A,0,)=J∫以x,y减c0b Thus, the one-dimensional Fourier transform of the projection of the linear attenuation function, ap(y), is equal to the two-dimensional Fourier transform of the original attenuation function evaluated along a line in the frequency domain(in this case the f=0 line) It can readily be demonstrated that if we rotate a function a(x,y) through an angle o in the x,y plane, its transform will be similarly rotated through an angle o [Castleman, 1979]. Thus as we rotate the source and detector around the object, each projected density function detected a(p, o can be Fourier transformed to provide one radial line of the two-dimensional Fourier transform of the desired reconstructed image, A(p, oi), where p is a radial spatial frequency. The set of all A(p, o )for small angular displacements o; form a set of spokes in the transform domain which can be interpolated to estimate A(y), the two-dimensional Fourier transform of the image in rectangular coordinates. The image can then be recovered by inverse transformation of A(ffr), which can readily be carried out digitally using fast Fourier transform algorithms, i. e, a(x,y)=F [A(ofr)I While the Fourier transform approach is mathematically straightforward, many commercial scanners utilize the equivalent but more easily implemented back-projection/deconvolution approach, where each ray is traced back along its propagation axis. When all rays have been back-projected and the result summed, one obtains an approximate(blurred)image of that plane. This image can then be sharpened(deblurred)through the use deblurring function. Refer to Macovski [1983]for the details of this procsse an appropriate two-dimensional of an appropriate filter, which is usually implemented by convolving with Clinically, the impact of computerized tomography was dramatic due to the vastly increased density resolu tion, coupled with the elimination of the superposition of overlying structures, allowing enhanced differenti- ation of tissues with similar x-ray transmittance, such as blood, muscle, and organ parenchyma. CT scans of the head are useful for evaluation of head injury and detection of tumor, stroke, or infection. In the body, CT is also excellent in detecting and characterizing focal lesions, such as tumors and abscesses, and for the evaluation of the skeletal system. [Axel et al., 1983]. In recent years the advent of magnetic resonance systems has provided even greater soft tissue contrast, and thus the role of Ct has been constrained by this at times competing modality Positron Emission Tomography Unlike computerized tomography, which relies on photons produced by an external source, in the modalities of positron emission tomography(PET) and single photon emission computed tomography(SPECT), the source of radiation is a radioisotope that is distributed within the body, and thus these modalities are sometimes referred to as forms of emission computed tomography(ECT). While conventional CT can produce images based upon anatomy of organs, emission Ct techniques can quantitate the distribution of tracer materials that can potentially elucidate physiologic function. The positron or positive electron is a positively charged particle that can be emitted from the nucleus of a radionuclide. The positron travels at most a few millimeters before being annihilated by interaction with a egative electron from the surrounding tissue. The product of this event is the emission of 511-keV gamma ray photons which travel in almost exactly opposite directions. The detectors themselves can be either discrete detectors or a modified Anger camera like those used in conventional nuclear imaging. A coincidence detector is employed to limit recorded outputs to cases in which events are detected simultaneously in both detector arrays, thus reducing the pickup of noise or scattering A possible detection scheme is illustrated in Fig. 116. 1(B). The detector arrays shown can be made energy selective to eliminate lower energy scattered gamma rays. While the distribution of radioactivity can be recon- structed using the reconstruction from projection techniques described in the section on CT Hurculak, 1987 e 2000 by CRC Press LLC© 2000 by CRC Press LLC Thus, the one-dimensional Fourier transform of the projection of the linear attenuation function, ap(y), is equal to the two-dimensional Fourier transform of the original attenuation function evaluated along a line in the frequency domain (in this case the fx = 0 line). It can readily be demonstrated that if we rotate a function a(x,y) through an angle f in the x,y plane, its transform will be similarly rotated through an angle f [Castleman, 1979]. Thus as we rotate the source and detector around the object, each projected density function detected ap(r,fi ) can be Fourier transformed to provide one radial line of the two-dimensional Fourier transform of the desired reconstructed image, A(r,fi ), where r is a radial spatial frequency. The set of all A(r,fi ) for small angular displacements fi form a set of spokes in the transform domain which can be interpolated to estimate A(fx,fy), the two-dimensional Fourier transform of the image in rectangular coordinates. The image can then be recovered by inverse transformation of A(fx,fy), which can readily be carried out digitally using fast Fourier transform algorithms, i.e, a(x,y) = F–1[A(fx,fy)] While the Fourier transform approach is mathematically straightforward, many commercial scanners utilize the equivalent but more easily implemented back-projection/deconvolution approach, where each ray is traced back along its propagation axis. When all rays have been back-projected and the result summed, one obtains an approximate (blurred) image of that plane. This image can then be sharpened (deblurred) through the use of an appropriate filter, which is usually implemented by convolving with an appropriate two-dimensional deblurring function. Refer to Macovski [1983] for the details of this process. Clinically, the impact of computerized tomography was dramatic due to the vastly increased density resolu￾tion, coupled with the elimination of the superposition of overlying structures, allowing enhanced differenti￾ation of tissues with similar x-ray transmittance, such as blood, muscle, and organ parenchyma. CT scans of the head are useful for evaluation of head injury and detection of tumor, stroke, or infection. In the body, CT is also excellent in detecting and characterizing focal lesions, such as tumors and abscesses, and for the evaluation of the skeletal system. [Axel et al., 1983]. In recent years the advent of magnetic resonance systems has provided even greater soft tissue contrast, and thus the role of CT has been constrained by this at times competing modality. Positron Emission Tomography Unlike computerized tomography, which relies on photons produced by an external source, in the modalities of positron emission tomography (PET) and single photon emission computed tomography (SPECT), the source of radiation is a radioisotope that is distributed within the body, and thus these modalities are sometimes referred to as forms of emission computed tomography (ECT). While conventional CT can produce images based upon anatomy of organs, emission CT techniques can quantitate the distribution of tracer materials that can potentially elucidate physiologic function. The positron or positive electron is a positively charged particle that can be emitted from the nucleus of a radionuclide. The positron travels at most a few millimeters before being annihilated by interaction with a negative electron from the surrounding tissue. The product of this event is the emission of 511-keV gamma ray photons which travel in almost exactly opposite directions. The detectors themselves can be either discrete detectors or a modified Anger camera like those used in conventional nuclear imaging. A coincidence detector is employed to limit recorded outputs to cases in which events are detected simultaneously in both detector arrays, thus reducing the pickup of noise or scattering. A possible detection scheme is illustrated in Fig. 116.1(B). The detector arrays shown can be made energy selective to eliminate lower energy scattered gamma rays. While the distribution of radioactivity can be recon￾structed using the reconstruction from projection techniques described in the section on CT [Hurculak, 1987], A f a x y dx e dy p y j x fyy (, ) (, ) –( ) –– 0 2 0 = + • • • • ÚÚ p