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The purpose of this assignment is to study an application of the Fourier Transform to medical imaging (see Ch.6, Section 5 of Stein-Shakarchi for context). Given t E R and 0 E (-7, 7], consider the line Lto = {(x, y) ER2 : x = tcos0 + ssin0, y = tsin0 - scos0, s ER). Ty (t cos 0, t sin 0) Lt,0 (1) For f E S(R2), define the X-ray transform of f to be the function xf (t, 0 ) := / f = / f(tcos0 + ssino, tsino-scos0) ds, (t, 0) ER x (-7, TT]. Lt , 0 Compute the X-ray Transform of the 2-D Gaussian, G(x, y) = e-T(x2ty?). (2) For a fixed 0 E (-7, 7], let Xf($, 0) be the 1-dimensional Fourier Transform of the function X f(t, 0) in the t variable. Prove the following identity: x f (5,0) = f(5 cos 0, { sin 0) for all & E R and 0 E (-7, T]. Note that the Fourier Transform on the right-hand side is the 2-dimensional Fourier Transform of the function f (x, y). HINT: Note that if 7 := (cos 0, sin @) and 71 := (sin 0, - cos 0), then Et = $7 . (ty + syl) for all , t, s ER. Use this observation to convert the 1-dimensional Fourier Transform to a 2-dimensional one. (3) Using the identity in (2), prove that if f E S(R2) satisfies X f (t, 0) = 0 for all (t, 0) ERx (-7, T], then f is identically zero on R2. This shows the map f - Xf is injective