Lecture Notes in MathematicsLecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1684
Advisor: Roberto Conti
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
C. A. Berenstein R E Ebenfelt S.G. Gindikin S. Helgason A.E. Tumanov
Integral Geometry, Radon Transforms and Complex Analysis Lectures given at the 1 st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venice, Italy, June 3-12, 1996
Editors: E. Casadio Tarabusi, M. A. Picardello, G. Zampieri
Fondazione
C.I.M.E.
Springer
Authors
Carlos A. Berenstein Institute for Systems Research 221 A. V. Williams Building University of Maryland College Park, MD 20742-0001, USA
Peter F. Ebenfelt Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden
Simon Gindikin Department of Mathematics Hill Center Rutgers University New Brunswick, NJ 08903-2101, USA
Sigurdur Helgason Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA
Alexander Tumanov Department of Mathematics University of Illinois 1409 West Green Street Urbana-Champaign, IL 61801-2943, USA
Editors
Massimo A. Picardello Dipartimento di Matematica Universith di Roma "Tor Vergata" Via della Ricerca Scientifica 00133 Roma, Italy
Giuseppe Zampieri Dipartimento di Matematica Pura ed Applicata Universit'~ di Padova Via Belzoni, 7 1-35131 Padova, Italy
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Die Deutsche Bibtiothek - CIP-Einheitsaufnahme
Integral geometry, radon transforms and complex analysis : held in Venezia, Italy, June 3-12. 1996 / C. A. Berenstein ... Ed.: E. Casadio Tarabusi ... - Berlin; Heidelberg; New York; Barcelona: Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998 (Lectures given at the ...session of the Centro lnternazionale Matematico Estivo (CIME) ... ; 1996,1) (Lecture notes in mathematics; vol. 1684: Subseries; Fondazione CIME) ISBN 3-540-64207-2 Centro Internationale Matematico Estivo <Firenze>: Lectures given at the ... session of the Centro lnternationale Matematico Estivo (CIME) ... - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong: Springer Friiher Schriftenreihe. - FriJher angezeigt u. d. T.: Centro lnternationale Matematico Estivo: Proceedings of the ... session of the Centro l nternationale Matematico Estivo (C1ME) 1996,1. Integral geometry, radon transforms and complex analysis. - 1998
Mathematics Subject Classification (1991): 43-06, 44-06, 32-06
ISSN 0075- 8434 ISBN 3-540-64207-2 Springer-Verlag Berlin Heidelberg New York
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PREFACE
This book contains the notes of five short courses delivered at the Italian Can- fro Internazionale Matematico Estivo (CIME) session Integral Geometry, Radon Transforms and Complex Analysis held at Ca' Dolfin in Venice (Italy) in June 1996.
Three of the courses (namely those by: Carlos A. Berenstein of the University of Maryland at College Park; Sigurdur Helgason of the Massachusetts Institute of Technology; and Simon G. Gindikin of Rutgers University) dealt with various aspects of integral geometry, with a common emphasis on several kinds of Radon transforms, their properties and applications.
The lectures by C. A. Berenste in , Radon transforms, wavelets, and applica- tions, explain the definition and properties of the classical Radon transform on the two-dimensional Euclidean space, with particular stress on localization and inver- sion, which can be achieved by the recent tool of wavelets. Interesting applications to Electrical Impedance Tomography (EIT) are also illustrated.
The lectures by S. Helgason, Radon transforms and wave equations~ give an account of Radon transforms on Euclidean and symmetric spaces, focusing atten- tion onto the Huygens principle and the solution of the wave equation in these environments.
The lectures by S. G. Gindikin, Real integral geometry and complex analysis, give an account of the deep connection between the two main themes of this CIME session, covering several variations of the Radon transform (RT): the projective RT; RT's taken over hyperplanes of codimension higher than 1; and RT's over spheres. An important and unifying tool is the ~" operator of Gel'fand-Graev-Shapiro, used to explain analogies between inversion formulas for the various RT's. This approach goes hand-in-hand with ~-cohomotogy and hyperfunctions, typical subjects in the field of complex a~alysis.
In related areas, the other two courses (namely those by: Alexander E. Tumanov of the University of Illinois at Urbana-Champaign; Peter F. Ebenfelt of the Royal Institute of Technology at Stockholm) share stress on CR manifolds and related problems.
The lectures by A. E. Tumanov , Analytic discs and the extendibility of CR functions, provide an introduction to CR structures and deal in particular with the problem of characterizing those submanifolds of C N whose CR functions are wedge-extendible. This property turns out to be equivalent to the absence of proper submanifolds which carry the stone CR structure. (The technique of the proof con- sists in an infinitesimal deformation of analytic discs attached to CR submanifolds.)
The lectures by P. F. Ebenfel t , Holomorphic mappings between real analytic aubmanifolds in complex space, deal with algebralcity of locally invertible holomor- phic mappings. Along with classical results, new criteria are introduced in terms of the behavior of these mappings on a real-analytic CR submanifold which is generic, minimal, and holomorphically non-degenerate in a suitable sense. To this end a fundamental tool is afforded by the so-called Segre sets.
VI
We wish to express our appreciation to the authors of these notes, and to thank all the numerous participants of this CIME session for creating a lively and stim- ulating atmosphere. We are particularly grateful to those who contributed to the success of the session by delivering very inspiring talks.
Enrico CASADIO TARABUSI Massimo A. P I C A R D E L L O Giuseppe ZAMPIERI
TABLE OF CONTENTS
BERENSTEIN, C. A.
EBENFELT, P. F.
GINDIKIN, S. G.
Holomorphic Mappings Between Real Analytic Submanifolds in Complex Space
Real Integral Geometry and Complex Analysis
Radon Transforms and Wave Equations
Analytic Discs and The Extendibility of CR Functions
35
70
99
123
Radon transforms, wavelets, and applications Carlos Berenstein
We present here the informal notes of four lectures 1 given at Cs Dolfin, Venice, under the auspices of CIME. They reflect the research of the author, his collaborators, and many other people in different applications of integral geometry. This is a vast and very active area of mathematics, and we try to show it has many diverse and sometimes unexpected applications, for that reason it would impossible to be complete in the references. Nevertheless, we hope that every work relevant to these lectures, however indirectly, will either be explicitly found in the bibliography at the end or at least in the reference lists of the referenced items. I apologize in advance for any shortcomings in this respect.
The audience of the lectures was composed predominantly of graduate students of universities across Italy and elsewhere in Europe, for that reason, the emphasis is not so much in rigor but in creating an understanding of the subject, good enough to be aware of its manifold applications. There are several very good general references, the most accesible to students is, in my view, [Hell. For deeper analysis of the Radon transform the reader is suggested to look in [He2] and [He3]. For a very clear explanation of the numerical algorithms of the (codimension one) Radon transform in R 2 and R 3, see [Na] and [KS]. There have also been many recent conferences on the subject of these lectures, for a glimpse into them we suggest [GG] and [GM].
Finally, I would like to thank the organizers, Enrico Casadio Tarabusi, Massimo Picardello, and Giuseppe Zampieri, for their kindness in inviting me and for the effort they exerted on the organization of this CIME session. I am also grateful to David Walnut for suggestions that improved noticeably these notes.
1. T o m o g r a p h i c imag ing of space p l a s m a
Space plasma is composed of electrically charged particles that are not uniformly distributed in space and are influenced by celestial bodies. The problem consists in determining the distribution function of the energy of these particles (or of their velocities) in a region of space. A typical measuring device will take discrete mea- surements (for instance, sample temperatures at different points in space) and then the astrophysicist will try to fit a "physically meaningful" function passing through these points. The procedure proposed in [ZCMB] is based on the idea that the mea- surements should directly determine the distribution function. We do it by exploiting the charged nature of the particles and using the Radon transform. (The recently launched Wind satellite carries a measuring device based on similar interaction prin- ciples and requires tomographic ideas for the processing of the data.)
The advantage of the tomographic principles that we shall describe presently is that each measurement carries global information and seems to have certain noise reduction advantages over the pointwise measurements of temperatures, which is the
|
](
Figure 1: Schematic detector.
usual technology. We will describe everything in a two-dimensional setting, but the more realistic three-dimensional case can be handled similarly.
The instrument we proposed in [ZCMB] is schematically the following. An electron enters into the instrument (a rectangular box in the figure below) through an opening located at the origin and is deflected by a constant magnetic field/~ perpendicular to the plane of the paper (see Figure 1). Under the Lorentz force, the electrons follow circular orbits and strike detectors lo- cated on the front-inside surface of the box (along the y axis). Those that strike a detector located at the point y have the property that
where m is the mass of the electron, e its charge, and B the magnitude of the magnetic field /~. In other words, all the electrons with the same first component v~ of their velocities strike the same detector located at the height y. The range of velocities over a segment of width a (width of the detector) is
Avx = (eB/2m)d
(In terms of the length of the detector plate D in Figure 1 and the maximum velocity vm~ we have Avx = (d/D) - Vm~). If f(v~,vy) represents the electron velocity dis- tribution, then the number dN of electrons counted by a detector in time dt is given by
dN = Anev~Avx i f(vx, vy)dvy, dr, - - 0 0
ne is electron density and A is the area of the entrance aperture. In other words,
1 dN f(v~, v~)dvy = AneAv~--~
- - o o
so tha t the count of hits provides the integral of f along a line vx = constant in the velocity plane. By rota t ing the detector or changing the orientation of the magnet ic field we obtain the Radon transform of f .
As a realistic example, consider a p lasma of nominal electron density ne = 10 c m -3 ,
velocity in the range Vmi n to Vmax of 1.2 X 10 s to 3.0 • 109 cm s -1, average velocity = 6.5 • 108 cm s -1, and we assume a Gaussian distr ibution function
so tha t
dt - const, e x p , , 2~2, ]
with individual detector area and aper ture of 0.04 c m 2 for a small instrument one gets tha t the dis tr ibut ion function f varies from 1 to 10 -5 while d N / d t varies from 102 to 105s -1. The s tandard measurement methods make the a priori assumption tha t f is the sum of a Gaussian centered at V and per turbed by adding a finite collection of Gaussians, often located in the region where f varies from 10 -4 to 10 -5, but the previously described instrument does not require any such assumption, on the other hand, experimental ly one sees that such large variations, like from 1 to 10 -5 as in the example, are realistic. We shall see in Section 2 tha t this is an embodiment of the Radon transform in R 2. The more realistic case of 3-d is handled by an instrument where there is a plane which contains the entrance aper ture and a 2-d array of detectors in the plane (x, y). One shows tha t at each detector location (x, y) one obtains an integral over a planar curve and that the addi t ion of overall elements with the same x component leads to a 2-d plane integral of the density dis t r ibut ion so tha t we have the Radon transform in R 3. (This is an observation we made joint ly with M. Shahshahani.)
Before concluding this section, let us remark tha t the large variations expected from the velocity density function f make the inversion of the Radon transform very ill-conditioned, even if f is assumed to be a smooth function. This is due to the continuity propert ies of the Radon transform and its inverse as seen in the next section. The remarkable point is that in medical applications, like CAT scans, the unknown density is natural ly discontinuous along some curves but otherwise it has small local variations, and it is this reason the inversion problem is u l t imate ly easier for medical applications.
Source
Figure 2: Schematic CAT scanner.
2. T h e R a d o n T r a n s f o r m in R ~
Let w E S 1, w = (cos 0, sin 0), and take p E R. The equation x-w = p represents the line l which has (signed) distance p fi'om origin and is perpendicular to the direction 02.
For any reasonable function f (e.g., continuous of compact support), we can com- pute the line integral, with respect to Euclidean arc length ds,
oo
:= / f(x)ds = / f(x0 Rf(w,p) + tw• )dt (1) X , o 2 ~ p - -00
where x0 is a fixed point in l, i.e., satisfying the equation x0 - w = p, and w • = ( - sin 0, cos 0) is the rotate of w by ~/2.
The map f ~-~ R f is called the Radon transform and R f is called the Radon transform of f . Clearly R f is a function defined on S 1 • R (that is, the family of all lines in R 2) with the obvious compatibility condition:
(R f )(-w, -p) = Rf(w, p). (2)
There are several reasonable domains of definition for R such as LI(R~), $(R2), etc., but in many applications it is enough to consider functions which are of compact sup- port, with singularities which are only jumps along reasonable curves, and otherwise smooth. This is obviously the transformation appearing in Section 1. The full 3-d instrument there corresponds to integration over planes in R 3, perpendicular to a unit vector w. A big source of interest of this transform lies in CAT (Computerized Axial Tomography) as a radiological tool where each planar section of a patient is scanned by X-rays as in Figure 2.
In this particular case it can be seen that
I0 f log ~ ~ / p d s (3)
J l
where I0 is the radiation intensity at the source and I is the intensity measured at the detector. The attenuation is a consequence of traversing a tissue of density #. So the data collected from this X-ray scanning appears in the form of the Radon transform R# of the density #, computed for a finite collection of directions wt, w2, . . . , w~v (usually equally spaced) and a finite collection of lines, i.e., values Pl,P2,...,PM for each direction. This is called a "parallel beam" CT scanner. The configuration that it is now most used but we shall not discuss here is the "fan beam" CT scanner, we refer to [Na], [KS] for a discussion of the differences of these two cases, they really only appear at the implementation level of the inversion algorithms because only a limited amount of data can be obtained in the real world.
Some easy properties of the Radon transform are obtained by observing that Rf can be written using distributions. In fact, if we introduce the unit density 5 ( p - x . w ) which is supported by the line x 9 w = p, then
Rf(w, p) = f f(x)~(p - x . w)dx (1') R ~
with the usual abuse of language. It is also convenient to write
R~f(p) = Rf(w,p). (4)
Formula (1') can be used to extend Rf to (R2\{0}) x R, using the fact that 5(p-x.w) is homogeneous of degree -1 ; indeed, one defines
nf(~, s) = ~ n s , (5)
One can therefore take derivatives of (1') with respect to the variables ~j(~ = (~1, ~2)) and obtain
~---~jRf(~,s)= f f(x)~-~j~i(s-x.~)dx (j = 1,2), (6)
but
and
O f f (x)x jh(s- :c'~)dx
0 Os (R(xjf)(~, s)).
On the other hand the Radon transform of the derivative of f is:
In particular, for
= ~jff---~(RJ)(s).
(7)
Rr = (~ + ~ ) (RJ ) ( s ) . (7')
When ~ is restricted to be an element co E S 2, we get,
02 ( R A f )(co, s) = -~s2s2 R f (co, s); (S)
0 2 In other words, R intertwines A and ~ when the arguments are restricted to S 1 • R. Another useful property is the following:
P ~ ( f 9 g) = P ~ f | P~g, (9)
where the symbol 9 on the left side of (9) denotes the convolution in R 2 and | denotes the convolution product in R. The easiest way to verify (9) is via the Fourier Slice Theorem, which we recall here:
Let ~1 denote the Fourier transform of a function in R and f or F2 the Fourier transform of a function f in R 2. Then
.T'l(P~f)('r) = f(Tco). (10)
oo
= f e-2'm"f( tco + scoX)dsdt, R 2
Letting now x = tco + sw • one has t = x- w and dtds = dx, the Lebesgue measure in R 2, in the previous equation we obtain
.T'I(/~/)(T) = f e-2"(:"')'V(z)dz It.2
9 F'2(f * g)(~) = ](~)~(~),
we can easily prove (9). Indeed, from (10) we have
~I (P~( f 9 g))(.) = ~2( : 9 g)(.~) = / ( . ~ ) ~ ( . ~ )
= y l (P~/ ) ( . )~: l (p~g)(~) = ~:l(p~f | p~g)( . )
and therefore, by the injectivity of 5vl, we get P ~ ( f 9 g) = P ~ f | Let us also note that if Ta denotes the translation by a, i.e., T , f ( x ) = f ( x - a),
then
R ( 7 - , ~ f ) ( w , p ) = R i f ( x - a ) ] ( w , p ) = P ~ f ( p - w . a) = % . a R f ( p ) .
We now proceed to state some inversion formulas, which give different ways to recover f from R f .
F o u r i e r I n v e r s i o n F o r m u l a :
oo
(nj)O-)d~ (11) 0 $1
The proof is clear, we begin with the Inversion Theorem for the Fourier transform. We have
f (x) = [ R2
oo
We now apply the Fourier Slice Theorem to get
oo
0 wES1
This inversion formula can be implemented numerically using the Fast Fourier Transform (EFT) (see [Na]). Quite often the points T w where the data J : l ( R ~ , f ) ( ~ - )
is known do not have a lat t ice structure. This causes problems for the F F T but we can use rebinning algori thms like [ST] to obviate this problem.
To obtain another kind of inversion formula we observe the following:
f i P~f(s)g(s)ds= fi ; S(sw + tw• - o o - o o - o o
Let x -- sw + tw • so that s -- x 9 w, dx = dtds, and therefore
- r 1%2
i.e., the adjoint of P~ is the operator R~ defined by
Rig(x) = 9(x. ~). (12)
We now consider for an arbi t rary function g(w, s), having the symmetry g(-w, -s) =
f Rf(w,s)g(w,s)dwds= fsl dw fi R~f(s)g(w,s)ds S l x R - ~
oo
(with the usual subst i tut ion, x = sw + tw • etc., we get)
= f d ~ f f ( x ) g ( ~ , ~ . x ) d 2 x $1 R 2
= /f(x)R#g(x)d2x. l:t2
(13)
The opera tor R # defined by (13) is known by the name of "backprojection operator" . Note, in this regard, tha t g(w, s) is a function of "lines" and that R#g(x) is its integral over all lines passing through x. It is easy to prove the following useful proper ty of the backproject ion operator
(R#g) 9 f = R#(g | Rf), (14)
where the convolution | in the second member clearly takes place in the second variable. This identi ty plays an impor tant role in the numerical inversion of the Radon transform.
Finally, we get to the following impor tant result:
2 9 f . (15) R # R f =
Indeed,
R#Rf (x ) = S R f ( w ' w ' x ) d w S~
= Sd,~z S f ( ( m - x ) m + s w • S1 --00
S~ 0
By set t ing y = s~o • s = lyl, dy = s&ods we get
R#RS(x) = ~ S ~S(x § y)<~y R 2
1 d = 2ST;--m~s(~,)y,
R2
1 1 (see e.g. [He2, page 134]), one deduces which is exactly (15). Since one has 2 ~ = tha t
m2(n#n/)(~) = ~](~)
One can therefore conclude tha t the inversion operator A is such tha t
X(~) =
'S f(x) = ~ e2~'~l~l(R#Rf)~(~)d~ = AR#Rf(x) , (16) R2
which is sometimes called the backprojection inversion formula. Reorganizing the terms in the last formula one can rewrite it to obtain a more
s tandard form, where the filtering is one dimensional.
10
where
Hg(s)= l-Tr ? f(~t-)tdt' - - 0 0
where the last integral is understood in the sense of principal value. In other words,
1 (Rf)'(O,t_)dtdO (18) f (~)- (2~)~ L, f x .O- t 9
R
No introduction to the Radon inversion formula can be complete without at least mentioning the inversion formula due to Radon, which among other things, is akin to the inversion formula for the hyperbolic Radon transform due to ttelgason, which will be mentioned below. Consider for a fixed x E R 2, the average of Rf over all lines at a distance q > 0 from x, namely, let
1 fRf(~,w.x+q)dw. Fz(q) := S 1
Radon found that
0
We refer to [GM] for the original 1917 paper and commentaries.
An approximate implementation of (17) can be given by using the Fourier inversion formula
71-
where, as above, w = (cos 0, sin 0) and
o(t) b
o o
1]
This last approximation constitutes a band limiting process, and it can also be ob- tained from (14) as follows: Let wb be a "band-limiter ' , i.e, supp(~b) C [-b, b] and Wb = R#wb. Then (by letting g = wb in (14)), we obtain
Wb * f = R#(wb | R f)
that is, we want Wb to be an approximate di-function (cf. [Na, Ch. 5]). To begin with, choose Wb radial, e.g.,
27r " b "
where 0 < •(a) < 1, + = 0 for a > 1; this implies that wb = const. I~l+(~b ), The previous example is given by the ideal lowpass filter defined by
and so
1 i f 0 < a < l ~ ' = 0 i f a > _ l
Wb(x) lbzJ~(blxl) = 2 ~ (bill) '
where J1 is the Bessel function of the first kind and order one. We shall see below that one of the wavelet-based inversion formulas is inspired by
(20). The formulas (15), (16), (17) allow for rather precise estimates of the degree in
which the Radon transform and its inverse preserve the smoothness of the function f and data Rf. One way to measure this is to do it using Sobolev norms defined in an obvious way in the space of functions in the space of all lines. For instance, if f E C ~ ( B ) , where B is the unit disk in R 2, then one can find in [Na, Theorem 3.1], that for any real a, there are constants c, C > 0 such that
cllf[lH~'(m) ~ Ilnf[[~.§ ~ CIIflIH~'(B) (21)
In the particular case of a = 0, we see that for f E L0~(R 2) and supp(f) CC B one cannot expect better than control of one-half derivatives of Rf.
Useful variations of the estimates (21) for other kinds of Radon transform can be found using that R#R is a Fourier integral operator of elliptic type [GS].
12
3. L o c a l i z a t i o n o f t h e R a d o n t r a n s f o r m
Returning to the problem of p lasma physics that s tar ted these lectures, besides the fact t ha t the functions we are t rying to detect seem to have a very large variation, that is, a large H 1 norm, we have the added difficulty that the amount of da t a one can process or send down to Earth is fairly limited. One knows experimental ly that , on a first approximat ion, all the variations from being Gaussian occur in the region where the values of f have gone down by 4-5 orders of magnitude. That is, if f has a value 1 for the bulk velocity, then we are interested in the region where the values of f lie between 10 -4 to 10 -5 . The way this problem was t radi t ional ly solved (for conventional measuring devices) was to assume f had the form of a linear combination of a small number of Gaussians, and one just tries to est imate the variances and coefficients of these per turba t ion terms. If one does not want to impose these a priori restrict ions on f , and we have only l imited amount of da ta to use, a natural idea is to jus t use those lines tha t cross only the annular region where the main Gaussian varies between 10 -4 and 10 -s . (In the case of dimension 3 we would be dealing with a shell instead of an annulus.) This requires a localization of the Radon Transform. There are two ways to proceed. One, the most obvious (or naive) way is to try to localize the Radon transform as follows:
Reconstruct a function f in a disk B(a, r) from the data Rf(g), using only lines g passing through B(a, r).
This cannot be true in dimension 2, as observed already by F. John. The reason is the well-known fact tha t waves cannot be localized in 2-dimensions, namely, if we drop a pebble in the water, the ripples propagate along ever-expanding disks with time. In other words, an arbi t rary per turbat ion confined to a disk at t ime t = 0 does not necessarily remain confined to the same disk (or any concentric disk, for tha t mat te r ) at all future times. On the other hand, as F. John pointed out [J], if u is a solution of the wave equation Au - utt = 0, then its Radon transform v = Ru is a solution of the one dimensional wave equation vss - vu = 0, as it is seen immedia te ly from the relat ion (8). For the one-dimensional wave equation with initial conditions at t ime t = 0, v(s, O) = Vo(S) and ~ 0) = vl(s) , we have
1 I r~+t v(8, t) = (vo(s - t) + vo(s + t)) + I
S--t
If we th ink the pebble as being given by u(x ,0) = no(s) = 0 for {x I _> c, 0 < ~ < < 1, and ut(x,O) = ul(x) =--- O, then for a fixed w E S 1 and any later t ime t > 0 we would have with v(s,t) = R(u(.,t))(w, s) that vo(s) = v(s,O) = 0 for {s I > e while vl(s) = vt(s, 0) - 0. Thus, at any later t ime t > c we have that v(s, t) is only different from 0 for t - e < Is{ < t + e. Thus, the strict localization of the Radon transform would impose tha t the support of u(x, t) be in the annulus t - ~ < {x I < t + e, which contradicts our observations. (Nevertheless, we shall see that some sort of localization takes place.)
The other al ternative, which fits the p lasma problem, is to try to see whether we could reconstruct the values of f outside of a disk from the values of Rf(g), with never crossing tha t disk. This turns out to be possible! It is the exterior problem for the Radon transform. We follow here the work of Quinto [Q]], [Q2] (and references therein.)
13
The starting point, as recognized in the pioneering work of A. M. Cormack, is to expand both the function f and its Radon transform g = Rf in a Fourier series. (For R n, n > 2, one uses spherical harmonics [Na, p. 25 ft.]) That is,
f(~) = ~ f~(r)~ ~~ x= (rcosO, rsinO),
l~--oo
Then, the Fourier coefficients ft and g~ are related by the two formulas
oo 2
O0
:~(r) -- ~ ~i ~ - ~)-~%i(~)d,, (23) T
where T~ is the Chebyshev polynomial of the first kind. One of the consequences of the Fourier Slice Theorem is that g cannot be an arbitrary function in the space of lines, it must satisfy certain compatibility conditions, usually called the moment conditions,
j s'~-lg(w, s)ds e span{e 'k~ Ikl < m}, w = e ~.
This allows for a modification of (23) that makes it far more practical for numerical purposes [Na, p. 29-30]. This pair of equations show that the values of Rf(i) over all lines exterior to the disk B(0, r) are thought to determine f in the exterior of B(0, r). In particular, if one has
supp f C B(0, P0),
then the values of f in the annulus Pl < Ixl < P0 are entirely determined by the mea- surements of the Rf(~), only for lines ~ that intersect this annulus. (The uniqueness of the exterior problem and its variants is usually called the support theorem. It was first proved by Helgason in 1965, we refer to [Hell, [He2] for details and generalizations.)
Quinto [Q1], [Q2] has successfully used this kind of ideas to obtain a very effective tomographic algorithm to determine cracks in the exterior shell of (usually large) circular objects, for instance, rocket nozzles. The method of Quinto is based on two things. First, the known characterization of the kernel of the exterior Radon transform in L 2 spaces with convenient radial weights (this is due to Perry for n -- 2 and Quinto for n > 3). For the case of interest at hand, n = 2, we consider the kernel of the exterior Radon transform in L2(B~, rdx), then the Fourier coefficients f~(r) must be given by the rule
fe(r) = linear combination of r 2-k, 0 ~ k ~_ I~1, [~[- k even. (24)
For instance,
14
fo = O, f l = O , / 2 = c r - 2 , f3 = c r - 3 , f4 = c l r - 2 + c 2 r - 4 , . . .
The second observation is the fact that the Radon transform maps/-/1 := L2(B[, r(1 - r)U2dx) into H2 = L2(S ~ x (1, cr g -~) and one has an explicit diagonalization procedure for R, so that there are orthonormal bases ~j and Cj, respectively of / /1 @ ker R and of ImR c H2, so that
R~j = gj~bj with aj > 0
and aj explicitly computable. Thus, for a given f of L2c(B~, dx) (i.e., of compact support), we have
oo
f = ~ , aj~j + ] with ] E ker R N n2c(S~, dx) j - - 1
so that
1 aj = - - < R f , Cj >H2
~j
This determines exactly ~ = f - ] . One expands now f in a Fourier series ~ f~(r)e ~e, with f~ of the form (22) as mentioned earlier. Now for r > > 1 we know that f -- 0,
so that f = -~ , thus, for r > > 1 we have f~(r) = - ~ e-~~176 but the --Tr
coefficients f~(r) are polynomials in 1/r, so they are completely determined everywhere (up to r = 1) by their values for r > > 1. It is here that one uses a sort of analytic continuation, so it is fairly unstable, but Quinto has modified further this algorithm if one assumes f to be known in the small annulus 1 < Ix] < 1 + e, to give it further stability [Q2].
In the context of the plasma problem, we compared numerically the use of the same number of data measurements Rf(e), either spread throughout the whole disk versus the measurements taken only (and thus more densely) in the annulus of interest. We found the surprising result (to us) that the standard algorithm, with more thinly spread measurements did better. It was this numerical observation that led to the search of a different way to localize the Radon transform using wavelets.
Let us first review briefly two other localization methods that had appeared earlier in the literature.
The first one is the following. Let us assume that the unknown function f has support in the disk B1 of center 0 and radius 1, but that we are only interested on the values of f in Bb, 0 < b < 1, while we collect data on Ba, 0 < b < a < 1. (Note that all the disks are centered at 0). This is the situation considered for the interior Radon transform [Na, VI.4]. The basic idea is to make Rf(g) = 0 when e doesn't intersect B , and apply the standard reconstruction algorithm. In other words, we
]5
want to obtain (even approximately) the values f(x) , Ixl S b, from R(fxa), where Xa stands for the characteristic function of the disk B~. The first problem is that there are many non-zero functions f that have R(fx~) -- 0. Luckily, these functions do not vary much on Bb [Na], so one could just try to find f up to an additive constant (and try to find that constant by other means). One can see from the table or the formula (4.4) in [Na, p. 170], that one needs a = 4b to obtain a maximum L ~ error of 1.6% of the L 2 norm of f in B1. In particular, this procedure could not be applied if we are interested in f(x), for x E B(xo,a) C B1 with x0 close to cOB1. A typical such example is that of spinal chord studies. Usually, one study involves 40-60 CAT scans, that is, 40-60 scans along body sections perpendicular to the spine at different heights. The spinal chord area is about 15% of any such cross section of the body, and there would be a substantial reduction of radiation received by the patient if one localizes the CAT scan to only those lines passing through or near the spinal chord area.
Another alternative that has been proposed is that of A-tomography [FRK], where one only attempts to reconstruct to discontinuities of the function f , i.e., perform edge detection in the image. The principle is based on the formula (15) namely, consider the "approximate" inversion
] = AR#Rf ,
so that / = 4~rAf. This formula preserves the "edges" (= discontinuities of f ) but not the actual values of f . A variation of this formula has been implemented in the Mayo Clinic to study angiograms [FRK].
Another interesting consequence of this kind of approximate formula is that it can also be applied to the attenuated Radon transform,
o o
R,f (w, s) = / f(sw + tw• § tw • ?1))dr -oo
where #(x,w) is assumed to be real analytic in R 2 • S 1 and nonnegative. This appears in SPECT tomography and, usually, both f and # are unknonwn. As observed by Kuchment and collaborators [KLM] the function A R # R j will have the same singularities as f . The point is that R#R, is still an elliptic Fourier integral operator. This fact had already been used effectively by many people, most notably Boman and Quinto [BQ], and it is the key observation in the work of Quinto [Q3], Ramm and Zaslavsky [RZ], and others.
The method of localization we want to discuss here with a bit more detail is that of using wavelets to invert and localize the Radon transform in dimension 2. This general principle, which is joint work with David Walnut, was presented first in a 1990 NATO conference [W], and independently in [Ho]. Since then, similar ideas have appeared elsewhere in the literature (see, for instance, the recent volume [AU], the papers [BWl], [BW2], [DB], [DO], [O], and references therein.) True localization using discrete wavelets and filter banks is clearly developed in [FLBW]. (See also [FLB] for the fan beam case.)
There are many excellent books on the subject of wavelets, at all levels of so- phistication and different points of view, the following is a very partial list [M], [D],
16
[Ka]. There are actually two different, albeit related concepts, the continuous wavelet transform (CWT) (easier to understand) and the discrete wavelet transform (DWT) (easier to work with).
The idea of CWT originates from the standard properties of the Fourier transform representation of nice functions. For f 6 L2(R) or f E S(R), we have both
f(~) = f f ( x ) e-2"ix~dx - - 0 0
O0
II/II = IIfII : ( If(x)12dx) ~/2 - o o
If we translate f by b 6 R, Tbf(x) := f ( x -- b), then (vbf)'(~) = e2"ib~/(~), and for dilations we have D~f(x) := -~af(x/a)(a > 0), so that IIfJI2 = IID~f[12 and
(D~f)'(~) = D(1/~)/(~).
In other words, the group x --+ ax + b (a > O, b q R) operates via unitary operators in L2(R), and has a corresponding representation on the space of Fourier transforms (which happens to coincide with L2(R)). The "problem" of the Fourier transform rep- resentation is that the behavior of f at a point ~ depends on the values of f everywhere, for that reason, the idea of a "windowed" Fourier transform has been introduced long ago, namely, introduce a cut-off function g (say, a "smooth" approximation of X[-1,1]) and consider
9 T'l((Tbg)f)(~) = f g(x -- b)f(x)c-2"i~dx. - o o
Note that 5rl((7-bg)f)(~) is r 9 f(~), where r is the wavelet r = g(x)e2~iz~,(b(x) = r If we want to consider also the behavior at f at different scales we are led naturally to the CWT: Given a wavelet r 6 L2(R), and f 6 L2(R) we define
4- A ../+
-oo
for 0 < a < cx), b 6 R, r denotes the complex conjugate of r and <, > denotes the L2-scalar product. We assume the wavelet is "oscillatory", that is, it is an arbitrary function in L2(R) which satisfies the condition
j I (012 _ cr :=-oo - - - ~ a r < oo.
17
OO
This condition implies that f r = O. (For instance, when r is continuous at - - C O
= 0, which occurs if r E LI(R) N L2(R).) In fact, later on we will be interested in wavelets with many vanishing moments
f xkr = O, - - 0 0
A typical wavelet is the Haar wavelet
0 < k < N .
so that
D1/2r = v~(X[0:/4] - X[1 /4 ,1 /2 ] )
which shows that for k -+ co, D2-~r "analyzes" smaller and smaller details of the "signal" f .
Moreover, Wcf determines f as seen from the following relation valid for any pair f ,g C L2(R)
j f dadb Wr = cr < f,g > [1r - - 0 0 - - 0 0
usually called Calderon's identity. If 11r = 1 one also has the L2-approximation property
1 b) DaTbr I I (26) Ilf-~ f Wcf(a, ---~0 A l ~ l a l < _ A 2
Ibl_<B
as A2 ---+ 0+,A2 ~ +co, B --+ +co.
The generalization to R ~ is easy. A function r E L2(R ~) is a wavelet if
For a radial wavelet r E L2(R ~) and f C L2(R ~) we define the CWT by
Wr f o r a e R \ ( 0 } , b C R ~,
where this time, D~r = lal-~/2r The interest of the CWT for tomograpy lies in the following two propositions from
[BW2].
P ropos i t i on 1. Let p r L2(R) be real valued, even, and satisfying
OO ^ 2
18
Define a radial function r in R 2 by 3v2r = 2~([~1)/1~1, then r is a wavelet a n d
w,d(a, b) = a-'/2 f (w,P~/)(a, b. o) ) do3 (28) S a
P r o o f . Using the Fourier Slice Theorem we have for "y 9 R
It is then easy to verify that (27) implies that r is a wavelet in R 2. Recall that the Riesz transform of order a, I"~ , of a function ~ 9 S ( R ) is defined
by ( I~) ' (9 ' ) -- h ' l -~ (9 ' ) , thus the identity (29) can be rewritten as
p(t) = l : - ~ ( P ~ r
Extend p to a function in the space of lines by making it independent of the slope of the line, p(w, t) = p(t) for every w 9 S ~, then we have
= 8 9 1 6 2 = r R#p(x)
since the last formula is a rewritting of formula (16) in terms of the Riesz transform. More generally, for any a > 0 and every w E S 1, we have
(R#Dapo,)(x) = aV2Df,(b(x),
so that, using identity (14) and the fact that p is real valued, we obtain
Wcf(a,x) = (f.D.(p)(x) = a-I/2(f. R#D~,:.,)(x)
= a-~/~R#(P~f | D~
= a - 1 / 2 / ( W p P ~ f ) ( a , x . w ) d w S 1
This concludes the proof of the proposition. 9
A similar relation between the Radon transform and the CWT can be found using "separable" wavelets in R 2.
P r o p o s i t i o n 2. Given a separable 2-dimensional wavelet of the form
r = r162
where each r satisfies Ir < (71(1 + 171) -1 for all 7 9 R, define the family of one-dimensional functions {P~}~es' by
1 ^1 ^2
where w = (wl,w2) 9 S t. Then, for every f 9 L I ( R 2) VI L2(R2),
19
0.,
0.6
0.4
0.2
C
-0.2
-0.4' -10
G a u s s i a n a n d I ts H i l b e r t t r a n s f o r m
'\] i,j/ I I I I I I J I I
~8 --6 --4 --2 0 2 4 6 8 10
Figure 3: Gaussian and its Hilbert transform.
( W r = a -1/2 J f)(a,x . ~ ) d ~
s 1
The point of Proposition 2 is the observation that the wavelet transform of a function f ( x ) with any mother wavelet and at any scale and location can be obtained by backprojecting the wavelet transform of the Radon transform of f using wavelets that vary with each angle, but which are admissible for each angle.
So far we have not yet shown that the inversion formulas of the Radon transform based on wavelets do a good localization job. Using Proposition 1 the problem is clear, find a function p such that p has small support and simultaneously r has small support. From the relation (29) we see that we have overcome the Reisz operator of order - 1 , its symbol is 171 = (sgnv)v, so it is the composition of the differentiation and the Hilbert transform. (This is exactly the content of the inversion formula (18).) The problem, of course, is the Hilbert transform, but if we choose p with many vanishing moments, then we can overcome the difficulty. For the sake of comparison we show in Figure 3 the Hilbert transform of a Gaussian, its effective support is about four times the effective support of the Gaussian (defined by making zero those points below 1% of maximum value), which tails exactly with the result about the interior Radon transform mentioned earlier in this section.
The key to explain the success of the wavelet method of localization is the following proposition [BW1], which in spirit is similar to the general principles about Calderon- Zygmund operators stated in [BCR].
P r o p o s i t i o n 3. Suppose that n is an even integer and the compa~iy supported function h E L2(R) is such that for some integer m >_ 0 we have that h is n + ra - 1 times differentiable and satisfies
20
9O
80
~, 70
10 i L 20 40 60 80 10~ 120 140
Radius of the region of interest in pixels
Figure 4: Exposure versus the radius of the ROI.
(a) 7jh(k)(7 ) E L~(R) N L2(R) for 0 __ j _~ m, 0 < k < m + n - 1
(b) ~ tJh(t)dt = 0 for 0 < j _< m - - o o
Then
and
t~+m-1II-~h C L2(R).
The proof is rather elementary, it depends on the fact that if h is a function of compact support with m + 1 vanishing moments then 171~-1h(7) has n + m - 1 continuous derivatives.
For ease of application it is better to work with the discrete wavelet transform (DWT). This is basically obtained by diseretizing the CWT or appealing to the mul- tiresolution analysis of Mallat and Meyer [D], [M]. We have done this in detail in [FLBW] using coiflets [D] in order to be able to implement the inversion process using filter banks. One can show that to obtain a relative error of 0.5~o one only needs a margin of security of 12 pixels around the region of interest (ROI). For instance, to recover within this error bound an image occupying a disk of radius 20 pixels in a 256 • 256 image, one only needs about 25% of exposure, as shown in Figure 4.
Figure 5 below is the Shepp-Logan phantom and its reconstruction from global fan beam data using the standard algorithm, in Figure 6 we use local data and our wavelets algorithm.
The following figures are the reconstruction of a heart from real CAT scanner data using our wavelet method, and the reconstruction of the central part from local data and our wavelet method is found below.
21
(a) (b)
Figure 5: (a) The Shepp-Logan head phantom; (b) the standard filtered backprojec- tion in fan beam geometry (4).
Figure 6: Reconstruction from wavelet coefficients.
22
Figure 7: Reconstruction of heart from wavelet coefficients.
Figure 8: The local reconstruction of of central portion of heart.
23
We leave to the discussion and references in [FLBW] and [BW2] the comparison with other methods of inversion of the Radon transform using wavelets. One should add to the references in those two papers, the very recent work of Rubin [R], which is based on a systematic use of the Calderon reproducing formula and it is thus a development of the original ideas in [Ho].
24
4. T h e hype rbo l i c R a d o n t r a n s f o r m and Elec t r ica l I m p e d a n c e Tomography
In this section we discuss the role tomography plays in a classical problem of Ap- plied Mathematics, the inverse conductivity problem. Several of the earlier attempts to solve this problem involve generalizing the Radon transform to other geometries, that is, integrating functions over other families of curves beyond straight lines in the Euclidean plane. There are many examples of such transforms, in fact, the integration over great circles in S 2 was a transform considered by Minkowski and which inspired Radon in his work. The two we shall introduce presently are the generalized Radon transform of Beylkin [By] and the Radon transform on the hyperbolic plane [He1].
Let ~ be an open subset of R 2 and r E C ~ ( ~ • (R 2 \ (0))) be such that
(a) r A~) ---- Ar ~) for )~ > 0
(b) V=r r 0 for all (x,~) E ~ x (R 2 \ {0})
Then, for any s E R and w E S ~ we can define the smooth curve
= {x e a : r = s} ,
that is, the level curves of ~b. We let da denote the Euclidean arc length in such a curve. For u E C~(~) define the "Radon transform"
R+u(~,s) = f u(x)l V: r 5)ld,~(x) / ' / s ,~
Let h(x,~) be the Hessian determinant of r with respect to the second variables, 0 2 x h(x, ~) = d e t f ~ l then the "backprojection" operator Rr # is defined by
h(x,~) R v(x) = f I
w E S 1
Introducing K as the operator of convolution by 1/Ixl, Beylkin proved the following approximate inversion formula for the Radon transform as an operator
R+: Lc2(a) --+ L~or
R#cKRr = I + T (30)
T : L~(~) -+ L~or )
is a compact operator. In fact, Beylkin gives a recipe for a family of backprojection operators and generalized convolution operators K so that a decomposition of the type (30) holds. This gives his transform great flexibility and applicability to many problems, especially inverse acoustic problems, of course, the reader can easily verify
25
that for convenient choices of r the transform Re yields the Euclidean Radon trans- form studied earlier and the hyperbolic one, which we now introduce. (The reader should consult [Hell, [He3] for more details on this subject.)
Let D, the unit disk of the complex plane C, be endowed with the hyperbolic metric of arc-length element ds given by
4'dzl~ (31) ds2 - (1 -Iz[2) 2'
where Idzl denotes the Euclidean arc-length element. This metric is clearly conformal to the Euclidean metric but has constant curvature
-1 . The geodesics of this metric are the diameters of D and the segments lying in D of the Euclidean circles intersecting the unit circle COD perpendicularly. One can introduce geodesic polar coordinates z ++ (w, r), where w = z / H , r = d(z, 0). Note that Iz[ = tanh(r/2). In these coordinates the metric (31) can be rewritten as
ds 2 = dr 2 + sinh 2 r dw 2
where dw 2 indicates the usual metric on cOD. The hyperbolic distance between two points is given by
Iz- ! d(z ,w) arcsinh ((1 -I~I~),~(I -l l )V
The Laplace-Beltrami operator AH on D can be written in terms of the Euclidean Laplacian A as
a . - 0-1zP)2a 4
02 O 02 - Or 2 + c o t h r + s i n h - ~ r o w 2. (32)
The classical Moebius group of complex analysis is the group of orientation preserving isometrics of the hyperbolic plane D.
One can define the hyperbolic Radon transform RH by
Rf( '~) = RHf (7 ) = ~ f ( z )ds ( z ) , 7 geodesic in D (33)
which is well defined for, say, continuous functions of compact support, or functions decaying sufficiently fast. Observe that to be integrable on the hyperbolic ray [0, oc[ (which is just the straight line segment from 0 to 1 in the complex plane C), f has to decay a bit faster than e -r. We denote by F the space of all geodesics in D, then the dual transform R # (or backprojection operator) is given by
R#r -- fr~ r (34)
where Fz is the collection of geodesics through the point z and dttz is the normalized measure of Fz. Since a geodesic through z is determined by its starting direction w C S 1, then Fz ~ S 1 and d#z is naturally associated to ~ d w when we use this particular parameterization of Fz.
26
In order to invert R H one can proceed in the spirit of Radon's inversion formula (19). This was done by Helgason [He2, p. 155]. Or one can try to find a filtered backprojection type formula like (16). For that purpose we need to define convolution operators with respect to a radial kernel k. For k E L~oc([0, cxD)) and f E Co(D) we define
f
k * f (z) = k *H f(z) := ]D f(w)k(d(z, w))dm(w) (35)
where dm(w) stands for the hyperbolic area measure, which in polar coordinates is given by
d m = sinh r drdw.
Corresponding to the Euclidean formula (15) we have
R#HRHf : k * f, where k(t) - 1 Tr sinh t
One can prove [BC1] that if
(36)
which is the exact analogue of (16).
It is convenient to recall here that in the hyperbolic disk D we have a Fourier
transform [He2]. It is easier to work it out for "radial" functions as we interpret our kernel k, then the Fourier transform is defined with the help of the Legendre functions P~(r) by means of the following formula
o o
For radial functions k, m, we have
(k * m)'(1) : k(~)~(~)
So that, if/~(A) # 0 for all I E R, in principle, that is, for a convenient class of functions f , the convolution operator f , > k *H f is invertible.
We refer to [He2], [BC1], [BC2], [Ku] for corresponding inversion formulas in the higher dimensional hyperbolic spaces, and the characterization of the range of the Radon transform. In particular, [Ku], [BC2] exploit the "intertwining" between RH and the Euclidean Radon transform as well as the Minkowski-Radon transform on spheres.
Let us cxplain now what the above hyperbolic Radon transform has to do with Electrical Impedance Tomography (EIT) and what EIT is.
Let us consider the following tomographic problem: using a collection of electrodes of the type used in electrocardiograms (EKG) uniformly distributed around the breast
27
of a patient and all lying in the same plane, introduce successively (weak) currents at each one of the electrodes (as done in EKG) and measure the induced potential at the remaining ones. The objective is to obtain an image of a cross section of the lungs to determine whether there is a collapsed lung or not. This was what Barber and Brown set up to do in 1984 [BB1], [BB2]. The point being that this equipment is cheap, transportable and provides a non-intrusive test (that is, no punctures have to be done to the chest cavity). Similarly, one can try to determine the rate of pumping of the heart using this kind of equipment. Notc that the pulse only determines the rate of contracting and expanding of the heart but not how much blood is being pumped by it. Another completely different problem arises in the determination of the existence and lengths of internal cracks in a plate, by using electrostatic measurements on the boundary [FV], [BCW], [W]. These three are examples of the following inverse problem. (The best reference for the general facts about this problem is the supply [SU]. See also the nice explanation for the general public [C], [S]):
Assume/5 is a strictly positive (nice) function in the closed unit disk D. If we were to introduce a current at the boundary OD, represented by a function ~ satisfying fOB Cds = 0, then the Neumann problem
div (/3 g radu) = 0 i n D (39)
/3~ = r on OD
has a solution u which is unique up to an additive constant. If r is a nice function then ~ (that is, the tangential derivative of u) is well defined on OD, so we have the input-output map
0u
which is a linear continuous map from the Sobolev space H~(OD) into itself. (This statement holds for any domain D with nice boundary, not just the disk.)
Consider now the (very non-linear) map
/3, >A/~ (40)
is it injective? Can one find the inverse to this map? This problem was originally posed by A. Calder6n, who proved that (40) was locally invertible near/3 = constant, more recently Nachman IN1], IN2] proved global invertibility. Since/3 is usually called the conductivity and 1//3 the impedance, this is the reason for the name EIT of this inverse problem. In the biological applications we know the value/3 for the different constituents like blood, lung tissue, etc., so one only looks for a profile of the areas occupied by them. In the determination of cracks, one can assume/3 "known", except for curves where/3 = 0, and one wants to determine this curve, or whether any exists. One can find in [SU] many important inverse problems that are equivalent to EIT: in acoustics, radiation scattering, etc. Note that in the problem of the rate of pumping of the heart, we can think that all we want to determine is just a single number, this rate. Isaacson, Newell and collaborators have in fact patented [C], [I] a device that measures this rate with the help of EIT. We also know that this problem, being an inverse elliptic problem is very ill-conditioned, so in any case one is willing to restrict oneself to find the deviation of/3 from an assumedly known conductivity/30. In the simplest case we assume/30 --= 1, so that/3 = 1 +5/3, 15/31 < < 1, and we further assume
28
5/3 = 0 on OD (One can always reduce matters to this case). Thus u = U + 5U, where U is the solution of (39) for the same boundary value, and/3 = 1. In other words
AU = 0 in D (41) OU = r on OD
Here A is the Euclidean Laplacian. The perturbation 5U then satisifes
A(SU) = - < grad (&3), grad U > in D o(~u) (42)
o~ -- -(8/~)~b on OV
We have at our disposal the choice of inputs r Their only restriction is that rOD ~bd8 : O. For that reason, they can be well approximated by linear combinations of dipoles. A dipole at a point oo c OD is given by o
It turns out that the solution Uw of
{ AU~ = 0 i n D (43) -r~~ on cOD
has level curves which are arcs of circles passing through w and perpendicular to 0D. That is, the level curves of U~, are exactly the geodesics of the hyperbolic metric. This fact passd unnoticed to Barber and Brown but they definitely realized that the value
O(SU) (44) # - Os
at a point a E OD must be some sort of integral of 53 over the level curve of U~ that ends at a, precisely the geodesic starting at w and ending at a. In other words, # is a function in the space of geodesics in D considered as the hyperbolic plane, all the geodesics are obtained this way by changing w and a. Without expressly stat- ing this, Barber and Brown introduced a "backprojection" operator that turned out to be exactly R#H and gave the approximation to ~ as R#H#. Santosa and Vogelius recognized explicitly that some sort of Radon transform was involved and used the generalized Byelkin transform and a convenient choice of K in (30) to stabilize nu- merically the inversion of EIT. Casadio and I, prodded by a question of Santosa and Vogelius, saw that RH was involved and developed the inversion formula (38) for this purpose. As it turns out, all of these approaches are just approximations to the lin- earized problem. Only in [BC3], [BC4], we realized the fact that the exact formulation of the linearized problem in terms of hyperbolic geometry requires also a convolution operator! Namely, let
~r = c~ - 3 cosh-4(t) (45) 8 r
and # the boundary data (44) considered as a function on the space of geodesics in D, then one has that the exact relation between 5/? and # is given by
RH(a *H ~/~) = # (46)
R# # = R# RH(g *H ~1~) (47)
29
1 G a H ( S *H (n~,,)) = ~ *H 6fl (48)
which requires to invert the convolution operator of symbol ~. One can compute its hyperbolic Fourier transform k exactly and find out that k(.~) :fi 0 for every A E R, so that the operator ~. is, in principle, invertible, but the numerical implementation of this inversion has proven difficult so far. (Although Kuchment and his students have made in [FMLKMLPP] some progress towards implementing a numerical Fourier transform in D, which we hope will prove useful to compute 6ft.) One can recognize in (47) and (48) the same principle that lead to the numerical approach in [BB1], [SV] and others. Due to the importance of this problem there have been many other interesting approximate inversion formulas, under special assumptions on the conductivity fl, for instance, fl is "blocky", that is the linear combination with positive coefficients of a finite number of disjoint squares [DS]. Their approach is variational, and one may wonder whether one could not use some version of the Mumford-Shah edge detection algorithms [MS] to obtain a rather sharp solution of the inverse conductivity problem (40).
5. F inal r e m a r k
The objective of these short notes (and the corresponding CIME course) was only to indicate how, beyond the well-known applications of tomography to Medicine, there are many other possible ones. Moreover, even to solve them approximately, they require deep mathematical tools, showing once more that the applicability of "pure" and "abstract" mathematics is not a fairy-tale but a concrete reality. It also indicates that it pays to "invest" one's time trying to communicate with those, be they physicists, or physicians, etc., that have the ready made applications. A lesson often lost by graduate students in Mathematics.
30
6. References
[AA] S. Andrieux and A. Ben Alda, Identification de fissures planes par une donn~e de bord unique, C.R. Acad. Sci. Paris 315 I(1992), 1323-1328.
[AU] A. Aldroubi and M. Unser, editors, "Wavelets in Medicine and Biology," CRC Press, 1966, 616 pages.
[BB1] D. C. Barber and B. H. Brown, Recent developments in applied potential, in "Information processing in Medical Imaging," S. Bacharach (ed.), Martinus Nijhoff, 1986, 106-121.
[BB2] D. C. Barber and B. H. Brown, Progress in Electrical Impendance Tomog- raphy, in "Inverse problems in partial differential equations," D. Colton et al. (eds.), SIAM, 1990, 151-164.
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Institute for Systems Research University of Maryland College Park, MD 20742 carlos@src.umd.edu
H O L O M O R P H I C M A P P I N G S B E T W E E N R E A L
A N A L Y T I C S U B M A N I F O L D S IN C O M P L E X S P A C E
PETER EBENFELT
I00 44 Stockholm Sweden
1. Introduction 2. Preliminaries on algebraic mappings and CR geometry
2.1. Algebraic mappings 2.2 Real analytic CR submanifolds in C N 2.3 Finite type and minimality 2.4 Normal forms for generic submanifolds 2.5 The complexification of a real analytic submanifold
3. Holomorphic nondegeneracy, finite nondegeneracy, and re- flection identities for holomorphic mappings 3.1. Holomorphic nondegeneracy of real analytic CR sub-
manifolds 3.2. Finite nondegeneracy of real analytic CR submanifolds 3.3. Reflection identities for holomorphic mappings
4. The Segre sets 4.1. The Segre sets of a real analytic CR submanifold 4.2. Homogeneous submanifolds of CR dimension 1 4.3. Proof of Theorem 4.1.21 (CR dimension 1 case)
5. An application to holomorphic mappings between real al- gebraic submanifolds 5.1. A reformulation of Theorem 1.1 5.2. Proof of Theorem 5.1.1
6. Other applications and concluding remarks 6.1. The algebraic equivalence problem 6.2. Uniqueness of biholomorphisms between real analytic,
generic submanifolds
1. INTRODUCTION
In this paper we shall present, in fairly self-contained form, some recent ideas and concepts concerning real analytic submanifolds in C N. One of the main construc- tions described here is an invariant sequence of sets--called the Segre sets--at tached to a real analytic submanifold in cN; this sequence of sets was introduced in joint work by the author together with Baouendi and Rothschild [BER1]. The first Segre set coincides with the so-called Segre variety, introduced by Segre [Seg] and suc- cessfully used in mapping problems for real analytic hypersurfaces by a number of authors: e.g. Webster [W1], Diederich-Webster [DW], Diederich-Fornaess [DF], Huang [H2], and others. Subsequent Segre sets turn out to be unions of Segre varieties.
One of the merits of the Segre sets is that they allow one to analyze "reflection identities" (see section 3.3) for higher codimensional submanifolds--the idea of using reflection identities to analyze CR mappings goes back to e.g. Lewy ILl, Baouendi-Jacobowitz-Treves [BJT], Baouendi-Rothschild [BR1]. The Segre sets also allow a new characterization of the notion of finite type, as introduced by Bloom-Graham [BG]. In fact, the Segre sets provide a way of constructing the CR orbits of a real analytic CR submanifold without solving any differential equations (see Theorem 4.1.21). As a result, one finds e.g. that the CR orbits of a real algebraic CR submanifold are algebraic (Corollary 4.1.31).
As an application and illustration of these techniques, we shall prove the following result (which we shall also reformulate in terms of more c]assical CR geometry, Theorem 5.1.1) from [BER1].
T h e o r e m 1.1 ( [BER1]) . Let A C C N be an irreducible real algebraic set, and Po a point in A such that Po C Ares. Suppose the following two conditions hold.
(1) There is no hoIomorphic vector field (i.e. a vector field with holomorphic coe~cients and values in T I '~ which is tangent to an open piece of
Areg. (2) If f is a germ, at a point in A, of a holomorphic ]unction in C N such that
the restriction of f to A is real valued, then f is constant.
Then, if H is a holomorphic map from an open neighborhood in C N of po E A into C N, with Jac H ~ O, that maps A into another real algebraic set A' with dim A' = dim A, necessarily the map H is algebraic.
The first result along these lines goes back to Poincar6 [P] who proved that a biholomorphic map H: C 2 ~-* C 2 defined near a point on a sphere S C C 2 and mapping S into another sphere S' C (;2 is rational. This was later extended to mappings between spheres in C g by Tanaka ITs]. Webster [W1] then proved that a biholomorphic mapping H: C g H C N, defined in some open subset of C N, tak- ing a real algebraic, somewhere Levi-nondegenerate (i.e. with nondegenerate Levi form at some point) hypersurface M into another real algebraic hypersurface M ' is algebraic. Recently, Baouendi Rothschild [BR3] showed that if the real algebraic hypersurface M satisfies condition (1) of Theorem 1.1 above (which is a weaker condition than being Levi nondegenerate somewhere) then any holomorphic map- ping H : C N ~-* C N, defined in some open subset of C N and with Jac H ~ 0, taking M into another real algebraic hypersurface M I is algebraic. Moreover, they show
37
that this condition is also necessary for such a conclusion to hold in the sense that if (1) is violated then there is a non-algebraic biholomorphism of M into itself. The sufficiency of condition (1) in this result by Baouendi-Rothschild is contained in Theorem 1.1 above, because a real analytic hypersurface that satisfies condition (1) automatically satisfies condition (2). We would also like to mention that the conditions (1) and (2) in Theorem 1.1 are essentially necessary for the conclusion of the theorem to hold. We refer the reader to [BER1] for details on this (see also section 6).
Condition (1) was first introduced, and named holomorphic nondegeneracy (see section 3.1 for a detailed treatment of this notion), by Stanton [Stl] in connection with the study of infinitesimal CR automorphisms of real hypersurfaces. It deserves to be mentioned here that holomorphic nondegeneracy is fairly easy to verify be- cause it turns out to be closely related to another property (finite nondegeneracy, see section 3.2), which is very computational and is a direct generalization of Levi nondegeneracy.
Results of the type above for mappings between hypersurfaces in different di- mensional spaces have been obtained by e.g. Webster [W2], Forstneric IF], and Huang [H1]. For higher codimensional submanifolds, work has been done by e.g. Senkin-Tumanov [TH], Tumanov [Wu2], and Sharipov-Sukhov [SS].
Other applications to rigidity properties of holomorphic mappings between real analytic submanifolds will be briefly discussed in the last section of this paper.
The paper is organized as follows. In chapter 2, we give the basic definitions and facts, mostly without proofs, concerning algebraic mappings and CR geometry. More recent concepts such as holomorphic nondegeneracy and finite nondegener- acy, as well as reflection identities for holomorphic mappings, are introduced and discussed in chapter 3. The construction of the Segre sets and proofs of the main results concerning these are given in chapter 4. In chapters 5-6, applications of the techniques are discussed. A reformulation of Theorem 1.1 and a proof is given in chapter 5. Applications to uniqueness questions and some open problems are discussed in the final chapter.
A c k n o w l e d g e m e n t . The author would like to thank Professors M. S. Baouendi and L. P. Rothschild for agreeing to have results and arguments from our joint papers [BER1-3] included in these notes. As the reader will no doubt notice, the results presented here, for which the author can claim any credit, are due to this above mentioned joint work.
2. PRELIMINARIES ON ALGEBRAIC MAPPINGS AND CR GEOMETRY
2.1. A lgeb ra i c m a p p i n g s . We denote by ON(po) the ring of germs of holomor- phic functions in C g at p0, and by .AN (P0) the subring of ON (Po) consisting of those germs that are also algebraic, i.e. those germs for which there is a nontrivial poly- nomial P(Z,x) e C[Z,x] (with Z E C N and x C C) such that any representative f(Z) of the germ satisfies
(2.1.1) P(Z, f(Z)) =_ O.
In particular, any function in .AN(Po) (throughout this paper we shall, without comment, identify a germ of a function with some representative of it) extends
38
as a possibly multi-valued holomorphic function in C N \ V, where V is a proper algebraic variety in C N. We list here some basic properties of algebraic holomorphic functions that will be used in the proof of Theorem 1.1. We use the notat ion AN for AN(O).
L e m m a 2.1.2. The following holds:
(i) If f E `AN then O~" f E r for any multi-index a. (ii) I f f G `AlV and gj E ,AN with gj(O) = O, for j = 1, ..., I(, then
f (g l (Z) , ...,9K(Z)) E `AN.
(iii) (The Algebraic Implicit Function Theorem) Let F( Z, x) be an algebraic holo- morphic function near 0 in C y x C, i.e. F E .,4N+1, and assume that
OF F(o, o) = o , -g-;(o, o) # o.
Then there is a unique function f ~ .AN such that x = f ( Z ) solves the equation F ( Z , x ) = O, i.e.
F(Z, f ( Z ) ) = 0
The arguments needed to prove this lemma are standard (see e.g. [BM], and also [BR3] for further properties of algebraic functions), and the proof is omitted.
We say that a germ of a holomorphic mapping H : C N --~ C K at p0 is algebraic if the components of H (we write H = (H1,..., Hh')) are all algebraic. It follows from Lemma 2.1.2 (ii) that this property is invariant under algebraic changes of coordinates in C x and C K at P0 and p~ = H(p0), respectively.
2.2. R e a l a n a l y t i c C R s u b m a n i f o l d s in C N. In sections 2.2-2.5, we shall set up the notation, and give the basic definitions and results from CR geometry needed for subsequent sections. Most facts and results in these sections will be stated informally, and without proofs. Unless otherwise specified, proofs can be found in e.g. [B].
We should point out that only real analytic submanifolds will be considered. The definitions presented in these sections can be made in the broader category of smooth (e.g. C ~176 submanifolds, but some of the facts stated fail to be true in that general setting. For instance, the two notions "finite type" and "minimality" presented in section 2.3 coincide for real analytic CR submanifolds, but do not coincide in general for merely smooth CR submanifolds.
Let M be a real analytic submanifold in C N and P0 a point in M. Let m be the (real) codimension of M. We may describe M near P0 as the zero locus
(2.2.1) M = {Z c cN: p(z, 2) = 0},
where p = (Pl ..... Pro) are real valued, real analytic functions near P0 with linearly independent differentials dpl, ..., dpm; we use the notation h(Z, 2) for a real analytic function in C N to indicate that we think of such objects as restrictions to ~ = Z of
39
holomorphic functions of (Z, 4) 6 C N • C N. We say that M is real algebraic if the pj can be taken to be real valued polynomials in Z and 2.
The complex tangent space of M at p 6 M is defined as
(2.2.2) T;(M) = Tp(M) A Jp(TR(M))
where Jp: Tp(C N) ~ Tp(C N) denotes the complex structure in C N. The (real) dimension of Tp(M) is even and satisfies
(2.2.3) 2N -- 2m <_ d im~T; (M) < 2N - m.
If M is a hypersurface, i.e. a real codimension 1 submanifold, then d i m i T y ( M ) is necessarily 2N - 2. In general, the dimension varies with p 6 M.
D e f i n i t i o n 2.2.4. M is said to be CR at p G M if dim~Tq(M) is constant for all q in some neighborhood ofp in M.
We decompose d as d = 0 + c3. The real analytic submanifold M is CR at p 6 M if and only if the rank of the covectors Opl, ..., Opm is constant at all q 6 M near p. Hence, any real analytic submanifold M C C N is CR outside a proper real analytic subset V C M. We say that a connected submanifold is CR if it is CR at every point.
The complexified complex tangent space CTp(M) = C | T;(M) decomposes as
the direct sum T(I '~ + T(~ where
T(I '~ = CTp(M) A Tp(l'~ N) (2.2.5)
T(~ : CTp(M) N T(~
here, T( I '~ and T(~ N) denote the spaces of (1,0)-vectors and (0,1)- vectors in C N respectively, and CTp(M) denotes the eomplexified tangent space of M, G | Tp(M). If M is CR then
(2.2.6) T(~ = O Tp (~ pcM
forms a complex vector bundle over M, called the CTl bundle. Sections of the CR bundle are called CR vector fields. A function (or distribution) defined on (a piece of) M is said to be a CR function (or CR distribution) if it is annihilated by all the CR vector fields on (that piece of) M. The restriction to M of a holomorphic function defined in a neighborhood of M is CR, and any real analytic CR function on M is the restriction of such a holomorphic function. In general, there are smooth (C ~ ) CR functions on M that are not restrictions of holomorphic functions.
D e f i n i t i o n 2.2.7. M, of codimension m, is said to be generic at p E M if the dimension of T;(M) is minimal, i.e. d i m ~ T ; ( M ) = 2 N - 2m.
The real analytic submanifold M is generic at p 6 M if and only if the rank of the covectors Opl,...,Opm equals m at p. In particular, if M is generic, then it is also CR.
40
If M is CR, we call the complex dimension of T(~ (= d imiTy(M) /2 ) the CR dimension of M and the real dimension of T~(M)/T;(M) the CR codiraen- sion of M. For a generic submanifold M C C N, the CR codimension equals the codimension rn and
(2.2.8) N = CR dim (M) + CR codim (M).
Another important fact is that a real analytic CR submanifold M is generic if and only if it is not contained in a proper complex submanifold of C N. If _~4 is not generic, then there is proper complex submanifold 2( C C N (unique if we consider X as a germ of a submanifold) such that M is generic as a real analytic submanifold of X. We shall refer to X as the intrinsic complexifieation of M.
The Levi form of a CR submanifold M at p ~ M is the (vector valued) quadratic
form on T(p~ defined as follows
(2.2.9) T(p~ ~ Lp ~ ~r,([L,L]) E CTp(M)/CT;(M),
where rr, is the projection 7r,: CTp(M) ~ CTp(M)/CT;(M) and L is some section
of T(~ that equals Lp at p. A real hypersurface M is said to be Levi nonde- generate at p E M if the quadratic form (2.2.9) is nondegenerate. In this paper, we shall not impose any conditions on the Levi form of a CR submanifold. Instead, we shall introduce (in section 3.1) a weaker condition, which also turns out to be essentially necessary in most applications we shall consider.
We conclude this section by giving a few examples of generic, CR, and non-CR submanifolds.
E x a m p l e 2.2.10. A complex subrnanifold of C N is CR but not generic.
E x a m p l e 2.2.11. A real hypersurface in C N is generic.
E x a m p l e 2.2.12. Consider the real analytic 4-dimensional submanifold M C C 3 defined by
(2.2.13) I m Z 3 - l Z l ] 2 - l Z 2 1 2 = 0 , I m Z 2 = 0 .
If we write pl(Z, Z) = 0 for the first equation and p2(Z, 2) = 0 for the second, then it is easy to verify that
1 Z 21dZ1 Z~dZ2 , Op2 ldz2. (2.2.14) Off1 = ~ d 3 - - = 2
The rank is two and, hence, M is generic.
E x a m p l e 2 .2 .15. Consider the real analytic 2-dimensional submanifold M C C 2 defined by
(2.2.16) Im Z~ -]Z112 = 0 , Re Z2 = 0.
As above, we write pl(Z ,Z) = 0 for the first equation and p2(Z,Z) = 0 for the second. We find that
(2.2.17) Opl = ~dZ2 - 2~dZ~ , Op2 = dZ2.
Outside {Z~ --- 0} the rank is two, and along {Z1 --= 0} the rank is one. Thus, M is generic outside the origin but is not even CR at the origin.
41
2.3. F i n i t e t y p e a n d m i n i m a l i t y . Let M C C N be a real analytic CR subman- if old.
Definit ion 2.3.1. M is said to be of finite type at Po E M if the CR vector fields, the complex conjugates of the CR vector fields, and their commutators, evaluated at Po, span CTp0(M ).
Equivalently, M is said to be of finite type at P0 E M if the sections of TO(M) and their commutators, evaluated at p0, span Tp0 (M).
More generally, we define the Harmander numbers of M at P0 E M as follows. We let E0 = T~o(M) and Pl the smallest integer > 2 such that the sections of TO(M) and their commutators of lengths < #1 evaluated at P0 span a subspace E1 of Tpo(M ) strictly bigger than E0. The multiplicity of the first Hhrmander number #1 is then gl = dim~E1 - dim~E0. Similarly, we define/*2 as the smallest integer such that the sections of of TO(M) and their commutators of lengths </*2 evaluated at P0 span a subspace E~ of Tpo(M) strictly bigger than El , and we let ga = dim~E2 - dimt~E1 be the multiplicity of/*2. We continue inductively to find integers 2 < / .1 < #2 < . . . </**, and subspaces T~o(M ) = Eo C E1 C . . . C E, C Tp0(M), where E , is the subspace spanned by the Lie algebra of the sections of Tr evaluated at P0. The multiplicity gj of each /*j is defined in the obvious way as above. It is convenient to denote by ml < m2 _< ' " _< mT the H6rmander numbers with multiplicity by taking ml = ms . . . . . mt~ =/ .1 , and so on. The number r coincides with the CR codimension of M if and only if M is of finite type
at P0.
Definit ion 2.3.2. M is said to be minimal at Po E M if M contains no proper CR submanifold through po with the same CR dimension as M.
We define the (local) CR orbit of po in M as the Nagano leaf of the CR vector fields through P0. The CR orbit of p0 is a minimal CR submanifold Wpo C M through P0 such that T;o(M ) C Tpo(Wpo). In fact, using the notation above, the tangent space Tpo(Wpo ) equals the space E, . For a real analytic CR submanifold M, the notions introduced above are related as follows:
M is minimal at po ~=:=>
Wp0 contains an open neighborhood of P0 in M r
M is of finite type at P0.
We refer the reader to [Tul] and the paper by ~hamartov in these Proceedings for further reading on minimality and its connection to wedge extendibility of CR functions.
E x a m p l e 2.3.3. Consider the real analytic hypersurface M C C 2 defined by
(2.3.4) Im Z2 = (Re Z2)IZll 2.
Note that the complex hyperplane {Z2 = 0} is contained in M. It follows that M is not minimal (not of finite type) at the points (Z1,0) E M. However, M is minimal (of finite type) at all other points.
42
If M is a real anMytic hypersurface and M is not minimal at a point P0 E M, then M contains a complex hypersurface through tha t point. This follows from the fact tha t the CR orbit of P0 in M, being a proper CR submanifold of M with CR dimension N - 1 , necessarily has dimension 2 N - 2 and, hence, is a complex manifold by the Newlander-Nirenberg theorem. Also, if a real analyt ic hypersurface M is not minimal at most points (outside a proper analyt ic subvariety), then M is Levi flat (i.e. M is locally biholomorphical ly equivalent to a real plane or, which is the same, the Levi form of M vanishes identically). Both of these facts fail in general for higher codimensionat CR submanifolds as the following example i l lustrates.
E x a m p l e 2.3.5. Consider the codimension two, real analytic , generic submanifold M C C a defined by
(2.3.6) Im Z 3 ---- IZ112 , Im Z2 = 0.
M is fol iated by the CR submanifolds Nx, of the same CR dimension as M , defined by
(2.3.7) Im Z 3 : I N 1 12 , Z 2 : X
for x C R. Hence, it is not minimal at any point. Also, M is not locally biholomor- phically equivalent to a real plane, and is therefore not Levi flat.
2.4. N o r m a l f o r m fo r g e n e r i c s u b m a n i f o l d s . In this section, we shall describe a convenient normal form for generic real analyt ic submanifolds. Let M C C N be a connected such submanifold. We write rn = CR codim (M) ( = codim (M) ) and n = C R d i m ( M ) , so that N = n + r n . We let P0 be a p o i n t in M. Then there are holomorphic coordinates Z = (z, w) vanishing at P0, with z E C ~ and w E C m, such tha t M is given near P0 = 0 by
(2.4.1) Im w = r z, Re w),
where r X,s) is a Cm-valued holomorphic function, Nm-valued for X = 2 and s E R m, such tha t
(2.4.2) r -- r X,s) _= 0.
Such coordinates are called normal coordinates, and (2.4.1) is called normal form for M; note that all examples given so far have been in normal form. We shall sketch a proof of the existence of normal coordinates at the end of this section. We refer the reader to [CM] or [BJT] for a detai led proof of the existence.
If M is real algebraic, then there are algebraic normal coordinates at P0, i.e. the change of coordinates is algebraic and the function ~(z, 2, s) is algebraic. This follows readi ly from the proof of the existence of normal coordinates; as we shall see, the proof is based on an appl icat ion of the implicit function, which preserves algebraici ty in view of Lemma 2.1.2.
By wri t ing Re w = (w + t~)/2 and Im w = (w - w) /2 i , we can solve for w in (2.4.1) by the implicit function theorem. We find that M consists of the set of points (z, w) for which
(2.4.3) w = Q(z, 2, w),
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where Q(z, x , T) is a Cm-valued holomorphic function. It is straightforward to check that (2.4.2) implies
(2.4.4) O(z, O, v) =_ Q(O, x, v) - r.
Note that (2.4.3) is not a R'~-valued equation for M. However, there is an m x m matrix-valued function a(z, w, X, r) such that
(2.4.5) w - Q(z, 2, ~) = a(z, w, 2, e ) ( I m w - r 2, Re w)).
By complex conjugating (2.4.3), we can also describe M by the equation
(2.4.6) ~ = O(~, z, w);
(2.4.7) h(Z) = h(Z)
for a holomorphic function h(Z). An explicit basis L1, ...,L,, for the CR vector fields on M near p0 can be given
in normal coordinates as follows
(2.4.8) Lj = ~zj + ~)k,~(2, z ,w ) , j = 1 , . . . ,n, k = l
where we use the notation
(2.4.9) Ok,~ (7, z, w) = ~ ( ~ , z, w).
Observe that the vector fields L1,..., Ln all commute. We conclude this section by sketching a proof of the existence of normal coordi-
nates. We assume that M is given by (2.2.1) near P0. We may assume, by applying an affine change of coordinates in C N, that p0 = 0 and that
i - (2.4.10) dpj(0, 0) = ~(dZ~+ i - dZn+j).
We write z = (Z1,..., Z,,), w = (Zn+l,. . . , Z~+m), and Z = (z, w). I