmoothing its projectionsR g ( t ) can be shown to be related by also applies here. Step 3 (Fan Backprojection): This last step consists ofimplementingthe remaining integration (13). in In terms of filtered projectionsti step may be expressed as hs (23) where the function h ( t ) is the same as in ( 6 ) . As was the case with (13),equation(23) possesses a weighted filtered-backWhen the number of projections Mproj is large and uniformly projection implementation. Except different for weighting distributed over 360°, for digital implementation (19) may be factors and fiter function, 'the steps are almost identical to expressed as those in the preceding case and will not be repeated here. .

111. X-RAYTRANSMISSION COMPUTED TOMOGRAPHYwhere x= r cos$ and y= r sin$. The function?(x, y ) is the reconstructedapproximation tothe image f ( r,$). To find the contribution of the filtered projection Qsi to the reconstruction at a pixel ( x,y ) we must first find L and y' using (1 1) and (1 (This is in sharp contrast to the case of parallel 2). projections wh

ere in order to compute such a contribution we only had to find x cos Bi+ y sin B i . ) Computation of such contributions at all pixels from one Qai is called a fan backprojection. The sum o,f all such fanbackprojectionsmultiplied by 2n/Mproj gives f ( x,y ) .C. Reconsttuction Algorithm forFan-Beam Projections Taken with Equispaced Detectors on a Straight Line

Consider a hypothetical parallel beam of monoenergetic X-rays propagatingthrough tissue as shownin Fig. 3. Let Ni, be thetotalnumber of photonsthatentertheobject (withinthemeasurementtime interval) through the beam from side A; andlet Nd be thetotalnumber of photons exiting (within the same time interval) through the beamon side B. In the plane shown in the figure the tissue everywhere may be characterized by its linear attenuation coefficient. Let p ( x, y ) be the attenuation coefficient at location ( x, y ) in the plane shown. When the width T of the beamis sufficiently small, the numbers N d and Ni, are related by[ 621,[ 1121:=Nin Nd

exP[ - J P ( x, Y ) d~]

(24)

or, equivalently,Nd Jayp(x,y)ds=1n- Nin

A reconstructionalgorithmfor t i casewas f i t derived hs by Lakshminarayanan[ 851 (see also[ 651 ). It can also be more simply derived[ 81 I by using a method similar to that of Scudder[ 1061 for the equiangular case. Fig. 2(b) shows the geometry for data collection. Although theprojections are measuredon a linesuch as DlD2, for theoreticalpurposes it is moreefficient ty assume the existence of an imaginary detector line DlDz passing through the origin. We now associate a ray integral along SB with point F on D; Dh as opposed to point B on Dl Dz. Each projection is now denoted by R&) where the distance t along the imaginary detector line D; Dh identifies the location of a ray in the fan projection at angle fl. Again, let D be the source to center distance; and f ( r,$) the image we seek to reconstruct in polar coordinates. For ease in presentation

(25)

where ds is an element of length and where the integration is carried out along line A B shown in the figure. The left-hand side precisely constitutes a ray integral for a projection. Now an ideal detector placed on side B will produce a signal directly proportional to N d . Therefore, measurementslike In (Ni,/Nd) taken for different raysat different angles may be used to generate projection data for the function p ( x, y ) . We would like to remind the reader that this is strictly true only under the assumptionthatthe X-ray beam consists of monoenergetic photons. This assumption is necessary because the linear attenuation coefficient is in general a function of photon energy.

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