Translation invariant Radon transforms

Translation Invariant Radon Transforms

Arp´´ ad Kurusa

Abstract. E. T. Quinto proved that for a generalized Radon transformRon Rⁿthe translation invariance of the operatorR^t◦Rimplies the invertibility of R. In this paper an other concept of the translation invariance is defined. We investigate the relation of these two concepts and determine the translation invariant Radon transforms to be a certain generalization of the Tretiak-Metz exponential Radon transform. Finally we give inversion formula and prove the support theorem for these transforms.

1. Introduction

Since in the first part of this century Radon created the classical Radon trans- form on points and lines inR²several new Radon transforms was invented and two main ranges of the investigations of the Radon transforms have been developed.

In the first one the Radon transforms are concretely considered [2] while in the other the classes of the Radon transforms are studied [6]. Our paper belongs to the second ranges. We investigate the class of translation invariant generalized Radon transforms on Rⁿ.

Our concept for translation invariance is based on the well known identity Rf_a(ω, p) = Rf(ω, p+ha, ωi) for classical Radon transform [2]. There is another concept in [6], where Quinto defined the translation invariance by the translation invariance of R^t◦R. However it is obvious that Quinto’s definition covers a larger class in general then ours, we show out that the double fibration model to construct the generalized Radon transform introduced by Gelfand and used in [6] restricts his results for a smaller class.

One goal of this paper is to prove in Section 3. that the translation invariant Radon transforms, by our concept, are not other but a certain generalization of the Tretiak-Metz exponential Radon transforms.

AMS Subject Classification(1980): 44A05, 46F10.

Math. Balkanica, 5(1991), 40–46. c A. Kurusa^´

Our another goal in Section 4. is to give inversion formula, which proves the injectivity of our transforms, and the so-called support theorem.

These latter results are only known in dimension two [3,8] and may prove useful in the practice as A. Markoe [5] proved that the inversion of the variably at- tenuated X-ray, which occurs in single photon emission tomography, can be reduced to the inversion of the exponential transform. See [7] for application in diagnostic medicine. On the other hand the exponential Radon transform can be regarded as a first order approximation to a general attenuation.

2. Preliminaries and definitions

Let µ ∈ C^∞(Rⁿ ×Sⁿ⁻¹ × R) be a strictly positive function such that µ(x, ω, p) =µ(x,−ω,−p). Then the generalized Radon transform is

(1) Rµ:D(Rⁿ)→D(Sⁿ⁻¹×R), Rµf(ω, p) = Z

H(ω,p)

f(x)µ(x, ω, p)dx, where H(ω, p) = {x ∈ Rⁿ:hx, ωi = p} is the hyperplane with normalvectorω ∈ Sⁿ⁻¹and distancepfrom the origin. dxis the surface element ofH(ω, p) andh., .i is the standard inner product on Rⁿ.

Let λ ∈ C^∞(Rⁿ ×Sⁿ⁻¹ × R) be a strictly positive function such that λ(x, ω, p) =λ(x,−ω,−p). Then the generalized Radon transform is

(2) R_λ^t:D(Sⁿ⁻¹×R)→C^∞(Rⁿ), R^t_λf(x) = Z

Sⁿ⁻¹

f(ω,hω, xi)λ(x, ω,hω, xi)dω, where dω is the surface measure on Sⁿ⁻¹. One can see here that our starting situation is more general then E. T. Quinto’s at (20) and (21) in [6].

We call a generalized Radon transform exponential if µ(x, ω, p) =µ1(ω, p)×exp(hµ2(ω), xi), where µ1∈C^∞(Sⁿ⁻¹×R) andµ2:S^n−1C

∞

→Rⁿ. Similarly R^t_λ is called exponential if

λ(x, ω, p) =λ1(ω, p)×exp(hλ2(ω), xi), where λ₁∈C^∞(Sⁿ⁻¹×R) andλ₂:S^n−1C

∞

→Rⁿ.

Now we present our concept for translation invariance of generalized Radon transforms. Our idea is based on two simple observations on the classical Radon transformR andR^t(see [2] and [4]) :

Rf_a(ω, p) =Rf(ω, p+hω, ai) and (R^tf)_a =R^t(f(ω, p+hω, ai)),

where f_a denotes the translation of the function f with a ∈ Rⁿ. The following definitions generalize these properties. We call Rµ respectively R^t_λ translation invariant, if there is a ν respectivelyη inC^∞(Rⁿ×Sⁿ⁻¹×R) for which

R_µf_a(ω, p) =ν(a, ω, p)R_µf(ω, p+hω, ai) respectively

(R^t_λf)a=R^t_λ(f(ω, p+hω, ai)η(a, ω, p)).

In the classical case ν ≡η≡1. At the same time it is obvious that ifν ≡η then the operatorR^t◦R is translation invariant.

3. Determination of the translation invariant transforms

Theorem 3.1. The generalized Radon transformationR_µonC_c^∞(Rⁿ)is translation invariant if and only if it is exponential.

Proof. Because of the simplicity of the sufficiency we only prove the necessity. It is immediate from our definition of translation invariance that

(3) µ(x−a, ω, p) =ν(a, ω, p)µ(x, ω, p+hω, ai),

where x∈H(ω, p+hω, ai). Leta=αω+βω¯a and x=pω+a+κ¯ωx, where ¯ωa

and ¯ω_xare perpendicular toω∈Sⁿ⁻¹. Takingκ= 0 one can rejectν(a, ω, p) from (3). Forα= 0 and ¯ωa= ¯ωx the result is

µ(pω+ (β+κ)¯ωa, ω, p)

µ(pω, ω, p) =µ(pω+βω¯a, ω, p)

µ(pω, ω, p) ×µ(pω+κ¯ωa, ω, p) µ(pω, ω, p) . Thus the map

β→µ(pω+βω¯a, ω, p)/µ(pω, ω, p)

is a continuous homomorphism from (R,+) to (R+,•) and so we have µ(pω+βω¯a, ω, p) = ¯µ1(ω, p) exp(βµ(¯¯ ωa, ω, p)),

where ¯µ1(ω, p) = µ(pω, ω, p). Writing back this formula into the previous equa- tion and letting ¯ω_a 6= ¯ω_x we get the homogenity of ¯µ in its first parameter i.e.

βµ(¯¯ ωa, ω, p) = hβω¯a,µ¯2(ω, p)i, where ¯µ2(ω, p) ⊥ω. Again let a =αω and write backµin its present form into (3). We obtain

¯

µ₁(ω, p) exp(hκ¯ω_x,µ¯₂(ω, p)i) =ν(αω, ω, p)¯µ₁(ω, p+α) exp(hκ¯ω_x,µ¯₂(ω, p+α)i).

Math. Balkanica, 5(1991), 40–46. c A. Kurusa^´

Because of the dependence on κ this implies the orthogonality of ¯ω_x and

¯

µ2(ω, p+α)−µ¯2(ω, p). Since ¯µ2 ⊥ ω and ¯ωx is arbitrary in ω^⊥, the orthogo- nal complement of ω, this yields to the independence of ¯µ₂ from its second ar- gument. Let ˜µ2(ω) = ¯µ2(ω,0), ˆµ2(ω)∈ C(Sⁿ⁻¹), µ2(ω) = ˜µ2(ω) +ωµˆ2(ω) and µ1(ω, p) = ¯µ1(ω, p) exp(−p˜µ2(ω)). Then we conclude

µ(x, ω, p) =µ1(ω, p) exp(hµ2(ω), xi), which was to be proved.

Theorem 3.2. The generalized dual Radon transformation R^t_λ on C_c^∞(Sⁿ⁻¹×R) is translation invariant if and only if it is exponential.

This theorem can be proven by imitation of the previous proof so we leave it to the reader. The following theorem shows how to chooseR^t_λso that the operator R^t_λ◦R_µ is translation invariant.

Theorem 3.3. If R_µ andR^t_λ are exponential, the operator R^t_λ◦R_µ is translation invariant if and only ifµ1(ω, p)λ1(ω, p)does not depend onpandµ2(ω)+λ2(ω)≡0.

Proof. It is clear thatR^t_λ◦Rµ is translation invariant if and only ifν ≡η i.e.

µ₁(ω, p) exp(h−µ2(ω), ai)

µ1(ω, p+ha, ωi) = λ₁(ω, p+ha, ωi) exp(hλ2(ω), ai)

λ1(ω, a) .

Leta=αω+βω, where¯ ω⊥ω. The dependence on¯ β gives our second statement from which the other statement of the theorem easily follows.

We note here that Quinto has got by his Proposition 4.1. of [6] the following formulas in (20) and (21) for the translation invariant Radon transforms

Rf(ω, p) = Z

H(ω,p)

f(x)µ1(ω, p) exp(hz, xi)dx, (20^∗)

R^tf(x) = exp(h−z, xi) Z

Sⁿ⁻¹

f(ω,hω, xi) a(ω) µ1(ω,hω, xi)dω, (21^∗)

where we used Quinto’s notations and µ₁(ω, p) = p

m(0)a(ω)/n(hω, pi). Since z constant this is clearly less general than our formulas. In the proof of the following theorem we will show the point where the double fibration model, used by Quinto, simplifies his result.

Theorem 3.4. The operatorR^t_λ◦R_µ is translation invariant if and only if for all y ∈Rⁿ andω∈y^⊥∩Sⁿ⁻¹ the function

λ(x, ω,hω, xi)(µ(x+y, ω,hω, xi) +µ(x−y, ω,hω, xi)), does not depend on xforx⊥y.

Proof. We follow Quinto’s method and notations [6]. LetK be SO(n),L be the isotropy subgroup ofe_x=x/|x|andM be the isotropy subgroup ofe∈Sⁿ⁻¹∩e^⊥_x. Furthermore let dk, dl and dm be the invariant measures on these groups with total measure one. Then

R^t_λ◦Rµf(x) = Z ∞

K

Z ∞

0

Z ∞

L

(rⁿ⁻²f(x+rkle)λ(x, kex,hkex, xi)×

×µ(x+rkle, kex,hkex, xi))dldrdk, where C =|Sⁿ⁻¹||Sⁿ⁻²|. Let ¯a(x, y, ω) =λ(x, ω,hω, xi)µ(x+y, ω,hω, xi). Using

¯

a, reversing the integrations with respect to dl and dk, the substitutionkl⁻¹ for kgives

R^t_λ◦Rµf(x) =C Z ∞

0

Z

K

(rⁿ⁻²f(x+rke)¯a(x, rke, kex))dk dr

by the right invariance of dk. Let dkm be the K invariant measure on K/M satisfyingdk=dm dkm. Then

(4) R^t_λ◦Rµf(x) =C Z ∞

0

Z

K/M

rⁿ⁻²f(x+rke) Z

M

¯

a(x, rke, kex)dm dkmdr.

The inner integral over M multiplied by |Sⁿ⁻²| is clearly the integral of

¯

a(x, rke, ke_x) over the great sphere Sⁿ⁻¹∩H(ke,0) in its standard measure. We denote it by ˆa(x, rke, ke). Thus we have

(5) R^t_λ◦Rµf(x) = Z

Rⁿ

f(x+y)ˆa x, y, y

|y|

/|y|dy.

Using the translation invariance and evaluatingR^t_λ◦R_µf on two specific distribu- tions Quinto proved that

(6) ˆa(x,0, ω) = ˆa(0,0, ω) and if x⊥ω then ˆa(x, rω, ω) = ˆa(0, rω, ω),

Math. Balkanica, 5(1991), 40–46. c A. Kurusa^´

whereω∈Sⁿ⁻¹. Writing the second equation into (5) one can easily see that these equations are not only necessary but sufficient too. At the same time the second equation clearly implies the first one.

Since the transformation ‘ ˆ ’ is invertible on even functions ofω ∈ Sⁿ⁻¹[2], the first equation of (6) is equivalent to ¯a(x,0, ω) = ¯a(0,0, ω) =a(ω). Using the speciality of the double fibration model Quinto could calculate the measureµhere, substantially from the square root of a. We can not do this in our situation.

Unfortunately the integrand in the second equation of (6) may be not even and so it is equivalent to the even part of the integrand being zero. This is just the condition in the theorem.

One can easily findµandλforR^t_λ◦Rµbeing translation invariant, but noRµ

norR^t_λare exponential ifµandλare allowed to be not strictly positiv. For example µ(x, ω, p) = cos(hx, ϕ(ω)i) andλ(x, ω, p) = 1/cos(hx, ϕ(ω)i), whereϕ(ω)⊥ω. But if I insisted on the strictly positivity of µand λI could only find the exponential µand λwhile I was unable to prove the uniqueness of this type.

4. Inversion formula and support theorem

LetR_µ andR^t_λ be exponential i.e.

Rf(ω, p) = Z

H(ω,p)

f(x)µ1(ω, p) exp(hµ2(ω), xi)dx, R^t_λf(x) = exp(h−z, xi)

Z

Sⁿ⁻¹

f(ω,hω, xi)exp(hµ₂(ω), xi) µ1(ω,hω, xi) dω.

We know then that the operator R_λ^t ◦Rµ is translation invariant on C_c^∞(Rⁿ).

Furthermore one can easily prove thatR^t_λ◦Rµis continuous linear map ofC_c^∞(Rⁿ) into C^∞(Rⁿ). Thus there must exist a temperate distribution T [1] for which R^t_λ ◦R_µf = T ∗f, where ∗ is the convolution operator. This gives inversion formula through the Fourier transformF [1] as

(7) f =R^t_λ◦Rµf ∗F⁻¹(1/F T).

Theorem 4.1. If Rµ is an exponentiali.e. translation invariant Radon transform, then R^t_λ◦Rµ is invertible onC_c^∞(Rⁿ)with the formula (7), where

T(x) = Z

Sⁿ⁻²⊥x

exp(hµ2(ω), xi)dω.

The proof of this theorem is immediate from (5) and left to the reader.

To finish the paper now we prove the so called support theorem for translation invariant Radon transforms. Our proof is a generalization to higher dimensions of A. Hertle’s proof in [3].

Theorem 4.2. Let R_µ be exponential Radon transform and f be a Lipschitz con- tinuous function of compact support on Rⁿ. If r > 0 and Rµf is supported in {(ω, p)∈Sⁿ⁻¹×R:|p| ≤r}, then f is supported in{x∈Rⁿ:|x| ≤r}.

Proof. If U is a rotation, then Rµf ◦U = Rµ¯(f ◦U), where ¯µ1 = µ1◦U and

¯

µ₂=U⁻¹◦µ₂◦U. Since ¯µis also exponential it suffices to show thatf is zero on every hyperplane perpendicular to the first coordinate axis withx1=p > r. Since µ₁ is strictly positive we can omit it. Thus it remains to prove

(8)

Z

H(ω,p)

xif(x) exp(hµ2(ω), xi)dx= 0 1≤i≤n,

where ω= (1,0, . . . ,0) andx= (x1, . . . , xn). Then by induction the factorxi can be replaced by any polynomial inx1, . . . , xn that implies f ≡0 onH(ω, p) by the Weierstrass theorem. To show (8) let U_ϕⁱ be the rotation in the plane of the first and i-th coordinate axis by angleϕ i, e,

U_ϕⁱ(x₁, . . . , x_n) = (x₁cosϕ+x_isinϕ, x₂, . . . , x_i−1,−x1sinϕ+x_icosϕ, x_i+1, . . . , x_n).

Differentiating the equationR_µf(U_ϕⁱω, p) = 0 with respect toϕand puttingϕ= 0 we obtain for i≥2 that

0 = Z

[−x_i∂₁f(x) +p∂_if(x)] exp(hµ₂(ω), xi)dx+

+ Z

f(x) exp(hµ2(ω), xi)[h∂iµ₂(ω), xi+pµⁱ₂−x_iµ¹₂]dx, where the integration is over H(ω, p) andµⁱ₂ is thei-th coordinate function ofµ2. Let the left hand side function in (8) be denoted by fi(p). Then this result gives the differential equation system

d

dpf_i(p) =

n

X

j=2

∂_iµ^j₂(ω)f_j(p), from which

(f₂(p), ..., f_n(p)) = (vector) exp([∂_iµ^j₂(ω)]p)

Math. Balkanica, 5(1991), 40–46. c A. Kurusa´

follows for allp≥r, where [∂_iµ^j₂(ω)] is (n−1)×(n−1) matrix. Since the left hand side has compact support so does the right hand side too , i.e. vector = 0 and the theorem is proved.

References

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[2] S. HELGASON,Groups and geometric analysis, Academic Press, Inc., 1984.

[3] A. HERTLE, On the injectivity of the attenuated Radon transform, Proc. AMS, 92 (1984), 201–205.

[4] ´A. KURUSA, New unified Radon inversion formulas, Acta Math. Hung., 60(1992), 283–290.

[5] A. MARKOE, Fourier inversion of the attenuatedX-ray transform, SIAM J. Math.

Anal., 15(1984), 718–722.

[6] E. T. QUINTO, The dependence of the generalized Radon transform on defining measures, Trans. AMS, 257(1980), 331–346.

[7] L. A. SHEPand J. B. KRUSKAL, Computerized tomography: The new medicalX-ray technology, Amer. Math. Monthly, 85(1978), 420–439.

[8] O. TRETIAKand C. METZ, The exponential Radon transform, SIAM J. Appl. Math., 39(1980), 341–354.

A. K´ URUSA, Bolyai Institute, Aradi v´ertan´uk tere 1. 6720 Szeged, Hungary; e-mail:

kurusa@math.u-szeged.hu