By Aninash C. Kak;Malcolm Slaney
ISBN-10: 089871494X
ISBN-13: 9780898714944
For somebody with a bit history and many selection, this e-book presents an exceptional easy grounding within the problems with tomographic reconstruction and the fundamental mathematical instruments concerned. dialogue begins slowly, with a bankruptcy that establishes the vocabulary and notation of the sign processing concerned. the following 3 chapters speak about non-diffracting circumstances, the place the radiation that senses the physique constructions isn't substantially deflected via them, as is the case for CAT, puppy, and SPECT. This comprises dialogue of the sensors, illuminators, and their geometries, on as much as helical scans and intricate sensor geometries. it's also confounding results, just like the wavelength established nonlinearities in absorption of X-rays and the way they impact beam transmission and the ultimate snapshot produced.
This bankruptcy comprises in basic terms short menton of MRI, due to the very diverse physics in the back of it, and of ultrasonography, as a result diffractive and refractive good points of the radiator and tissues being tested. Likewise, little point out is made from the explanations for various modalities or recommendations for merging their effects.
The ultimate chapters handle the precise difficulties of ultrasound, digging as a long way in because the wave equations and the typical approximations that make the wave equations no less than slightly sensible as instruments for resolution. those chapters additionally tackle extra complex and computationally exhorbitant algorithms, even though now not in approximately the aspect that back-projection obtained within the past chapters.
This e-book first seemed in 1988, which sounds like centuries in the past within the time scale of tomography set of rules improvement. Even the 2001 replace is getting older, and it by no means particularly went into the Feldkamp algorithms now extensively in use. The dialogue of sonography turns out sketchier than dialogue of the X-ray dependent modalities, and MRI more moderen exotica get little if any recognition. that is effective, even though. it is a gigantic box, and the authors do quite good at defining and addressing the world they meant to hide. The operating set of rules developer will not get a lot from this vintage. the objective viewers this present day is maybe a grad scholar or commercial practitioner who is been thrown in on the deep finish. so long as its limits stay transparent, it is a worthy advent for readers with the mathematics talents and time had to extract its value.
-- wiredweird
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Extra resources for Principles of computerized tomographic imaging
Example text
18 summation is equal to xoy0 but the second term is equal to x-1y1. Although in the original formulation of the finite Fourier transform, the x sequence was only specified for indices from 0 through N - 1, the periodicity property in (48) implies that x-1 be equal to xN-1. This leads to the name circular convolution since the undefined portions of the original sequence are replaced by a circular replication of the original data. The effect of circular convolution is shown in Fig. 9(a). Here we have shown an exponential sequence convolved with an impulse.
If , however, we use the same equation to evaluate F(±u, ± v), we discover that the periodicity properties 2 To be consistent with the notation in the one-dimensional case, we should express the space and frequency domain arrays as f m , n and Fu,v. However, we feel that for the two-dimensional case, the math looks a bit neater with the style chosen here, especially when one starts dealing with negative indices and other extensions. 3 where they represent continuously varying spatial frequencies.
If x and y represent spatial coordinates, (127) can be used to give a physical interpretation to the Fourier transform F(ut u) and to the coordinates u and v. Let us first examine the function The real and imaginary parts of this function are cos 2 (ux + vy) and sin 2 (ux + vy), respectively. In Fig. 18(a), we have shown cos 2 (ux + vy). y-axis goes through v cycles per unit distance. This is the reason why u and v are called the spatial frequencies along the x- and yaxes, respectively. Also, from the figure it can be seen that the spatial period of the pattern is (u2 + u 2 ) -1/2 .
Principles of computerized tomographic imaging by Aninash C. Kak;Malcolm Slaney
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