The Radon Transform Joshua Benjamin

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 ∈ 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 ∆x J ζ, ζ, x dζ 8π S2

Where J( , ) is the integral of f over the hyperplane hx, ζi = p and ζ is a unit vector, and δ 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 α + y sin α we can write Z f˜(p, α) = R f = f (x, y)ds (2.1.2) L

2 Now if we rotate by α and label the new axis s are le with x = p cos α − s sin α y = p sin α + s cos α Z ∞ f˜(p, α) = R f = f (p cos α − s sin α, p sin α + s cos α)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 ζ = (cos α, sin α) ζ⊥ = (− sin α, cos α) Since we can nd a scalar t such that x = pζ + tζ⊥, we now have Z ∞ f˜(p, ζ) = R f = f (pζ + tζ⊥)dt (2.1.4) −∞ Note that the equation of a line can be wrien as p = ζ · x = x cos α + y sin α is allows the transform to be expressed as an integral over R2 where the Dirac-delta function selects the line p = ζ · x from R2 ZZ f˜(p, ζ) = R f = f (x)δ(p − ζ · x)dxdy (2.1.5) R2 Which when dx = dxdy we have Z f˜(p, ζ) = R f = f (x)δ(p − ζ · 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, ζ) = R f = f (x)δ(p − ζ · x)dx (2.2.1)

is shows that we integrate over each hyperplane in the given space. is can be expressed as Z f˜(ξ) = f (x)dm(x) (2.2.2) ξ Where dm is the Euclidean measure on the hyperplane ξ

3 2.3 Dual Transform

e dual transform is dened as follows Z R∗ ϕ = ϕ˜(x) = ϕ(ξ)dµ(ξ) (2.3.1) x∈ξ

Where dµ is the measure of the compact set {ξ ∈ Pn : x ∈ ξ}. 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 δ(p − ζ1x − ζ2y)dxdy −∞ −∞

Take ζ = (ζ1, ζ2) and with an orthogonal linear transformation

u x = ζ1 ζ2 v −ζ2 ζ1 y We have Z ∞ Z ∞ 2 2 f˜(p, ζ) = e−u −v δ(p − u)dudv −∞ −∞

Z ∞ 2 = e−p e−v dv −∞ √ 2 = πe−p

We see that with for every n ∈ N dimensions we will integrate n − 1 Gaussian distributions. So for arbitrary n √ −x2−x2...−x2 n−1 −p2 R{e 1 2 n } = ( π) e

4 Chapter 3

Properties

Most of the properties of the transform will follow directly from Z R f = f (x)δ(p − ζ · 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, tζ) = R f = f (x)δ(tp − tζ · x)dx Z = |t|−1 f (x)δ(p − ζ · 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, tζ) = |t|−1 f˜(p, ζ) (3.1.1)

3.2 Linearity

e Radon transform is in fact a linear transformation, which can be seen by the following calculations. Take f , g to be functions and c1, c2 to be constants. Z R{c1 f + c2g} = [c1 f (x) + c2g(x)]δ(p − ζ · x)dx

= c1 f˜ + c2g˜

R{c1 f + c2g} = 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 R{ f (Ax)} = f (Ax)δ(p − ζ · x)dx

Z = | det(B)| f (y)δ(p − ζ · By)dy Z = | det(B)| f (y)δ(p − BTζ · y)dy

R{ f (Ax)} = | det(B)| f˜(p, BTζ) (3.3.1) e two special cases are when | det(B| = 1 and when A = λI (is a multiple of the identity. For each of those cases we have,

R{ f (Ax)} = f˜(p, Aζ) (3.3.2)

1 ζ R{ f (λx)} = f˜(p, ) (3.3.3) λn λ 3.4 Shi ing

Shiing is a direct result Z R{ f (x − a)} = f (x − a)δ(p − ζ · x)dx

Z = f (x − a)δ(p − ζ · a − ζ · y)dy

R{ f (x − a)} = f˜(p − ζ · a, ζ) (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) = Cn∆x f˜(ζ · x, ζ)dζ (4.1.1) |ζ|=1 Z Z ∞ ∂ ˜ C ∂p f (p, ζ) f (x) = n dζ (4.1.2) iπ |ζ|=1 −∞ p − ζ · x Where (−1)(n−1)/2 1 1 Cn = = 2(2π)(n−1) 2 (2πi)n−1 Now dene the adjoint operator Z R†ψ = ψ(ζ · x, ζ)dζ |ζ|=1 Note that on a Hilbert space, the dual is the adjoint. Now dene the second order dif- ferential operator (from Helgason as 2, but the author decides to denote with L),

∂2ψ(p, ζ) Lψ(p, ζ) = ∂p2 From these, we have the intertwining property of the Radon transform and its dual/adjoint.

R∆ f = LR f (4.1.3)

7 R∗Lϕ = ∆R∗ ϕ (4.1.4) R†Lψ = ∆R†ψ (4.1.5)

4.2 Unication

Dene a function  n−1  = ∂ ( ) ( odd) Λ0g Cn ∂p g p p=t n Λg(t) = n−1 (4.2.1) Cn ∂ Λ1g = i H ∂p g(p) (t)(neven)

Where 1 Z ∞ g(p) gH(t) = Hg = dp π −∞ p − t is the Hilbert transform (from Bracewell) is gives us

f = R†ΛR f (4.2.2)

Where the adjoint can be replaced by the dual. Note the identity

R†ΛR = I

8 Chapter 5

Relation to the Fourier Transform

e following chapter presents a connection between the Fourier transform (denoted F) and the Radon transform R.

5.1 Fourier and Radon

e n-dimensional Fourier transform (Fn) can be wrien as follows: Z ∞ Z fˆ(η) = dt f (x)e−i2πtδ(t − η · x)dx −∞ Where t ∈ R and η is a vector in Fourier frequency space. Substitute η = sζ and t = sp where s ∈ R and ζ is a unit vector as before. Now note Z ∞ Z fˆ(sζ) = |s| dp f (x)e−i2πspδ(sp − sζ · x)dx −∞ Z ∞ Z = e−i2πspdp f (x)δ(p − ζ · x)dx −∞ Z ∞ fˆ(sζ) = f˜(p, ζ)e−i2πspdp (5.1.1) −∞ e right-hand side gives the Fourier transform in 1 dimension on the radial direction of the Radon transform denoted by F1.

9 5.2 Another Inversion

Observe that the preceding equation sets up the relation

f R f˜

Fn F1 fˆ

Stated in terms of operators: −1 R f = F1 Fn f (5.2.1) Which yields inversion formula

−1 f = F1 FnR f (5.2.2)

10 Chapter 6

Application

ere are too many applications to list in such a short exposition, but for a more com- prehensive treatment of applications of the Radon transform, consult Stanley Deans book, e Radon Transform and Some of Its Applications. e author has decided to give the example of tomography. Tomography is a non-invasive imaging technique allowing for the visualization of the internal structures of an object without the superposition of over and under-lying structures that oen are a part of traditional projection images. For example, in a conventional chest radiograph, the heart, lungs, and ribs are all super- imposed on the same lm, whereas a computed tomography (CT) slice captures each organ in its actual three-dimensional position. Tomography has found widespread application in many scientic elds, including physics, chemistry, astronomy, geophysics, and medicine. While X-ray CT may be the most familiar application of tomography, tomography can be performed using other imaging modalities, including ultrasound, magnetic resonance, nuclear-medicine, and microwave techniques. Each tomographic modality measures a dierent physical quantity: CT: e number of x-ray photons transmied through the patient along individual projection lines. Nuclear medicine: e number of photons emied from the patient along individual projection lines. Ultrasound diraction tomography: e amplitude and phase of scaered waves along a particular line connecting the source and detector. e task in all cases is to estimate from these measurements the distribution of a particular physical quantity in the object. e quantities that can be reconstructed are: CT: e distribution of linear aenuation coecient in the slice being imaged. Nuclear medicine: e distribution of the radiotracer administered to the patient in the slice being imaged. Ultrasound diraction tomography: the distribution of refractive index in the slice being imaged. e measurements made in each modality can be converted into samples of the Radon transform of the distribution that needs to be reconstructed. For example, in CT, divid-

11 ing the measured photon counts by the incident photon counts and taking the negative logarithm yields samples of the Radon transform of the linear aenuation map. e Radon transform and its inverse provide the mathematical basis for reconstructing tomographic images from measured projection or scaering data.

12 Bibliography

Bracewell, R.N. and Riddle, A.C.

1967 Inversion of fan beam scan in radio astronomy, Astrophys. J. 150 (1967), 427-

434.

Deans, S.R.

1983 e Radon Transform and Some of Its Applications. New York: Wiley

Funk, P.

1916 U¨ ber eine geometrische Anwendung der Abelschen Integral gleichnung,

Math. Ann. 77 (1916), 129-135.

Gelfand, I.M., Graev, M.I., and Vilenkin, N.

1966 Generalized Functions, Vol. 5:Integral Geometry and Representation eory,

Academic Press, New York, 1966.

Helgason, S.

1965a e Radon transform on Euclidean spaces, compact two-point homoge

neous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153-180.

1965b Radon-Fourier transforms on symmetric spaces and related group repre

sentation, Bull. Amer. Math. Soc. 71 (1965), 757-763.

13 Radon, J.

1917 U¨ ber die Bestimmung von Funktionen durch ihre Integralwerte la¨ngs gewis

serMannigfaltigkeiten, Ber. Verh. Sa¨chs. Akad. Wiss. Leipzig. Math. Nat.

Kl. 69 (1917), 262-277.