Integral Geometry and Radon Transforms

308 Pages·2011·2.232 MB·English

by  Sigurdur Helgason (auth.)

Integral Geometry and Radon Transforms
Sigurdur Helgason
Integral Geometry
and Radon Transforms
1 C
Sigurdur Helgason
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
USA
[email protected]
ISBN 978-1-4419-6054-2 e-ISBN 978-1-4419-6055-9
DOI 10.1007/978-1-4419-6055-9
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010938299
Mathematics Subject Classification (2010): 53C65, 44A12
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TO ARTIE

Preface
Thisbook dealswith a special subjectin the widefield ofGeometricAnal-
ysis.ThesubjecthasitsoriginsinresultsbyFunk[1913]andRadon[1917]
determining, respectively,a symmetric function on the two-sphereS2 from
its great circle integralsand an integrable function on R2 from its straight
line integrals. (See References.) The first of these is related to a geometric
theorem of Minkowski [1911] (see Ch. III, §1).
While the above workof Funk and Radon lay dormantfor a while, Fritz
Johnrevivedthesubjectinimportantpapersduringthethirtiesandfound
significant applications to differential equations. More recent applications
to X-raytechnologyand tomographyhave widened interest in the subject.
This book originated with lectures given at MIT in the Fall of 1966,
basedmostlyon mypapersduring 1959–1965on the Radontransformand
its generalizations. The viewpoint of these generalizations is the following.
The set of points on S2 and the set of great circles on S2 are both acted
ontransitivelybythegroupO(3).Similarly,thesetofpointsinR2 andthe
set P2 of lines in R2 are both homogeneous spaces of the group M(2) of
rigidmotionsofR2.ThismotivatesourgeneralRadontransformdefinition
from [1965a] and [1966a], which forms the framework of Chapter II:
Given two homogeneous spaces X = G/K and Ξ = G/H of the same
groupG,twoelementsx=gK andξ =γH aresaidtobeincident(denoted
x#ξ) if gK∩γH (cid:3)=∅ (as subsetsof G). We then define the abstract Radon
transform f →f(cid:2)from C (X) to C(Ξ) and the dual transform ϕ→ ˇϕ from
c
C (Ξ) to C(X) by
c
(cid:3) (cid:3)
f(cid:2)(ξ)= f(x)dm(x), ˇϕ(x)= ϕ(ξ)dμ(ξ)
x#ξ ξ#x
with canonical measures dm and dμ. These geometrically dual operators
f → f(cid:2)and ϕ → ϕˇare also adjoint operators relative to the G-invariant
measures dg , dg on G/K and G/H.
K H
IntheexampleR2,onetakesG=M(2)andK thesubgroupO(2)fixing
the origin x and H the subgroup mapping a line ξ into itself. Thus we
o o
have
X =G/K =R2, Ξ=G/H =P2
andhereitturnsoutx∈X isincidenttoξ ∈Ξifandonlyiftheirdistance
equalsthedistancepbetweenx andξ .Itisimportantnotjusttoconsider
o o
the case p = 0. Also the abstract definition does not require the members
of Ξ to be subsets of X. Some natural questions arise for the operators
f →f(cid:2), ϕ→ ϕˇ, namely:
viii
(i) Injectivity
(ii) Inversion formulas
(iii) Ranges and kernels for specific function spaces on X and on Ξ
(cid:2)
(iv) Support problems (does f of compact support imply f of compact
support?)
Weinvestigatetheseproblemsforavarietyofexamples,mainlyinChap-
ter II. Interesting analogies and differences appear. One such instance is
when the classical Poisson integral for the unit disk turns out to be a cer-
tain Radon transform and offers wide ranging analogies with the X-ray
transform in R3. See Table II.1 in Chapter II, §4.
In the abstract framework indicated above, a specific result for a single
example automatically raises a host of conjectures.
The problems aboveareto a largeextent solvedfor the X-raytransform
and for the horocycle transform on Riemannian symmetric spaces. When
G/K is a Euclidean space (respectively, a Riemannian symmetric space)
and G/H the space of hyperplanes (respectively, the space of horocycles)
the transformf →f(cid:2)hasapplicationsto certaindifferentialequations.If L
is a natural differential operator on G/K, the map f →f(cid:2)transfers it into
(cid:2)
a more manageable operator L on G/H by the relation
(Lf)(cid:2)=L(cid:2)f(cid:2).
Then the support theorem
f(cid:2) compact support ⇒f compact support
implies the existence theorem LC∞(G/K) = C∞(G/K) for G-invariant
differential operators L on G/K.
On the other hand, the applications of the original Radon transform on
R2 to X-ray technology and tomography are based on the fact that for
(cid:2)
an unknown density f, X-ray attenuation measurements give f directly
and thus yield f itself via Radon’s inversionformula. More precisely, let B
be a planar convex body, f(x) its density at the point x, and suppose a
thin beam of X-rays is directed at B along a line ξ. Then, as observed by
(cid:2)
Cormack, the line integral f(ξ) of f along ξ equals log(I0/I) where I0 and
I, respectively, are the intensities of the beam before hitting B and after
(cid:2)
leaving B. Thus while f is at first unknown, the function f (and thus f)
is determined by the X-ray data. See Ch. I, §7,B. This work, initiated
by Cormack and Hounsfield and earning them a Nobel Prize, has greatly
increased interest in Radon transform theory. The support theorem brings
ina certainrefinementthatthe densityf(x) outsideaconvexsetC canbe
determined by only using X-rays that do not enter C. See Ch. I, §7, B.
ix
This book includes and recasts some material from my earlier book,
“The Radon Transform”, Birkh¨auser (1999). It has a large number of new
examples of Radon transforms, has an extended treatment of the Radon
transform on constant curvature spaces, and contains full proofs for the
antipodal Radon transform on compact two-point homogeneous spaces.
The X-raytransform on symmetric spaces is treated in detail with explicit
inversion formulas.
In order to make the book self-contained we have added three chapters
at the end of the book. Chapter VII treats Fourier transforms and distri-
butions, relying heavily on the concise treatment in Ho¨rmander’s books.
We callparticularattentionto hisprofoundTheorem4.9,which inspite of
its importance does not seem to havegenerallyentered distribution theory
books.Wehavefoundthisresultessentialinourstudy[1994b]oftheRadon
transformonasymmetricspace.ChapterVIIIcontainsashorttreatmentof
basicLiegrouptheoryassumingonlyminimal familiaritywiththe concept
of a manifold. Chapter IX is a short exposition of the basics of the theory
of Cartan’s symmetric spaces. Most chapters end with some Exercisesand
Further Results with explicit references.
Although the Bibliography is fairly extensive no completeness is at-
tempted. In view of the rapid development of the subject the Bibliograph-
ical Notes can not be up to date. In these notes and in the text my books
[1978]and [1984]and [1994b]are abbreviated to DS and GGA and GSS.
IamindebtedtoT.H.Danielsen,S.JensenandJ.Orloffforcriticalread-
ingofpartsofthemanuscript,toR.MelroseandR.Seeleyforsuggestions,
to F. Gonzalez, J. Hilgert, A. Kurusa, F. Rouvi`ere and H. Schlichtkrull
for concrete contributions mentioned at specific places in the text, and for
various textual suggestions. Once more my sincere thanks to Jan Wetzel
for skillful preparation of the manuscript and to Kaitlin Leach at Springer
for her patient cooperation.
MIT Sigurdur Helgason
May 2009
Contents
Preface vii
CHAPTER I
The Radon Transform on Rn
§1 Introduction 1
§2 The Radon Transform of the Spaces D(Rn) andS (Rn). The
Support Theorem 2
§3 The Inversion Formula. Injectivity Questions 16
§4 The Plancherel Formula 25
§5 Radon Transform of Distributions 27
§6 Integration over d-planes. X-ray Transforms. The Range of the
d-plane Transform 32
§7 Applications 46
A. Partial Differential Equations. The Wave Equation 46
B. X-ray Reconstruction 52
Exercises and Further Results 56
Bibliographical Notes 60
CHAPTER II
A Duality in Integral Geometry
§1 Homogeneous Spaces in Duality 63
§2 The Radon Transform for the Double Fibration 67
(i) Principal Problems 68
(ii) Ranges and Kernels. General Features 71
(iii) The Inversion Problem. General Remarks 72
§3 Orbital Integrals 75
§4 Examples of Radon Transforms for Homogeneous Spaces in Duality 77
A. The Funk Transform 77
B. The X-ray Transform in H2 80
C. The Horocycles in H2 82
D. The Poisson Integral as a Radon Transform 86
E. The d-plane Transform 88
F. Grassmann Manifolds 90
G. Half-lines in a Half-plane 91
H. Theta Series and Cusp Forms 94
I. The Plane-to-Line Transform in R3. The Range 95
J. Noncompact Symmetric Space and Its Family of Horocycles 103
Exercises and Further Results 104
Bibliographical Notes 108
xii
CHAPTER III
The Radon Transform on Two-Point Homogeneous Spaces
§1 Spaces of Constant Curvature. Inversion and Support Theorems 111
A. The Euclidean Case Rn 114
B. The Hyperbolic Space 118
C. The Spheres and the Elliptic Spaces 133
D. The Spherical Slice Transform 145
§2 Compact Two-Point Homogeneous Spaces. Applications 147
§3 Noncompact Two-Point Homogeneous Spaces 157
§4 Support Theorems Relative to Horocycles 159
Exercises and Further Results 167
Bibliographical Notes 168
CHAPTER IV
The X-Ray Transform on a Symmetric Space
§1 Compact Symmetric Spaces. Injectivity and Local Inversion.
Support Theorem 171
§2 Noncompact Symmetric Spaces. Global Inversion and General
Support Theorem 178
§3 Maximal Tori and Minimal Spheres in Compact Symmetric Spaces180
Exercises and Further Results 182
Bibliographical Notes 183
CHAPTER V
Orbital Integrals and the Wave Operator for Isotropic Lorentz Spaces
§1 Isotropic Spaces 185
A. The Riemannian Case 186
B. The General Pseudo-Riemannian Case 186
C. The Lorentzian Case 190
§2 Orbital Integrals 190
§3 Generalized Riesz Potentials 199
§4 Determination of a Function from Its Integral over Lorentzian
Spheres 202
§5 Orbital Integrals and Huygens’ Principle 206
Bibliographical Notes 208
CHAPTER VI
The Mean-Value Operator
§1 An Injectivity Result 209
§2 A´sgeirsson’s Mean-Value Theorem Generalized 211
§3 John’s Identities 215
Exercises and Further Results 217
Bibliographical Notes 219