A Computerized Tomography(CT) Detector Array Detector Array Source B. Positron Emission Tomography(PET collimator C. Single Photon Emission Computed Tomography FIGURE 116. 1 Comparison of three photon-based tomographic imaging modalities. Fig. 116.1(A)and assuming parallel rays, the intensity picked up by the detector array can be expresse L,()=Io exp[-a(x,y)dx] where a(x,y) represents the linear attenuation coefficient to x-ray photons within the body as a function of x,y position, and Io is the source intensity. Rearranging, we see that a, (y)=a(x, y)dx=In(L (y)/Iol where a(y)is the projected attenuation function Taking a one-dimensional Fourier transform of this projected density function we see that Fa(y)=A(f)=∫∫以xy减e形yb where A, (fy) is the Fourier transform of a single line of detected data. But this can also be written c2000 by CRC Press LLC© 2000 by CRC Press LLC Using the coordinate system of Fig. 116.1(A) and assuming parallel rays, the intensity picked up by the detector array can be expressed as Id(y) = I0 exp[–Úa(x,y)dx] where a(x,y) represents the linear attenuation coefficient to x-ray photons within the body as a function of x,y position, and I0 is the source intensity. Rearranging, we see that where ap(y) is the projected attenuation function. Taking a one-dimensional Fourier transform of this projected density function we see that where Ap(fy) is the Fourier transform of a single line of detected data. But this can also be written FIGURE 116.1 Comparison of three photon-based tomographic imaging modalities. a y a x y dx I y I p d ( ) ( , ) ln[ ( )/ ] – = = • • Ú 0 F a y A f a x y dx e dy p py j fyy [ ( )] ( ) ( , ) – –– = = • • • • ÚÚ 2p