Uniqueness theorems for the exponential X-ray transformsharafutdinov/files/articles/JIIPP1993.pdf · 2019-03-13 · E(x) = is called attenuation corresponding to absorption ε. Let

/. Inv. Ill-Posed Problems, Vol.1, No. 4, pp. 355-372 (1993)© VSP 1993.

Uniqueness theorems for the exponential X-ray transform

V.A. SHARAFUTDINOV*

Abstract — We consider the exponential X-ray transform with absorption which does not depend on thepoint and arbitrarily depends on the direction. The problem of the smallest set of projections sufficient fordetermining a function is investigated. We prove, for a finite distribution J, that any countable family ofprojections is sufficient if the projection centers do not accumulate to the convex hull of the support of J.A similar result is established for incomplete projections.

1. INTRODUCTIONLet

Ω^ίΤ- 1^ = «1 , . . . , in)€Rn | | i | = l}be a unit sphere in the space Rn. We consider a fixed (complex-valued) function ε G(7°°(Ω) which is referred to as the absorption function. Let RJ = Rn \ {0}. A functionE G C°°(KS) defined as

E(x) =is called attenuation corresponding to absorption ε.

Let a function J be finite and summable in Rn. The exponential X-ray transform ofJ is a function /£ J(f) defined by

(/«J)(0 = J~E(tt)J(a + tO at (1.1)

for (α, ξ) G Rn χΩ. This transform is of considerable interest for emission tomography.When ε = 0, we use the term X-ray transform (without the attribute exponential).

For a fixed α G Rn, the exponential X-ray transform (/*»7)(£)» as a function of ξ G Ω>is called the projection of function J(x) centered at a (proper) point a (an alternative termis the divergent projection). Along with these projections, we can consider projectionswith improper centers (or parallel projections). Let us first give a suitable definition ofthe improper point.

We use the compactification Rn of the space Rn which is defined as follows. Let ΩΟΟdenote a specimen of the sphere Ω which is considered as nonintersecting with Rn. Forexample, we can assume ΩΟΟ = Ωχ{οο}, where {00} is a set containing a single elementoo. For £ G Ω, we denote the corresponding element of ΩΟΟ by £00 [i.e. f«, = (f,oo)].Let Rn = Rn U ΩΟΟ. The points of Rn are called the proper points of Rn, and those ofΩΟΟ the improper points.

We introduce topology in Rn so that (1) it_ induces on Rn and Ω«, the ordinarytopologies of these spaces, (2) ΩΟΟ is closed in Rn, (3) a sequence of proper points χ kconverges to an improper point &*> if and only if |z*| -> oo and Xk/\xk\ -* £·

* Institute of Mathematics, Siberian Branch of Russian Acad. Sei., Russia, 630090 Novosibirsk, Univer-sitetskii prospekt, 4.

The work was partially supported by the International Science Foundation (Soros foundation).

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356 V. A. Sharafutdinov

For ξ € Ω, a function (/£ J)(x) defined on a hyperplane ξ1 = {χ e Rn | (ζ, ξ) = 0}by

VLW) = Γ ÔJX* + ii)di, χ e £x (1.2)«/—oo

is called a projection of the junction J centered at an improper point &„. Hereafter(z>£) = Et-z'f denotes the scalar product.

This compactification of Rn is necessary by the following. For ε Ξ Ο we have I^J =^(0-0οο^> anc^ therefore we can consider the two end points of each diameter in the sphereΩ«, to be identical. In doing so we transform Rn into a projective space RP71, which isthe traditional compactification of Rn. For the general case ε φ 0 the integral in (1.2)depends not only on the line y = χ + ίξ but also on its orientation, and consequentlywe have to use die compactification Rn.

The integrals in (1.1) and (1.2) are defined for a finite summable function J. Weshow in Section 2 that these definitions can be extended to finite distributions J so that7*J is a distribution on the sphere Ω, for α ^ supp J, and I^J is a distribution on ahyperplane ζ·1.

What is the smallest set of centers of projections which uniquely determine a finitedistribution J? According to Theorem 2.1, this is any countable set of points belongingto Rn \U, where U is an arbitrary neighbourhood of the convex hull of the support of J.

Of appreciable interest is the problem of determining J by a family of projections/* J which are known only in a certain domain ω c Ω, and not on the whole sphere Ω.In what follows, this case is described by Theorem 2.2.

Theorem 2.1 has been proved by the author in [6,7] for ε = 0, and independently byHamacer, Smith, Solmon, and Wagner [4] for finite summable functions. The followingstatement due to Derevtsov [2,6] shows that Theorem 2.1 cannot be improved, i.e. forε = 0 and any finite set of points in Rn \B, where Β is a unit ball in Rn, there exists anonzero distribution J with support in Β and zero projections at all points in this set.We have formulated Theorem 2.2 by analogy with Lemma 2.11 in [5].

Our proofs of Theorems 2.1 and 2.2 are based on the so-called method of moments;its idea is as follows. Assuming that the projections of J with centers a*, k = 1,2,...,are zero and α is an accumulation point of the sequence a*, we obtain by induction onm that the distribution J has zero integral moments of degree m along all the straightlines passing through a. This method was used by Hadamard for proving the statementthat a continuous function in the half-plane y > 0 is uniquely determined by its integralsalong all semicircles whose centers are in the line y = 0. Hadamard's proof can befound in [1].

In Section 2 we extend definitions (1.1) and (1.2) to finite distributions and formulatethe main results of this work, Theorems 2.1 and 2.2. Section 3 is devoted to some topicsin the theory of distributions. We mention the basic concepts of this theory only tointroduce the notation and terminology and concentrate on the problems which arerarely considered in textbooks, such as the direct image of a distribution or parameter-dependent distributions. Section 4 contains the proofs of Theorems 2.1 and 2.2.

2. FORMULATION OF THE MAIN CONCEPTS AND RESULTSLet us recall the ordinary designations for distributions.

Here, the term manifold implies a smooth finite-dimensional manifold which mayhave a boundary; smooth is used as a synonym of the term infinitely differentiable. Thecomplex vector space of smooth functions on a manifold N is denoted by C°°(N), and its



Uniqueness theorems for the exponential X-ray transform 357

subspace containing finite functions by C§°(N). Let K c N be a compact, then Cdesignates a subspace of C™(N) containing finite functions φ for which supp φ c K. Thespace C°°(N) with the topology of uniform convergence on each compact is denotedby S(N), and its subspace C^(N) with the topology induced by S(N) is denoted byT>K(N). Let T>(N) denotes the subspace Cg°(N) with the inductive limit topology ofsubspaces T>K(N) (with respect to all compacts K c N). Let V(N) and S'(N) be therespective spaces of continuous linear functionals on T>(N) and S(N). Their elementsare referred to as distributions (or finite distributions) on TV. The value of F G V(N)[S'(N)\ on the function φ G Ί>(Ν) [Ε(Ν)] is denoted by (Ρ,φ) or (Ρ(χ),φ(χ)). Therestriction F G V(U) is defined for any F G V(N) and an open U C N.u

Let J G £'(Rn), α G Rn, α £ suppJ. The distribution /£ J G £'(Ω) defined by theequality

<2·»with attenuation Ε introduced in Section 1, is called a projection of distribution J centeredat a (proper) point a.

For ξ G Ω, let ρζ: Rn -» ξ-1- be an orthogonal projection. The distribution I^J Gf' ·1·) defined by

(It_J, p) = ( J(x), e-«··*0 φ(ρ,χ)) , *> G £(^) (2.2)

is called a projection of distribution J G £7(Rn) centered at an improper point &».In order to relate (1.1), (1.2) with (2.1), (2.2) we recall that any function T locally

summable on Rn can be put into correspondence with a distribution F = F(x) ax Gdefined by

Similarly, any function Τ summable on Ω corresponds to a distribution F£'(Ω) defined by

where d^ is an angular measure on Ω.Let J' be a finite function summable on Rn, and J = J(x)dx. It follows from (1.1)

and (2.1) that PaJ = (/£ J)df. Similarly, (2.1) and (2.2) imply that I^J = (7|ooJ)dx.

Theorem 2.1. Assume that η > 2, ε G (7°°(Ω), J G 8'(Rn) is a nonzero distribution,and K is a convex hull of the support of J. Then the set of points α G Rn \U at which/* J = 0 is finite for any open set U C Rn such that K c U.

For ω c Ω and α G Rn, C(a,w) = {x G Rn| χ = a + ίξ, ζ G ω, ί > 0} denotes acone with a vertex α and a generatrix set w. We say that the cone Ο(α,ω) is in a freeposition with respect to a set K c Rn if it contains a ray which begins at a and doesnot intersect K.

Theorem 2.2. Assume that η > 2, ε t £7°°(Ω), J G £'(Rn), α0 0 suppJ, and u; is adomain (an open connected set) in Ω such that the cone (7(α0,ω) is in a free position

with respect to supp J. If (/α«θ| = 0 f°r a11 α in a certain neighbourhood of a0, thenJ = 0.




Comparing Theorems 2.1 and 2.2 we can ask whether the condition (/£ J)| = 0____for all a in a certain neighbourhood of a0 in Theorem 2.2 can be replaced by a weakercondition (/a*^)| = ^ for a sequence of points ak which are different from c*o andconvergent to a0. In case of ω = Ω the answer is positive in accordance with Theorem 2.1.In case of ω fi Ω we can show that the answer is positive for η = 2 and negative forn>3 .

3. SOME CONCEPTS OF DISTRIBUTION THEORYThe spaces V(N) and S'(N) are considered as spaces with weak topology. Therefore, asequence Fk G £'C/V), k = 1,2,..., converges to zero if and only if the sequence (Fk, φ)converges to zero for any φ G £(TV). Distributions can be multiplied by functions fromS(N) according to the formula (φΡ,ψ) = (Ρ,φψ).

Lemma 3.1. Let U be a domain in a smooth manifold N and F G £'(JVxR); wedenote points in NxR by pairs (x, t\ where χ £ N and t G R. If (F, ίΙζφ(χ)) = 0 for anyinteger k > 0 and any φ G V(U), then F\ = 0.

IL/xR

This lemma is a corollary of the following two statements.(1) The set of functions which have the form

Σ ν>,·(*Μ·(θ , ψi e v(U) , & G £(R)t = l

is dense in V(U) <g> £(R) [3].(2) It is evident that the set of polynomials is dense in £(R).Let M and TV be smooth manifolds. If /: M -> TV is a smooth mapping, then

the homomorphism /*: £(7V~) -> £(M) defined by the formula f*<p = φ ο / is linearand continuous (here ο denotes composition). The homomorphism dual to /* will bedenoted by /': £7(M) -» £;(7V). Thus, the distribution f'F G £7(7V) (called the directimage of F in the mapping /) is defined for F G £'(M ) by the equality

If #: TV -» L is another smooth mapping, then (# ο /)' = #'/'. The direct image isconnected with the multiplication of distributions by smooth functions as follows:

(3.1)

In particular, if / is a diffeomorphism, then

ί\φΡ) = (φ ο /-^/'F , φ G £(M) . (3.2)

If /: Μ — > TV is a proper mapping, i.e. the inverse image of any compact is a compact,then /*: V(N) —> V(M) and consequently /' can be extended to a homomorphism

Let /7 be a domain in Rn. The operators d/dx\ i = 1, . . . , n, are defined on V(U)as




According to the usual rule of differentiation of a product,

By convention, the same subscript and superscript in a single term imply summationfrom 1 to n.

Lemma 3.2. Assume that F E £'(Rn) and m > 1 is an integer. If there exist numbersa1,..., an, b e R such that at least one of them is not zero and

then F = 0.

Proof. We are to consider two possible cases.(1) b - Ο, α = (α1,...,^) ^ 0. It is enough to consider m = 1 since the proof for

the general case can be obtained by induction. Here (3.3) reduces to aidF/dxi = 0. Wechoose an affine coordinate system (yl,...,yn) in Rn so that a is the first basic vectorin the system. Then dF/dyl = afdF/dx* = 0, and hence F = 0.

(2) b ? 0. If we divide (2.3) by 6m, we obtain

(re''1 + c'1)... (xim + c'm) .fmF .m = 0 (3.4)

where c' = α'/δ. Let G = f'F, where /: Rn -> Rn is defined by the formula f(x) = χ + c.It is enough to establish that G = 0. Applying /' to (3.4) and using (3.2) we obtain

- = 0 . (3.5)

We first show that (3.5) implies the inclusion

suppGc{0}. (3.6)

To do this we have to prove that (Ο,φ) = 0 for any φ G £(Rn) such that 0 £ suppy?.Assume this φ to be fixed. Then there exists ε > 0 such that the set {x e Rn| |x| < ε}and suppv? do not intersect. We define V>< G £(Rn) for t e R by putting ψ^χ) = φ(^χ).Note that supp ψ{ C e"' supp φ. The function

f(t) = (G,lM (3.7)

is smooth on R If we denote R = sup{|z| | χ e supp G} and choose 20 so that ε e~<0 > R,then supp i/>t Π supp G = 0 for ^ < ^0, and therefore

0, t<tQ. (3.8)

Let us show that the equality

p=0




is valid for any integer k > 0, with constants c£ dependent only on k and p. We prove(3.9) by induction on k. For k = 0, we see that (3.9) follows from (3.7). Assuming that(3.9) is valid for some k, we differentiate this equation with respect to t

According to the definition of ψι we have d^tjdt = x'difrt/dx'. Substituting this expres-sion in (3.10) we find

YV f^(t\ = (-ΙΫ/χ^ xik 9kG a··*-» d^t \P \ " θϊ«...Οϊ··*'Χ ar'W

If we replace the second term by its expression using the induction hypothesis (3.9), weobtain

ί-1 + (* + *)

which completes the proof of (3.9).Taking k = m in (3.9) we use (3.5) to verify that f(i) satisfies the homogeneous

equationm-l

p=0

This and (3.8) yield /(ί) Ξ 0, in particular, /(O) = (G, ̂ ) = 0. We have thus proved(3.6).

It is known that a distribution whose support is the point 0 can be represented as afinite linear combination of derivatives of the ^-function, i.e.

. (3.11)M<*

We can easily verify by induction on m that

(n,m3a)Da^ (3.12)... . , 3ox11 . . . oxlm

for any multiindex α and a(n,m,a) ^ 0. Substituting (3.11) in (3.5) and using (3.12) weobtain

]T a(n,m,a)caDa6 = 0. (3.13)




Different derivatives of the ^-function are linearly independent. Therefore (3.13) impliesthat all ca are zeros, i.e. G = 0. The lemma is proved.

We are to consider distributions dependent on several real parameters (e.g. thedistribution PaJ defined on Ω by (2.1) depends on α G Rn), and, in particular, todifferentiate distributions with respect to parameters. Here we give the general rulesfor these calculations.

Let Λ be a domain of the space R*. The space R* is considered as a space ofparameters, therefore we designate its points by Greek letters, e.g. α = (α1,...,«*) GR*. We call a mapping F: A —> ε'(Ν) a distribution dependent on a parameter a G Aon the manifold N. We say that the distribution F is m times differentiable with respectto the parameter α G Λ (m < oo) if the function F has continuous partial derivatives oforder m and lower order. Recall that E'(N) is considered for weak topology. Thus, ifF, G: A —> E '(N) are two distributions dependent on a parameter α G A, the equalitydF/da* = G is equivalent to d(F(a)^)/dai = (G(a),y?) for any φ G S(N).

Distributions dependent on a parameter can be multiplied by functions dependenton a parameter. To be precise, if F: A —> S'(N) and φ = φ(χ, α) is a smooth functionon NxA, the distribution (?F: A —> £'C/V) is defined by the equation (<pF)(a) = <paF(a),where φα G ε(Ν) is given by φα(χ) = φ(χ,ά). The standard rule of differentiation of aproduct with respect to a parameter holds.

Let F: A —> £'(M) be a distribution dependent on a parameter and /: Μ χ Α —> Νbe a smooth mapping. For a G .4, we introduce a mapping fa: M -» N which isdefined by the formula fa(x) = /(ζ,α). The distribution /'F: Λ -> £'(ΛΟ defined bythe equality (f'F)(a) = f'aF(a) is called the direct image of the distribution F in themapping /.

The following statement is used below for L = ε1 (Ν).

Lemma 3.3. Let L be a linear topological space. Assume that an m-times differen-tiable function /: U —> L is given in a domain U c Rn and, for some XQ G i/, there existsa sequence ζ*, A = 1,2,..., convergent to x0 so that ZQ ^ %k and /(ζ*) = 0. Supposethat the sequence (xk - xo)/\Xk - XQ\ converges to a vector a = (a1,..., an) G Rn, and

for any 0 < p < m. Then

Proof. By condition, the Taylor expansion of / in the neighbourhood of ZQ is

/(z) = (z" - zj1) . . . (z- - 4m)dxid™f

dxim(*<>) + o(\x - zoD .

Consequently

dmf0 = /(ζ,) = (χϊ - z<>) . . . (ζ'Γ - z - - ) 0 . ( z 0 ) + o(\xk - zoD .

If we divide this equation by \xk - z0|m and take the limit k -> oo, we obtain thestatement of the lemma.




4. PROOFS OF THEOREMS 2.1 AND 2.2It is more convenient to prove Theorem 2.1 in the following formulation which is evi-dently equivalent to that of Section 2.

Lemma 4.1. Assume that J G £'(Rn+1), ε G <7°°(Ωη), η > 1, and tf is a convexcompact set in Rn+1 such that_supp J c K. Suppose that there exists a point o GRn+1 \K and a sequence G Rn+1, k = 1,2,..., convergent to A so that k ? o and7jk J = 0. Then J = 0.

Before proving this statement we are to give auxiliary constructions and lemmas.Let us first suppose that o in Lemma 4.1 is a proper point. For β G Rn+1, we denote

the central projection with its center at β by p/?: Rn+1 \ {β} -> Ωη. This projection isdefined by the formula P (y) - (y - )/\y - \. Then, it follows from (2.1) that, forβ i supp J,

E(y- )~We choose a rectangular Cartesian coordinate system (y1,... ,yn+1) = (z,0 =

(z1,... ,rcn,i) in Rn+1 so that the origin of coordinates is at o and the plane t = 0and K do not intersect. We also choose 8 > 0 such that the domain

U = {(z,*)eRn+1 | 6<t< 1/6}

contains K. LetA = {(a,r)GR n + 1 | r < S] .

We designate points of the domain A either by (a, r ) or by β = (βι,...,βη+ι). We canassume that all the points β k = (a k, rk) in Lemma 4.1 belong to A.

Let us introduce the mappings

p:U-+Rn, p(x,t) = x (4.1)

(4.2)

(43)

(4.4)

We define for (α, r) G A a diffeomorphism of 7 onto itself by the formula /(α,τ)(«, Ο =/((a?,0»(air)) ^^ designate it by /(α,τ): U -* U. Similarly, we define for a G Rn adiffeomorphism of U onto itself by the formula ga(x,t) = #((#, £),<*) and designate itby ga: U -+ U. Note that

ga = /(α,ο) , h = go. (4.5)Using these mappings we define a distribution F: A — > £7(Rn) dependent on the

parameter (α,τ) G A. We put




The distribution G\ Rn -» £'(Rn) depends on the parameter α e Rn and is defined as

(4-7)

Finally, for each integer ra > 0 we introduce a distribution Mm G £'(Rn) by the formula

<«>Note that both F and G are infinitely differentiable with respect to the whole sets oftheir parameters. It follows from (4.5) that

G(a) = F(a, 0) , M0 = G(0) .

According to the definition of function E in Section 1,

E(tx,t) = e-<M(r)

where the function μ e C°°(Rn) is defined by the formula

χ 1

(4.9)

μ(χ) = Λχ\2+Ιε+ ι νΉ2 +

Using the rule in (3.1) we see that (4.6)-(4.8) can be rewritten as

(4.10)

(4.Π)

(4.12)

Lemma 4.2. Assume that J G f ;(£/) and a distribution F: Λ -* i;(Rn) is defined for Jby (4.6). The equality 1% J = 0 is equivalent to F(a, r) = 0 for any point β = (α, r) e Λ.

Proof. According to the choice of the coordinate system

n: ={y = ( x , t ) e R n + 1 \ \ y \ = 1, t > 0 } .

Therefore, we can assume that Ie J e £'(Ω!£). We define a diffeomoφhism q: Ω^ -» Rn

by ς(ζ,0 = ^Λ· According to (2.1)

Ε(χ - α,ί-τ)



364 V. A. Sharafutdtnov

for any φ e £(Rn). Hence, if we define ψ e £(Rn) by the formula φ(χ)(1 + \χ\2)~η'2φ(χ) for ψ ς. £(Rn); then

(χ°'~ T) J(*. 0. y ° Ρ ° /(..r

/ , , (E(x-a,t-T)=

Comparing this equation with (4.11) we arrive at the statement of the lemma.

Lemma 4.3. Assume that J e £'({/) and distributions F(a,r), G(a) and Mm aredefined by (4.6)-(4.8). Then

(Q) ,·'*» ^ '

+ Σ Σ H^ k(*)D Mk , l < »ι, . . . , »m < η (4.15)

for any integers fc > 0 and m > 0 and some μil...im, k € (7°°(Κη) which are dependentonly on ε.

We will prove this lemma later. We are now to complete the proof of Lemma 4.1using (4.14) and (4.15) for a proper point o.

The coordinate system is chosen so that o = 0. Without loss of generality we cansuppose that the sequence k = (α*, τ*) is such that the sequence k/\ k\ converges tosome unit vector c = (c1 , . . . , cn + 1 ) = (a1 , . . . , an , b). According to Lemma 4.2

F(ak,rk) = 0. (4.16)

Taking the limit k -> oo we obtain F(0, 0) = 0. In view of (4.9) this implies

G(0) = Mo = 0. (4.17)

Then, we are to prove by induction on p that

Mp = 0 (4.18)

dpG^— ̂ -(0) = 0, l < M , . . . , i , < » (4.19)

for any integer p > 0. Indeed, suppose that (4.18) and (4.19) have already been provedfor all p such that 0 < p < m. Then, because of (4.14) and (4.15)

dpF— -(0,0) = 0 , 0 < p < m , l < i i , . . . , i p < n + l

(4.20)

de?1... dc?m dx^ ... dx*m ' ~~ ' ' ~~




Applying Lemma 3.3 we obtainn +1 am τρ

. ^"^ _1 u *1 ...u trn

which can be rewritten in terms of (α, τ) as

J^ /m\ , <9mFΣ(7J6 a't+1 ···a""ar*dJ*+....dQ'~ <°.°> = °·

Substituting the expressions for the partial derivatives of F from (4.14) and (4.20), weget

(a*1 — bxtl)... (atm — bxlm)-^—. ™ . = 0.

Using Lemma 3.2, we verify that Mm = 0. Then, using (4.20) we obtain (4.18) and (4.19)for p = m. Therefore, the induction step is completed and (4.18), (4.19) are proved forall p.

In view of (4.13), equation (4.18) can be rewritten as

Let r: U -* t/ denote a diffeomorphism which is defined by r(z,2) = (χ,ί"1). Takinginto account that r~l = r, we can rewrite (4.21) as

0, m = 0,1,.-.

Using (3.1) we rearrange this formula to get

0 - pV

This means that

o. pvfor any integer m > 0 and any </? G 5(Rn). Hence, using Lemma 3.1 we obtain that

Since r ο Λ is a diffeomorphism, this equation implies that t~nE(x,i)J = 0. The functiont~HE(x,t) does not vanish on (7, therefore the latter relation shows that J = 0.

Proof of Lemma 4.3. Recall that the distributions F(a,r), G(a) and Mm are definedfor the distribution J G ε1 (U) by formulae (4.6)-(4.8). The left- and right-hand sides of(4.14) and (4.15), if considered with respect to J for fixed a, are continuous operatorsacting from £'(i/) into £'(Rn). Since the set of distributions J = J(x,t)dxdt, withJ G Co°(i/), is dense in £'(i/), we see that it is enough to prove the lemma only forthese J.




Thus, let J = J(x,t)dxdt, where J G Ο°(£/)· Then, it follows from definitions(4.6)-(4.8) that

F(a, r) = F(a, r, x) ax , G(a) = 5(a, x) dx , Mm = Aim(x) dz

and the functions .F, Q and .Mm can be expressed in terms of J by the formulae

F(a, r, x) = /°Vi/z(r) J(te + α - rx, *)d* , x G Rn , (α, τ) e A (4.22)./o

0(a, x) = 1"« J(ii + a, i) di , z, a e R" (4.23)./o

Mm(x) = /°°rm e-fM(r) J(tx, t)dt, x e Rn , m = 0, 1, ... (4.24)«/O

where the function //(x) is defined in (4.10).Comparing (4.22) with (4.23), we see that f(a, r, x) = Q(a-rx, x). We differentiate

this equation to obtain

which is equivalent to (4.14).Differentiating (4.23), we get

(Ο τ\ - I C-WX) ^ (4Τ 4\Α4 (ά9Î V/j «*/ Ι Ι & _ . _ It/Χ } ι> ι \ΛΙ> . I *T.A/J ι

Then, if we differentiate (4.24) we obtain (using multiindices α, β and 7 of length Ώ)

D° Λ/f /''»Λ = ι /Ι0!-771^"'^) ^πα TVir f } n tL·/ J^*Tfl\'^') I ^ v'̂ j? ^s / \ ) "/ *·*7o

>7 TV/T / ^ H / ^4 9^^-, v/ /\'"*/j "/ ^*^ · l^.A»U I

By induction on |/3|, it is easy to verify that

101χ) , \β\>0 (4.27)

for some smooth functions μβρ(χ) which are determined by μ(χ). Substituting this ex-pression in (4.26) we obtain

W-MDaMm(x) = A^(x) + Σ Σ μ^Ρ(χ)Αΐη_ρ(χ) (4.28)

7<α ρ=1

whereΛ^(χ) = rtW-ke-t (x\D"J)(tx,t)dt. (4.29)

JO

Κ equations (4.28) are considered as a system of linear equations with respect to A^(x\\a\ < m, then the matrix of this system is triangular, and therefore the system has asolution

M-MA"m(x) = Z)aîm(x) + Σ Σ ^P(xWMm-p(x).

7<α ρ=1




If we put here |a| = m and recall the definition of A^(x) in (4.29), we obtain

ft™ Λ/f lml-Mm

/•of°

Jo

This and (4.25) yield

So:*1... dct*m dx*1... dx*m JT

which is equivalent to (4.15). Lemma 4.3 is proved.We have thus proved Lemma 4.1 provided that $> is a proper point.Let us now examine the case when βϋ in Lemma 4.1 is an improper point, i.e. o = &»

for ξ G Ωη, and the sequence k contains an infinite number of proper points. Withoutloss of generality we can consider all k to be proper points.

We choose a coordinate system in Rn+1 so that the vector ξ has coordinates(Ο,.,.,Ο,-1), and the set Κ is inside the above domain U. Let (ak/rk,l/rk) be thecoordinates of k. By condition, k -» £» as k —> oo, which implies that rk < 0 while(otk,Tk) -> 0. Besides, by condition we have Ic

kJ = 0, which in view of Lemma 4.2 isequivalent to the equalities

= 0, t = l,2,. . . . (4.30)

According to (4.6) and (4.10), we have for α € Rn and r < 0

In order to simplify this expression we introduce the diffeomorphism g(a>T): Rn -> Rn,0(α,τ)(ζ) = (ζ - a)/r. Applying q'(ct >r) to (4.31), we have

Note that <7(α>Τ) ο ρ = ρ ο Γ(α>τ), where the diffeomorphism Γ(α>τ): ί/ — > ί/ is defined bythe formula Γ(α>τ)(ζ,2) = ((α - x)/r,t). Therefore, the above equation can be rewrittenas

The diffeomorphism inverse to r(a>T) is given by r^(x, t) = (a - rx, t). Applying (3.2),we rearrange the previous equation:

(4.32)

where the diffeomorphism

/(α,τ) = r(ttfT) ο /(α/τ>ι/τ): U -> U

is defined by




The inverse diffeomorphism has the form

Let us again rearrange (4.32) to simplify the function μ. We make a substitutionJ = exp[(c, x) + co*] J, where c = (c1, . . . , cn) and co are constants which will be specifiedlater. Using the rule (3.2) we obtain

-i£(a-Tar)

(433)

where μ(ζ) = μ(χ) - (c, x) - c0. Let us choose the constants c and CQ so that

μ (0)=0, ^(°) = 0 > » = l, . . . ,n. (4.34)

Comparing (4.30) with (4.33) we can conclude that

*) = 0 (4.35)

where the distribution F: Λ -> £'(Rn) is dependent on the parameter (α,τ) G A anddefined by the equations

f ( ( x , t), (a, r)) = ( γ^, t) (4.37)

(the domain A was defined after Lemma 4.1).By analogy with (4.1)-(4.8), we define the mapping

g: U χ Rn -> U , ^((χ, ί), α) = (*-α<, Ο (4.38)

and the distribution G: Rn -* £(Rn) dependent on a parameter α G Rn:

G = p/(e-i^/J). (4.39)

For integer m > 0, we define Mm G £'(Rn) by

Mm =P'(tmJ). (4.40)

Lemmajl.4. Assume that μ G ̂ ^(R71) satisfies (4.34) and the distributions F(a,r),G(a) and Mm are defined in (4.36)-(4.40) for J G £'(Rn). Then

,0) = (-l)V....^^^- (4.41)' ^ ' ' ' v '

dmG = dmMm9aTl . . . daim dxil . . . dxim

m-l _

+ Σ Σ fa-.imjk(*)D Mk , 1 < ή, . . . , im < n (4.42)*=0 \ \<k

for any k > 0 and m > 0 and some /̂ ...,-,̂ Έ CîR71) which depend only on /i.




If we assume that this lemma is valid and repeat the argument given after the state-ment of Lemma 4.3, we can derive from (4.35) that J = 0 and hence J = exp[(c,z} +co]J = 0.

Proof of Lemma 4.4. Similar to the proof of Lemma 4.3, we are to verify that it isenough to consider the case when J = J(x,t)dxdt, where J G CQ°(U). In this case

F(a, r) = r (a, r, a;) ax , G (a) = £(a, x) dx , Mm = Mm(x) ax

where

F(a, r, x) = /"%-«««-«> J(x + (a - rx)t, t) at (4.43)JQ

β(α,χ) = i / ia J(z + e*M)di (4.44)Λ)

MmOO = JQ°°trnJ(x,t)dt. (4.45)

Comparing (4.43) with (4.44) we see that F(a, r, x) = Q(a - rx, x) which implies (4.41).We differentiate (4.44) with respect to α (using a variable α e Rn and multiindices

)S, 7 and 8) to get

<«, *) = Σ 4L Γ i|5|^[e-<A(a)](^.7)(z + αί, i)di - (4.46)Ί+δ=βΊ'°' J0

It is easy to prove by induction on \Ί\ that

Σ <W(«) (4-47)

where_

\ θα J V aV "· \da«and -ykfa) are smooth functions determined by the function μ. We substitute (4.47) in(4.46) and put α = 0 in the equation obtained. Taking into account (4.34) we have

,a) = D* rtWj(x,t)dt + Σ Σ fi(hk(x)Dl rtkj(x,t)dtJ° k<\ \ \y\<k J°

which, in view of (4.45), can be rewritten as

,z) = l%Mm(x) + Σ Σ fi(hk(x)DlMk(x).

This equation is equivalent to (4.42). The lemma is proved.Finally, let us consider the case in which all k in Lemma 4.1 are improper points.

We assume thatA = (&,ifo)oo, * = 0,1,...

where (£*,*?*) = ({ί>· · ·>ί?,»7*) € Πη. We choose the coordinate system so that thevector (£o, τ/ο) has coordinates (0, . . . , 0, 1), and the set Κ is inside the above domain U.Then

I6|2 + ̂ = l, & - > 0 , i f o - > l , ^ôo . (4.48)




According to the definition given in (2.2), we have for (ξ, 77) = (ξ1,... ,ξη,η) G Ωη

(4.49)

where p^ny. Rn+1 -> (ί,^)1" denotes an orthogonal projection. For (ξ, η) G Ωη, 77 > Ο,we define the diffeomorphism q^ny. (ξ, η)-1 -* Rn, ^,τ,)θΜ) = a; - ^/τ/. Applying theoperator qfa Λ) to (4.49), we get

It is easy to verify that q^) ο ρ^ιΤ?) = ρ ο ^/^, where g is given by (4.48). Consequently,the previous equation can be written as

(4.50)

We can readily obtain

[exp(-(£,z) - η ί ) ε ( ξ , η ) ] ο 9 ΰ ιη = αφ[-(ξ,χ)ε(ξ,η) - ίμ(ζ/η)] .

Therefore, using (2.1) and (2.2) we can rearrange (4.50) into the form

-(£,x)e(£,T?) / / re Ί\ - η'\^~ίμ·^Ι^η' 71C ^^ν^η)^) ~ Ρ le 9ξ/η·*\ '

Then, using the above technique we substitute J = exp[(c,x) + co]J, which yields

where /i G (7°°(Κη) satisfies (4.34). Comparing (4.52) with (4.39) we see that

e-<^W€,n) 4^(7^ J) = ο(ξ/η) . (4.53)

By condition, I^^^J = 0. In view of (4.53), this implies

G(ak) = 0 (4.54)

where a k = £k/Vk· According to (4.48), we have ak — > 0 for A; — > oo.Similar to the proof of (4.18), we begin with relations (4.42) and (4.54) and verify by

induction on m that Mm = 0 for all m. Hence, J = 0, and therefore J = exp[(c, x) +co]J = 0.

This completes the proof of Theorem 2.1.

Proof of Theorem 2.2. We say that a set ω c Ωη-1 lies in a hemisphere if thereexists ίο G Ωη~! and α, Ο < α < 1, such that (£,&) > α for all ί G ω. Note that it isenough to prove Theorem 2.2 for domains which lie in hemispheres. Indeed, supposethat Theorem 2.2 is valid for domains which lie in a hemisphere and u; is an arbi-trary domain in Ω"1"1 which satisfies the conditions of Theorem 2.2. Let us representω in the form ω = U?=i^:> where each ω, lies in a hemisphere, the cone Ο(α0,ωι)is in free position with respect to supp J, and the intersection of u;t, i = 2, . . . , k, ando;iU. . .IM-i is not empty. According to the assumption, J =0. Since ωιΠω2 ·£ 0,

C(a0,u;i)

then (7(α0,ω2) is in free position with respect to supp J, and therefore J\ =0 .



Uniqueness theorems for uie exponential X-ray transform 371

We can verify by induction that J = 0, i = 1,..., k. Since (7(α0, ω) = Ut C(a0, u>j),

JC(aou,) = 0 ·

Suppose that

J/ «_ £»/ /Ό714· 1 \ /Q f- DTI +1 O W oiirvr» 7G o (K ), ρο G Κ , Po ? SUpp J

and α; G Ωη is a domain lying in a hemisphere such that the cone Ο(βο,ω) is in a freeposition with respect to supp J'. Let (/| J')| = 0 for all /J in a certain neighbourhoodof A). Let us choose a coordinate system (z,$) in Rn+1 so that β0 = (Ο,.,.,Ο) andthe inequality η > a > 0 be valid for (ξ1,...,ξ",η) € ω· We fix a small 8 > 0 andsuppose that Xs € C°°(R) is such that \8(t) = 0 for t < 6 and Xs(t) = 1 for t > 26. LetJ = Xs(t)J'. Hence supp Jet / , where i/ is the domain introduced after the formulationof Lemma 4.1. Then, if δ is sufficiently small, the inequality

= 0 (4.55)ύ

is valid for all β in a certain neighbourhood of o.Using J G €r(U\ we define the distributions F(a, r), G(a) and Mm as after Lem-

ma 4.1. Repeating the argument in the proof of Lemma 4.2, we verify that the equationIpJ = 0 is equivalent to F(a, r) = 0 for any β = (α, r) G A. Here D is the image ofω in the diffeomorphism q: Ω+ —> Rn which was defined in this proof. Consequently, itfollows from (4.55) that F(a, r)| = 0 for all (a, r) in a certain neighbourhood of zero.Therefore, by using (4.9) and (4.15) we can verify that the equations

ft™ Μ m~1

_ϊ1_ί!^_ 4- γ γ Λ. . 0k(x)D0Mk = Ο (4.56)/77**1 /7T*m ^ ^ TO'f-' / \ /Ι/Χ . . .(/Χ £ = () ι

are valid in I).According to the condition of the theorem, the cone (7(0, ω) is in free position with

respect to supp J. This implies that there exists a domain ω' c ω such that C(0,u/) Πsupp J = 0. Suppose that D1 is the image of ω' in the diffeomorphism q: Ω^ -» Rn.Let us show that

Mm D / = 0, m = 0,1,.... (4.57)

Indeed, let φ G V(D'). In view of (4.13)

because the support of the function ψ(χ,ί) = <p(x/t) is in (7(0, ω').Since £>' c D, then (4.56) and (4.57) imply that Mm = 0, m = 0, 1, . . . . Repeating

the argument given after (4.21) we find that

(r'h'(E(x,t)J/tn),tm<p(x)) = 0

for any φ G V(D)9 where the diffeomorphism r: U -* t/ is defined by the formular(z,J) = CM"1). Applying Lemma 3.1, we obtain r'h'(E(x,t)J/tn)\ = 0. It is easyto see that the diffeomorphism r o h maps C(0,u;) onto D. Therefore, the previousequation implies that (E(x,t)J/tn)\ = 0, and consequently J' - 0. The

I C(0,u;) C"(0,u;)theorem is proved.




REFERENCES

1. R. Courant and D. Hubert, Methoden des Matematischen Physik, II. Springer, 1937.2. E. Yu. Derevtsov, Example of non-uniqueness of the solution of an inverse problem in photometry.

In: Mathematical Methods for Solving Direct and Inverse Problems of Geophysics, (Ed. A. S. Alexeev).Comp. Cent. Siber. Branch, USSR Acad. Sei., Novosibirsk, 1981, pp. 31-38 (in Russian).

3. G. De Rham, Varietes Differentiates. Paris, Hermann & C, 1955.4. C. Hamacer, K. T. Smith, D. C. Solmon, and S. L. Wagner, The divergent beam X-ray transform. The

Rocky Mountain J. of Math. (1980) 10, 253-283.5. S. Helgason, The Radon Transform. Boston-Basel-Stuttgart, Birkhauser, 1980.6. M. M. Lavrent'ev, E. Yu. Derevtsov, and V. A. Sharafutdinov, Determination of an optical body that

is in a homogeneous medium from its projections. Sov. Math. Dokl. (1981) 24, 342-345.7. V.A. Sharafutdinov, Determination of an optical body that is in a homogeneous medium from

its images. In: Mathematical Methods for Solving Direct and Inverse Problems of Geophysics,(Ed. A. S. Alexeev). Comp. Cent. Siber. Branch, USSR Acad. Sei., Novosibirsk, 1981, pp. 123-148(in Russian).



Uniqueness theorems for the exponential X-ray transformsharafutdinov/files/articles/JIIPP1993.pdf · 2019-03-13 · E(x) = is called attenuation corresponding to absorption ε. Let

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