We turn now to the underlying problem of reconstructing attenuation functions from line-integral data

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We turn now to the underlying problem of reconstructing attenuation functions from line-integral data. 11.4.1 The Radon Transform Our goal is to reconstruct the function f(x, y) ≥ 0 from line-integral data. Let θ be a fixed angle in the interval [0, π). Form the t, s-axis system with the positive t-axis making the angle θ with the positive x-axis, as shown in Figure 11.1. Each point (x, y) in the original coordinate system has coordinates (t, s) in the second system, where the t and s are given by t = x cos θ + y sin θ, and s = −x sin θ + y cos θ. If we have the new coordinates (t, s) of a point, the old coordinates are (x, y) given by x = t cos θ − s sin θ, and y = tsin θ + s cos θ. 90 CHAPTER 11. TRANSMISSION TOMOGRAPHY (CHAPTER 8) We can then write the function f as a function of the variables t and s. For each fixed value of t, we compute the integral Z L f(x, y)ds = Z f(t cos θ − s sin θ, tsin θ + s cos θ)ds along the single line L corresponding to the fixed values of θ and t. We repeat this process for every value of t and then change the angle θ and repeat again. In this way we obtain the integrals of f over every line L in the plane. We denote by rf (θ, t) the integral rf (θ, t) = Z L f(x, y)ds. The function rf (θ, t) is called the Radon transform of f. 11.4.2 The Central Slice Theorem For fixed θ the function rf (θ, t) is a function of the single real variable t; let Rf (θ, ω) be its Fourier transform. Then Rf (θ, ω) = Z rf (θ, t)e iωtdt = Z Z f(t cos θ − s sin θ, tsin θ + s cos θ)e iωtdsdt = Z Z f(x, y)e iω(x cos θ+y sin θ) dxdy = F(ω cos θ, ω sin θ), where F(ω cos θ, ω sin θ) is the two-dimensional Fourier transform of the function f(x, y), evaluated at the point (ω cos θ, ω sin θ); this relationship is called the Central Slice Theorem. For fixed θ, as we change the value of ω, we obtain the values of the function F along the points of the line making the angle θ with the horizontal axis. As θ varies in [0, π), we get all the values of the function F. Once we have F, we can obtain f using the formula for the two-dimensional inverse Fourier transform. We conclude that we are able to determine f from its line integral