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The Radon Transform Joshua Benjamin THE RADON TRANSFORM JOSHUA BENJAMIN III CLASS OF 2020 [email protected] Acknowledgments I would like to thank: Fabian Haiden for allowing me to create an exposition and for oering advice as to what topic I should explore. I would also like to thank Kevin Yang and Reuben Stern for aiding me in checking for errors and oering advice on what information was superuous. ii To My Haters iii THE RADON TRANSFORM JOSHUA BENJAMIN III FABIAN HAIDEN iv Contents 1 Introduction 1 2 e Radon Transform 2 2.1 Two Dimensions (R2)............................2 2.2 Higher Dimensions (Rn)..........................3 2.3 Dual Transform . .4 2.4 Gaussian distribution . .4 3 Properties 5 3.1 Homogeneity and scaling . .5 3.2 Linearity . .5 3.3 Transform of a linear transformation . .6 3.4 Shiing . .6 4 Inversion 7 4.1 Inversion formulas . .7 v 4.2 Unication . .8 5 Relation to the Fourier Transform 9 5.1 Fourier and Radon . .9 5.2 Another Inversion . 10 6 Application 11 vi Chapter 1 Introduction e reconstruction problem has long been of interest to mathematicians, and with the advancements in experimental science, applied elds also needed a solution to this is- sue. e reconstruction problem concerns distributions and proles. Given the pro- jected distribution (or prole), what can be known about the actual distribution? In 1917, Johann Radon of Austria presented a solution to the reconstruction problem (he was not aempting to solve this problem) with the Radon transform and its inversion formula. He developed this solution aer building on the work of Hermann Minkowski and Paul Funk while maintaining private conversation with Gustav Herglotz. His work went largely unnoticed until 1972 when Allan McLeod Cormack and Arkady Vainshtein declared its importance to the eld. Radon was concerned with R2 and R3, but his work was extended to Rn and some more general spaces. e Radon transform for Cn is known as the Penrose transform and it is related to integral geometry (the modern approach to integral geometry was largely inspired by integral transforms, specically the Radon transform). In this paper, the author has decided to use functions of certain classes dened in class. Unless stated otherwise, all f 2 S(Rn) or D(Rn). Respectively, Schwartz functions and functions of compact support. For reference on the generaliza- tions of the transform and applications to integral geometry, see e Radon Transform by Sigurdur Helgason. 1 Chapter 2 e Radon Transform e Radon Transfrom was originally used for determining a symmetric function on S2 from its great circle integrals and a function of the plane R2 from its line integrals. is was also extended by Radon to three dimensions. Radon found that Z ! 1 ( ) = − ( h i) (2.0.1) f x 2 Dx J z, z, x dz 8p S2 Where J( , ) is the integral of f over the hyperplane hx, zi = p and z is a unit vector, and d the laplacian operator. In the following sections are presented various forms of the Radon transform. Each one may be used to suit the purposes of the person computing and the information know regarding a specic problem. 2.1 Two Dimensions (R2) Since the Radon Transform is an operator like the Fourier transform we have two meth- ods of writing. Starting with R2 Z f˜ = R f = f (x, y)ds (2.1.1) L where ds is an increment of length along a line L. Radon proved that is f is continuous and has compact support, then R f is uniquely determined by integrating along all pos- sible lines L. e transform over a line L corresponds to a prole. is is called a sample of the Radon transform. Knowledge of the transform over all lines gives full knowledge of the transform. To add precision, let us consider the normal form a line and then shi the coordinates. With p = x cos a + y sin a we can write Z f˜(p, a) = R f = f (x, y)ds (2.1.2) L 2 Now if we rotate by a and label the new axis s are le with x = p cos a − s sin a y = p sin a + s cos a Z ¥ f˜(p, a) = R f = f (p cos a − s sin a, p sin a + s cos a)ds (2.1.3) −¥ We can also use vector notation which will come in handy in higher dimensions. Let x = (x, y) and take the unit vectors z = (cos a, sin a) z? = (− sin a, cos a) Since we can nd a scalar t such that x = pz + tz?, we now have Z ¥ f˜(p, z) = R f = f (pz + tz?)dt (2.1.4) −¥ Note that the equation of a line can be wrien as p = z · x = x cos a + y sin a is allows the transform to be expressed as an integral over R2 where the Dirac-delta function selects the line p = z · x from R2 ZZ f˜(p, z) = R f = f (x)d(p − z · x)dxdy (2.1.5) R2 Which when dx = dxdy we have Z f˜(p, z) = R f = f (x)d(p − z · x)dx (2.1.6) 2.2 Higher Dimensions (Rn) e Dirac delta function version of the transform is due to the work of Gel’fand, Graev, n and Vilenkin in their 1966 paper. In R we will use x = (x1, x2, ..., xn) and dx = dx1dx2...dxn so we still have Z f˜(p, z) = R f = f (x)d(p − z · x)dx (2.2.1) is shows that we integrate over each hyperplane in the given space. is can be expressed as Z f˜(x) = f (x)dm(x) (2.2.2) x Where dm is the Euclidean measure on the hyperplane x 3 2.3 Dual Transform e dual transform is dened as follows Z R∗ j = j˜(x) = j(x)dm(x) (2.3.1) x2x Where dm is the measure of the compact set fx 2 Pn : x 2 xg. e measure of the entire set is 1 making it a probability measure. 2.4 Gaussian distribution 2 2 Example 2.4.1. Let f (x, y) = e−x −y . e transform is Z ¥ Z ¥ −x2−y2 f˜ = R = e d(p − z1x − z2y)dxdy −¥ −¥ Take z = (z1, z2) and with an orthogonal linear transformation u x = z1 z2 v −z2 z1 y We have Z ¥ Z ¥ 2 2 f˜(p, z) = e−u −v d(p − u)dudv −¥ −¥ Z ¥ 2 = e−p e−v dv −¥ p 2 = pe−p We see that with for every n 2 N dimensions we will integrate n − 1 Gaussian distributions. So for arbitrary n p −x2−x2...−x2 n−1 −p2 Rfe 1 2 n g = ( p) e 4 Chapter 3 Properties Most of the properties of the transform will follow directly from Z R f = f (x)d(p − z · x)dx For more on the properties delineated here, Gel’fand, Graev, and Vilenkin is a reference that provides a discussion. 3.1 Homogeneity and scaling Note that Z f˜(tp, tz) = R f = f (x)d(tp − tz · x)dx Z = jtj−1 f (x)d(p − z · x)dx so long as t 6= 0. So the Radon transform produces an even function of degree −1. We know that it is even because of evaluating the following relation when t = −1. f˜(tp, tz) = jtj−1 f˜(p, z) (3.1.1) 3.2 Linearity e Radon transform is in fact a linear transformation, which can be seen by the fol- lowing calculations. Take f , g to be functions and c1, c2 to be constants. Z Rfc1 f + c2gg = [c1 f (x) + c2g(x)]d(p − z · x)dx = c1 f˜ + c2g˜ Rfc1 f + c2gg = c1R f + c2Rg (3.2.1) 5 3.3 Transform of a linear transformation Dene two vectors x, y and matrices A−1 = B so that x = A−1y = By. Note that, Z Rf f (Ax)g = f (Ax)d(p − z · x)dx Z = j det(B)j f (y)d(p − z · By)dy Z = j det(B)j f (y)d(p − BTz · y)dy Rf f (Ax)g = j det(B)j f˜(p, BTz) (3.3.1) e two special cases are when j det(Bj = 1 and when A = lI (is a multiple of the identity. For each of those cases we have, Rf f (Ax)g = f˜(p, Az) (3.3.2) 1 z Rf f (lx)g = f˜(p, ) (3.3.3) ln l 3.4 Shi ing Shiing is a direct result Z Rf f (x − a)g = f (x − a)d(p − z · x)dx Z = f (x − a)d(p − z · a − z · y)dy Rf f (x − a)g = f˜(p − z · a, z) (3.4.1) 6 Chapter 4 Inversion e inversion formula presented by Radon (given in chapter 1) hides the more general form. e author has decided to give the formulas for the even and odd dimension, and the unication formula as presented by Helgason. For proof of the existence of the formula, consult Deans or Helgason. e inversion formulas for the odd and even dimension Radon transform respectively are, 4.1 Inversion formulas Z (n−1)/2 f (x) = CnDx f˜(z · x, z)dz (4.1.1) jzj=1 Z Z ¥ ¶ ˜ C ¶p f (p, z) f (x) = n dz (4.1.2) ip jzj=1 −¥ p − z · x Where (−1)(n−1)/2 1 1 Cn = = 2(2p)(n−1) 2 (2pi)n−1 Now dene the adjoint operator Z R†y = y(z · x, z)dz jzj=1 Note that on a Hilbert space, the dual is the adjoint.
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