ProcIEEE_Kak_computerized_tomography_with_xray_emission_ultr(12)

e

variance{f(x,y))=

A

( C o J i*

-

n

ve(kT)ik

h,(x cos Bi

+ y sin d j - kr).

(54)

result only applies when compensators (such as wedges) are not used. They reduce the dynamic range of the detector output signal. In noise andysed their effect can be approximately modeled by using different N~’ for afferent rays. S

‘ hs Ti

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KAK: COMPUTERIZED TOMOGRAPHY

1259

and Logan headphantom[ 1071. (b) Reconstruction of thephantomfrom 120 projections and 101 raysineach Fig. 13. (a)TheShepp parallel projection. Display matrix: 64 X 64. (c) The relative-uncertainty image for the reconstruction in (b). (d) A graphical depiction of of the relative-uncertainty values through the horizontal middle line (c).

tional to the area under the square of the filter function used for reconstruction. This does not imply that t h i s area could be made arbitrarily small since any major departure from the variance{f(o,O)}= (60) Iwl function, will introduce spatial distortion in the image even thoughit may beless noisy.Equations(60)and(61) By Parseval's theorem this result may be expressed in the fre- were F i t derived by Shepp and Logan[ 1071. None of the domain quency as construed be above equations should the that to imply varis ance approaches zero as r i made arbitrarily small. Note from Fig. 3 that 7 is also the width of themeasurement beam. In variance{ f(0,O))= -"" IH(w)12 n2 dw (61) any practical system as r is reduced, No will decrease also. MprojNo 4/ 2 7 The preceding discussion has resulted in expressions for the where r is the sampling interval for the projection data. This variance of noise in reconstructions made with a filtered-backresult saysthatthe variance of noise at the origin ispropor-projectionalgorithmfor parallel projectiondata. As menh

The expression (5

8) for the variance may now be written as

'-

I

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1260

PROCEEDINGS OF THE IEEE,

VOL. 67, NO. 9, SEPTEMBER 1979

tioned before in the introduction filtered-backprojection algorithms have become very popular becauseof their accuracy. Still thequestion arises that given a set of projections can there by an algorithm that might reconstruct an image with a smaller error. The answer t o t h i s question has been supplied by some very theoretically elegantwork by Tretiak[ 1171. Tretiak hasderived an algorithmindependentlowerbound for the mean squared error in a reconstructed image and has argued that for the case of reconstructions from parallel projection data this lower bound is very close to the error estimates obtained by Brooks and DiChiro[ 17] for the filteredbackprojection algorithm, which leads t o the conclusion that very little improvement can be obtained over the performance of such analgorithm. The reader is also referred t o[ 61 and[ 11 1] for discussions on noise properties of the X-ray CT images.

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