ProcIEEE_Kak_computerized_tomography_with_xray_emission_ultr(10)

the solid curve in the figure is obtained. Most preprocessing corrections simply specify the selection of an“appropriate” absorber and then experimentally obtain the solid curve in Fig. 12. Thus should a ray integral be measured to be at A, it is simply increased to A’ for tomographic reconstruction. This procedure has the advantage of very rapid implementation and works well forsoft tissue cross sections because differences in the composition of various soft tissues are minimal (they are all approximately water like from the standpoint of X-ray attenuation). For preprocessing corrections the reader is also referred to references[ 151,[ 891,[ 901, and for a technique that combines preprocessing with image deconvolution see[ 341. Preprocessing techniques usually fail when bone is present in a cross section. In such cases it is necessary to postprocess the CT image. Methods for doing so may be subdivided into twotypes:i) iterative[781,[83], and i dual energy techi ) niques of Macovski and his associates[ 1],[ SO]. In the iterative scheme one first does a reconstruction (usually incorporating the linearization correction mentioned above) from .the projectiondata. This reconstruction is then thresholded to get an image that shows only the bone areas. This thresholded image is then“forwardprojected” to determine the contributionmade by bone to each ray integral in each projection. Based on this contribution a correction is applied to each ray integral. Joseph and Spital[78] and Kijewski and Bjangard[ 831 have obtained very impressive results with this technique.

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KAK: COMPUTERIZEDTOMOGRAPHY

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The dual energy technique proposed by Alvarez and Macovski[ 11 is theoreticallythe most elegant approach to eliminating the beam hardening artifacts. Their approach is based on modeling the energy dependence of the linear attenuation coefficient byP ( X, Y, E )= a1 ( x, y ) g(E)+ a2(x, Y ) fm(E).

and I2(-41,A2)=[s2(E)eXP[-(Alg(E)+AZfKN(E))] dE(40)

which gives us two (integral) equations for the twounknowns Thepart u1 (x, y)g(E) describes the contribution made by A 1 and A 2 . The two source spectraS1( E ) and S2 ( E ) may for photoelectricabsorption to the attenuation at point (x, y ); example be obtained by simply changing the tube voltage on a l ( x, y ) incorporatesthe material parameters at ( x, y ) and the X-ray source. This, however, does require that two scans g(E) expresses the (material independent) energy dependence be made for each tomogram. In principle, one can obtain equivalent results from a single scan with split detectors[ 201. of this contribution. The function g(E)is given by Alvarez and Macovski[2] have shown that statistical fluctua1 g(E)= 3. ( 3 4 ) tions in a1 ( x, y ) and a 2 ( x,y ) caused by the measurement erE rors in I and I2 are small com

pared to the differences of these 1 quantities for bodytissues. (See also Brooks[ 181. He has concluded thatg(E)=E-2’8.) ) The second part of ( 3 3 ) given by a 2 ( x, y )~ K N ( Egives the Compton contribution scatter to the attenuation. Again a 2 ( x, y ) depends upon the material properties whereas fKN(E), the Klein-Nishina function, describes the(material independent) energy dependence of this contribution. The function fKN(E) is given byc

(33)

D. Statistical Propertiesof Noise in CT ImagesLet again us consider the hypothetical parallel beam of monoenergetic X-ray photons propagating through tissue as shown in Fig. 3 . If the width 7 of the beam is small enough, the integral of p ( x, y ) along the dotted lineisgivenby (2) and (25): Pe(t)E

1+-ln(1 2a

+ 2a) -

(1+ 3 a ) (35) (1+ 2aI2

ray path A B

J

P ( x, y ) dS= ln Ni, -

ln Ne(k7)(41)

with a= E/ 5 10 975. The energy E is in kiloelectronvolts. The importance of ( 3 3 ) lies in the fact that all the energy dependence has been incorporated in known and material independent functions g ( E ) and fKN(E). Substituting this equation in (26) we get

where Ne(k7) denotes the value of N d for the ray at location shown in the figure. Randomness in the measurement of PO ( t ) is introduced by statistical fluctuations inNe(k7). Note that in practice only Ne(k7) is measured directly. The value of Ni, for all rays is inferred by monitoring the X-ray source with a reference detector and from the knowledge of the spatial distribution of emitted X-rays. It is usually safe to N d= s o ( E ) exP[-(AIg(E)+ A~~ K N ( E ) ) I d E( 3 6 ) assume that the reference X-ray flux islarge enough so that Ni, may be considered to be known with negligible error. In where the rest of the discussion here wewill assume that for each rayintegral measurement Ni, is a known deterministic conA1= al(x,y)ds ( 3 7 ) stant, while on the other hand the directly measured quantity ray path Ne(k7) is a random variable. The randomness of Ne(k7) is statistically described by the Poisson probability function and[601, 11081:

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