Radon Transform. With errata

199 Pages·1999·0.836 MB·English

by  Helgason S.

i
Sigurdur Helgason
Radon Transform
Second Edition
Contents
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . iv
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . v
CHAPTER I
The Radon Transform on Rn
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
x
2 The Radon Transform ...The Support Theorem . . . . . . . . . 2
x
3 The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . 15
x
4 The Plancherel Formula . . . . . . . . . . . . . . . . . . . . . . . 20
x
5 Radon Transform of Distributions . . . . . . . . . . . . . . . . . 22
x
6 Integration over d-planes. X-ray Transforms.. . . . . . . . . . . . 28
x
7 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
x
A. Partialdi(cid:11)erential equations. . . . . . . . . . . . . . . . . . . 41
B. X-ray Reconstruction.. . . . . . . . . . . . . . . . . . . . . . 47
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 51
CHAPTER II
A Duality in Integral Geometry.
1 HomogeneousSpaces in Duality . . . . . . . . . . . . . . . . . . . 55
x
2 The Radon Transform for the Double Fibration . . . . . . . . . . 59
x
3 Orbital Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
x
4 Examplesof RadonTransformsforHomogeneousSpacesinDuality 65
x
ii
A. The Funk Transform. . . . . . . . . . . . . . . . . . . . . . . 65
B. The X-ray Transform in H2. . . . . . . . . . . . . . . . . . . 67
C. The Horocycles in H2. . . . . . . . . . . . . . . . . . . . . . 68
D. The Poisson Integral as a Radon Transform. . . . . . . . . . 72
E. The d-plane Transform. . . . . . . . . . . . . . . . . . . . . . 74
F. Grassmann Manifolds. . . . . . . . . . . . . . . . . . . . . . 76
G. Half-lines in a Half-plane. . . . . . . . . . . . . . . . . . . . . 77
H. Theta Series and Cusp Forms. . . . . . . . . . . . . . . . . . 80
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 81
CHAPTER III
The Radon Transform on Two-point
Homogeneous Spaces
1 Spaces of Constant Curvature. Inversion and Support Theorems 83
x
A. The Hyperbolic Space. . . . . . . . . . . . . . . . . . . . . . 85
B. The Spheres and the Elliptic Spaces . . . . . . . . . . . . . . 92
C. The Spherical Slice Transform . . . . . . . . . . . . . . . . . 107
2 Compact Two-point Homogeneous Spaces. Applications . . . . . 110
x
3 Noncompact Two-point Homogeneous Spaces . . . . . . . . . . . 116
x
4 The X-ray Transform on a Symmetric Space. . . . . . . . . . . . 118
x
5 Maximal Tori and Minimal Spheres in Compact Symmetric Spaces119
x
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 120
CHAPTER IV
Orbital Integrals
1 Isotropic Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
x
A. The Riemannian Case. . . . . . . . . . . . . . . . . . . . . . 124
B. The General Pseudo-Riemannian Case . . . . . . . . . . . . 124
C. The Lorentzian Case . . . . . . . . . . . . . . . . . . . . . . 128
2 Orbital Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
x
3 Generalized Riesz Potentials. . . . . . . . . . . . . . . . . . . . . 137
x
4 Determination of a Function from its Integral over Lorentzian
x
Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5 Orbital Integrals and Huygens’ Principle . . . . . . . . . . . . . . 144
x
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 145
CHAPTER V
Fourier Transforms and Distributions.
A Rapid Course
1 The Topology of the Spaces (Rn), (Rn) and (Rn) . . . . . . 147
x D E S
iii
2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
x
3 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 150
x
4 Di(cid:11)erential Operators with Constant Coe(cid:14)cients . . . . . . . . . 156
x
5 Riesz Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
x
BibliographicalNotes . . . . . . . . . . . . . . . . . . . . . . . . 170
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . 186
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
iv
Preface to the Second Edition
The(cid:12)rsteditionofthisbookhasbeenoutofprintforsometimeandIhave
decided to follow the publisher’s kind suggestionto preparea new edition.
Manyexamples of the explicit inversionformulasandrangetheoremshave
been added, and the group-theoretic viewpoint emphasized. For example,
the integral geometric viewpoint of the Poisson integral for the disk leads
to interesting analogieswith the X-raytransform in Euclidean3-space.To
preserve the introductory (cid:13)avor of the book the short and self-contained
Chapter V on Schwartz’ distributions has been added. Here 5 provides
x
proofsofthe neededresultsaboutthe Rieszpotentialswhile 3{4develop
xx
thetoolsfromFourieranalysisfollowingcloselytheaccountinHo(cid:127)rmander’s
books [1963] and [1983]. There is some overlap with my books [1984] and
[1994b]which,however,relyheavilyonLiegrouptheory.Thepresentbook
is much more elementary.
IamindebtedtoSineJensenforacriticalreadingofpartsofthemanuscript
and to Hilgert and Schlichtkrull for concrete contributions mentioned at
speci(cid:12)c places in the text. Finally I thank Jan Wetzel and Bonnie Fried-
man for their patient and skillful preparation of the manuscript.
Cambridge, 1999
v
Preface to the First Edition
The title of this booklet refers to a topic in geometric analysis which has
its origins in results of Funk [1916] and Radon [1917] determining, respec-
tively,asymmetricfunctiononthetwo-sphereS2 fromitsgreatcircleinte-
gralsandafunctionoftheplaneR2 fromitslineintegrals.(Seereferences.)
Recentdevelopments,inparticularapplicationstopartialdi(cid:11)erentialequa-
tions,X-raytechnology,andradio astronomy,havewidened interestin the
subject.
These notes consist of a revision of lectures given at MIT in the Fall of
1966,basedmostyonmypapersduring1959{1965ontheRadontransform
and its generalizations. (The term \Radon Transform" is adopted from
John [1955].)
The viewpoint for these generalizationsis as follows.
The set of points on S2 and the set of great circles on S2 are both ho-
mogeneousspacesof theorthoginalgroupO(3). Similarly,thesetofpoints
inR2 andthe setof linesinR2 arebothhomogeneousspacesofthegroup
M(2) of rigid motions of R2. This motivates our generalRadon transform
de(cid:12)nition from [1965a, 1966a] which forms the framwork of Chapter II:
Given two homogeneous spaces G=K and G=H of the same group G, the
Radon transform u u maps functions u on the (cid:12)rst space to functions u
!
onthe secondspace.For(cid:24) G=H,u((cid:24)) isde(cid:12)nedasthe (natural)integral
2
of u overthe set of points x G=K which are incident to (cid:24) in the sense of
b b
2
Chern[1942].Theproblemofinvertingu uisworkedoutinafewcases.
b
!
It happens when G=K is a Euclidean space, and more generally when
G=K is a Riemannian symmetric space, that the natural di(cid:11)erential op-
b
erators A on G=K are transferred by u u into much more manageable
!
b
di(cid:11)erential operators A on G=H; the connection is (Au) = Au. Then the
theory of the transform u u has signi(cid:12)canbt applications to the study of
!
properties of A. b bb
On the other hand, the applications of the original Radon transform on
b
R2 to X-ray technology and radio astronomy are based on the fact that
foranunknowndensityu,X-rayattenuationmeasurementsgiveudirectly
and therefore yield u via Radon’s inversion formula. More precisely, let B
beaconvexbody,u(x)itsdensityatthepointx,andsupposeathinbeam
b
of X-rays is directed at B along a line (cid:24). Then the line integral u((cid:24)) of u
along(cid:24) equals log(I =I) where I and I, respectively, are the intensities of
o o
the beam before hitting B and after leaving B. Thus while the function u
b
is at (cid:12)rst unknown, the function u is determined by the X-ray data.
The lecture notes indicated abovehave been updated a bit by including
ashortaccountofsomeapplications(ChapterI, 7),byaddingafewcpro-
b
x
llaries (Corollaries2.8 and 2.12, Theorem 6.3 in Chapter I, Corollaries2.8
vi
and 4.1 in Chapter III), and by giving indications in the bibliographical
notes of some recent developments.
An e(cid:11)ort has been made to keep the exposition rather elementary. The
distribution theoryand the theoryof Rieszpotentials,occasionallyneeded
inChapterI,isreviewedinsomedetailin 8(nowChapterV).Apartfrom
x
the general homogeneous space framework in Chapter II, the treatment is
restricted to Euclidean and isotropic spaces (spaces which are \the same
in all directions"). For more general symmetric spaces the treatment is
postposed (except for 4 in Chapter III) to another occasion since more
x
machinery from the theorem of semisimple Lie groups is required.
I am indebted to R. Melrose and R. Seeley for helpful suggestions and
toF.GonzalezandJ.Orlo(cid:11)forcriticalreadingof partsofthe manuscript.
Cambridge, MA 1980
1
CHAPTER I
THE RADON TRANSFORM ON RN
x1 Introduction
It was proved by J. Radon in 1917 that a di(cid:11)erentiable function on R3
can be determined explicitly by means of its integrals over the planes in
R3. Let J(!;p) denote the integral of f over the hyperplane x;! =p, !
h i
denoting a unit vector and ; the inner product. Then
h i
1
f(x)= L J(!; !;x )d! ;
(cid:0)8(cid:25)2 x(cid:18)ZS2 h i (cid:19)
where L is the Laplacian on R3 and d! the areaelement on the sphere S2
(cf. Theorem 3.1).
We now observe that the formula above has built in a remarkable du-
ality: (cid:12)rst one integrates over the set of points in a hyperplane, then one
integrates over the set of hyperplanes passing through a given point. This
suggests considering the transforms f f;’ ’(cid:20)de(cid:12)ned below.
! !
The formula hasanotherinterestingfeature.Fora (cid:12)xed! the integrand
x J(!; !;x ) is aplane wave, that is abfunction constant oneachplane
! h i
perpendicular to !. Ignoring the Laplacian the formula gives a continu-
ous decomposition of f into plane waves. Since a plane wave amounts to
a function of just one variable (along the normal to the planes) this de-
composition can sometimes reduce a problem for R3 to a similar problem
for R. This principle has been particularly useful in the theory of partial
di(cid:11)erential equations.
The analog of the formula above for the line integrals is of importance
in radiographywhere the objective is the description of a density function
by means of certain line integrals.
InthischapterwediscussrelationshipsbetweenafunctiononRn andits
integrals over k-dimensional planes in Rn. The case k = n 1 will be the
(cid:0)
oneofprimaryinterest.WeshalloccasionallyusesomefactsaboutFourier
transforms and distributions. This material will be developed in su(cid:14)cient
detail in Chapter V so the treatment should be self-contained.
FollowingSchwartz[1966]wedenoteby (Rn)and (Rn), respectively,
E D
the space of complex-valued 1 functions (respectively 1 functions of
C C
compactsupport)onRn.Thespace (Rn)of rapidlydecreasingfunctions
S
onRnisde(cid:12)nedinconnectionwith(6)below.Cm(Rn)denotesthespaceof
mtimescontinuouslydi(cid:11)erentiablefunctions.WewriteC(Rn)forC0(Rn),
the space of continuous function on Rn.
ForamanifoldM,Cm(M)(andC(M))isde(cid:12)nedsimilarlyandwewrite
(M) for 1(M) and (M) for 1(M).
D Cc E C
2
x2 The Radon Transform of the Spaces (Rn) and
(Rn). The Support TheoremD
S
Let f be a function on Rn, integrable on each hyperplane in Rn. Let Pn
denote the space of all hyperplanes in Rn, Pn being furnished with the
obvious topology. The Radon transform of f is de(cid:12)ned as the function f
on Pn given by
b
f((cid:24))= f(x)dm(x);
Z(cid:24)
where dm is the Euclideanbmeasure on the hyperplane (cid:24). Along with the
transformationf f weconsideralsothedual transform ’ ’(cid:20)whichto
a continuous func!tion ’ on Pn associates the function ’(cid:20)on !Rn given by
b
(cid:20)’(x)= ’((cid:24))d(cid:22)((cid:24))
Zx2(cid:24)
where d(cid:22) is the measure on the compact set (cid:24) Pn : x (cid:24) which is
f 2 2 g
invariantunderthe groupofrotationsaroundxandforwhichthe measure
of the whole set is 1 (see Fig. I.1). We shall relate certain function spaces
on Rn and on Pn by means of the transforms f f;’ ’(cid:20); later we
! !
obtain explicit inversion formulas.
b
x
x
}
x
Æx,wæ
w
0
FIGUREI.2.
FIGUREI.1.
Each hyperplane (cid:24) Pn can be written (cid:24) = x Rn : x;! = p
2 f 2 h i g
where ; is the usual inner product, ! = (! ;:::;! ) a unit vector
1 n
h i
and p R (Fig. I.2). Note that the pairs (!;p) and ( !; p) give the
2 (cid:0) (cid:0)
same (cid:24); the mapping (!;p) (cid:24) is a double covering of Sn(cid:0)1 R onto
! (cid:2)
Pn. Thus Pn has a canonical manifold structure with respect to which
thiscoveringmapisdi(cid:11)erentiableandregular.Wethusidentifycontinuous
3
(di(cid:11)erentiable) function on Pn with continuous (di(cid:11)erentiable) functions
’ on Sn(cid:0)1 R satisfying the symmetry condition ’(!;p) = ’( !; p).
(cid:2) (cid:0) (cid:0)
Writing f(!;p) instead of f((cid:24)) and f (with t Rn) for the translated
t
2
function x f(t+x) we have
!
b b
f (!;p)= f(x+t)dm(x)= f(y)dm(y)
t
Zhx;!i=p Zhy;!i=p+ht;!i
so b
(1) f (!;p)=f(!;p+ t;! ):
t
h i
Taking limits we see thatbif @i =@=b@xi
@f
b
(2) (@ f)(!;p)=! (!;p):
i i
@p
b
Let L denote the Laplacian (cid:6) @2 on Rn and let (cid:3) denote the operator
i i
@2
’(!;p) ’(!;p);
! @p2
whichisawell-de(cid:12)nedoperatoron (Pn)= 1(Pn).Itcanbeprovedthat
E C
ifM(n)isthegroupofisometriesofRn,thenL(respectively(cid:3))generates
the algebra of M(n)-invariant di(cid:11)erential operators on Rn (respectively
Pn).
Lemma 2.1. The transforms f f;’ (cid:20)’ intertwine L and (cid:3), i.e.,
! !
(Lf)b=(cid:3)(f);b ((cid:3)’)_ =L(cid:20)’:
Proof. The (cid:12)rst relation follows fbrom (2) by iteration. For the second we
just note that for a certain constant c
(3) ’(cid:20)(x)=c ’(!; x;! )d!;
ZSn(cid:0)1 h i
where d! is the usual measure on Sn(cid:0)1.
The Radon transform is closely connected with the Fourier transform
f(u)= f(x)e(cid:0)ihx;!idx u Rn:
ZRn 2
In fact, if s R, !ea unit vector,
2
1
f(s!)= dr f(x)e(cid:0)ishx;!idm(x)
Z(cid:0)1 Zhx;!i=r
e
4
so
1
(4) f(s!)= f(!;r)e(cid:0)isrdr:
Z(cid:0)1
This means that the n-edimensional Fbourier transform is the 1-dimensional
Fourier transform of the Radon transform. From (4), or directly, it follows
that the Radon transform of the convolution
f(x)= f (x y)f (y)dy
1 2
ZRn (cid:0)
is the convolution
(5) f(!;p)= f (!;p q)f (!;q)dq:
1 2
R (cid:0)
Z
We consider nowbthe space b(Rn) of combplex-valued rapidly decreas-
S
ing functions on Rn. We recall that f (Rn) if and only if for each
2 S
polynomial P and each integer m 0,
(cid:21)
(6) sup xmP(@ ;:::;@ )f(x) < ;
1 n
jj j j 1
x
x denoting the norm of x. We now formulate this in a more invariant
j j
fashion.
Lemma 2.2. A function f (Rn) belongs to (Rn) if and only if for
each pair k;‘ Z+ 2 E S
2
sup (1+ x)k(L‘f)(x) < :
x2Rnj j j j 1
This is easily proved just by using the Fourier transforms.
In analogywith (Rn) we de(cid:12)ne (Sn(cid:0)1 R) as the space of 1 func-
S S (cid:2) C
tions ’ on Sn(cid:0)1 R which for any integers k;‘ 0 and any di(cid:11)erential
(cid:2) (cid:21)
operator D on Sn(cid:0)1 satisfy
d‘
(7) sup (1+ r k) (D’)(!;r) < :
!2Sn(cid:0)1;r2R j j dr‘ 1
(cid:12) (cid:12)
(cid:12) (cid:12)
The space (Pn) is then de(cid:12)(cid:12)ned as the set of ’ (cid:12)(Sn(cid:0)1 R) satisfying
S 2S (cid:2)
’(!;p)=’( !; p).
(cid:0) (cid:0)
Lemma 2.3. For each f (Rn) the Radon transform f(!;p) satis(cid:12)es
the following condition: For2kS Z+ the integral
2
b
f(!;p)pkdp
R
Z
can be written as a kth degree hombogeneous polynomial in ! ;:::;! .
1 n