# Inversion of the Attenuated X-Ray Transforms: Method of Riesz Potentials

Authors: Yu Yufeng
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

The attenuated X-ray transform arises from the image reconstruction in single-photon emission computed tomography. The theory of attenuated X-ray transforms is so far incomplete, and many questions remain open. This paper is devoted to the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functionsμ, integrable on any straight line of the plane. By constructing the symmetric attenuated X-ray transform Aμ on the plane and using the method of Riesz potentials, we obtain the inversion formula of the attenuated X-ray transforms on Lpℝ21≤p<2 space, with nonnegative attenuation functions μ, integrable on any straight line in ℝ2. These results are succinct and may be used in the type of computerized tomography with attenuation.


---

## Body

## 1. Introduction

Computerized tomography (CT) means the reconstruction of a function from its line or plane integrals. The Radon transforms are the bases of the mathematics of computerized tomography [1]. Denote the hyperplane in ℝn with normal vector θ∈Sn−1 and with distance tt∈ℝ from the origin by(1)Hθ,t=x∈ℝn:x⋅θ=t,where “·” is the standard inner product in ℝn. Then, the Radon transform [2–5] of a function f on ℝn is defined by(2)Rfθ,t=∫Hθ,tfydyH,where dyH is the Lebesgue measure on H (θ, t). When n equals 2, the hyperplane H (θ, t) is a straight line, denoted by l (θ, t), and the Radon transform becomes the X-ray transform:(3)Xfθ,t=∫lθ,tfxdxl,where dxl is the Lebesgue measure on the straight line l (θ, t). Of course, there are still X-ray transforms in n-dimensional spaces for n > 2, see [4], Chap. 1 and [1], Chap. 2. A generalization of the Radon transform is the k-plane transform ([5], Chap. 3), which integrates a function over translates of k-dimensional subspaces of ℝn. If k = n − 1, then this is precisely the Radon transform. If k = 1, this integrates function over lines and is just the X-ray transform. In dimension 2, there is no difference between the Radon transforms and the X-ray transforms, whereas in higher dimensions, there are significant differences.The Radon transforms are used not only in practical fields [1, 2, 5, 6] but also in theoretical fields, for example, integral geometry [3, 6]. For practical or theoretical purposes, varieties of inversion formulas of the Radon transforms are created. There are several classical methods for the inversion of the Radon transforms, such as the method of mean value operators [7, 8], the method of Riesz potentials [1, 7], the convolution-backprojection method [1, 2, 5, 6, 9, 10], and the continuous ridgelet transform method [10].The attenuated X-ray transforms arise from the single-photon emission computed tomography (SPECT) [1]. The theory of the attenuated X-ray transforms is so far incomplete, and many questions remain open. For about twenty years, it has been an open problem whether the attenuated X-ray transform Xμ is invertible on ℝ2. Until 1998, a positive answer was given by [11], in light of the theory of so-called A-analytic functions. In 2000, a breakthrough was made by Novikov [12], who found an explicit inversion formula for Xμ with general attenuation μ. And the injectivity of Xμ on ℝ2 can be derived from his inversion formula. His results were known in 2000 but were formally published in 2002. In 2001, Natterer [13] gave a reformulation of the Novikov formula and proved it in a simple and convenient way for sufficiently smooth and fast decaying attenuations μ and for continuously differentiable test functions f. But, he also indicated that it is difficult to determine exactly the class of functions f for which Novikov’s formula holds. In 2003, Boman and Strömberg [14] developed an inversion formula for a generalized Radon transform related to but more general than the attenuated X-ray transform. For the history of the attenuated X-ray transform, the readers could refer to [5], Sec. 5.3.In the mathematics of single-photon emission computerized tomography, the attenuation function (nonnegative)μ (x) on ℝ2 is given first, and the attenuated X-ray transform (see [2], Sec.8.8; [15]; [16], p.432; [13], p.113) is defined by(4)Xμfθ,s=∫x⋅θ=sfxexp−Dμx,θ⊥dxl,for θ,s∈S1×ℝ1, where Dμ is the divergent beam transform of μ,(5)Dμx,θ=∫0+∞μx+tθdt,forx,θ∈ℝ2×S1,f is the density function, dxl is the Lebesgue measure on the straight line lθ,s=x∈ℝ2:x⋅θ=s, and θ⊥ = (−θ2, θ1) for θ=θ1,θ2∈S1. When μ ≡ 0, the attenuated X-ray transform Xμ becomes the X-ray transform X, namely, the Radon transform R on ℝ2.Obviously,0≤exp−Dμx,θ⊥≤1 for all x and θ. From the existence of the Radon transforms ([17], Theorem 4.28), we have (Xμf)(θ, s) which exists and is finite for almost all θ,s∈S1×ℝ1 when f∈Lpℝ21≤p<2. From the mapping properties of the Radon transforms ([17], Theorem 4.34), we know that(6)∫S1∫ℝXμfθ,srdsq/rdθ1/q≤cp,q,rfp,for 1 ≤ p < 2, 1 ≤ q ≤ p/(p − 1) and r = p/(2 − p), with cp,q,r a constant depending only on p, q, r.This paper is devoted to the inversion of the attenuated X-ray transforms onLpℝ21≤p<2 space, with varying nonnegative attenuation functions μ, integrable on any straight line in ℝ2. Generally, (Xμf)(θ, s) ≠ (Xμf)(−θ, −s) for θ,s∈S1×ℝ1, that is, Xμf has two different values on the same straight line l (θ, s). This differs from that of the X-ray transforms. The X-ray transform Xf has unique value on the same straight line l (θ, s). In other words, the X-ray transforms have the symmetry property (Xf)(θ, s) = (Xf)(−θ, −s) for all θ,s∈S1×ℝ1. Therefore, Xf can be regarded as a function on the set of straight lines in the plane, denoted by ℙ2, whereas the attenuated X-ray transform Xμf cannot be seen as a function on ℙ2, but on S1×ℝ1 instead. In account of these facts, we construct the symmetric attenuated X-ray transform Aμ, and then by the method of Riesz potentials [1, 7], we obtain the inversion formula of the attenuated X-ray transform Xμ on Lpℝ21≤p<2 space, with nonnegative attenuation function μ, integrable on any straight line in ℝ2. These results are succinct and may be used in the type of computerized tomography with attenuation.


## 2. Preliminaries: Symmetric Attenuated X-Ray Transforms and Riesz Potentials

In this section, we construct the symmetric attenuated X-ray transformAμ on the plane and then convert it into an operator similar to the Riesz potentials I21 ([18], Sec. 25) on with the aid of the dual X-ray transform X∗ ([1], Chap. 2), where Inα is defined by(7)Inαfx=f∗kαx,with kαy=γnα−1yα−n and x,y∈ℝn, see [18], Sec. 25. These processes are the preliminaries for the inversion of Aμ in the next section.The symmetric attenuated X-ray transformAμ on the plane is defined by(8)Aμfθ,s=12Xμfθ,s+Xμf−θ,−s,(9)=12∫x⋅θ=sfxexp−Dμx,θ⊥+exp−Dμx,−θ⊥dxl,for θ,s∈S1×ℝ1 and nonnegative function μ on ℝ2. Throughout this paper, we suppose that the attenuation function μ is nonnegative on ℝ2. It is easily verified that, like the X-ray transforms, the symmetric attenuated X-ray transforms Aμ also have the symmetry property (Aμf)(θ, s) = (Aμf)(−θ, −s) for all θ,s∈S1×ℝ1, which is crucial to our inversion of the attenuated X-ray transforms. Due to this property, Aμf can be viewed as a function on the set of straight lines in a plane.Subsequently, based on the symmetry property, a natural idea is whether we can invert the symmetric attenuated X-ray transformsAμ in a similar way to the X-ray transforms. In this paper, we consider the method of Riesz potentials [1, 7] to the inversion of X-ray transforms, where the dual X-ray transform X∗ is used:(10)X∗gx=∫S1gθ,x⋅θdθ.We attempt to representX∗Aμ as some convolution operator with the Riesz kernel on ℝ2, namely, an operator similar to the Riesz potentials. First, some preliminaries are needed for our derivation.Lemma 1 (see [18], Sec. 25.3).
Forf∈Lpℝ21≤p<2 and for almost all x∈ℝ2, the Riesz potential of f,(11)I21fx=1γ21∫ℝ2fx+yydy,exists and is finite.Lemma 2.
Let(12)Cμx,θ=12exp−Dμx,θ+exp−Dμx,−θ,for x∈ℝ2,θ∈S1. Then,(13)Cμx,θ=Cμx,−θ,(14)0≤Cμx,θ≤1.Proof.
The proof is simple.Lemma 3.
The following equalities hold:(15)∫S1∫lθ,0y⋅θ⊥≥0fydyldθ=∫ℝ2fyydy,∫S1∫lθ,0y⋅θ⊥≥0fydyldθ=∫ℝ2fyydy,∫S1∫lθ,0fydyldθ=2∫ℝ2fyydy,provided that either side of these equalities is finite when f is replaced by |f|.Proof.
Applying the formula of polar coordinate transforms for integrals, we can get the proof of this lemma.
Next, we give the main results of this section.Theorem 1.
Forf∈Lpℝ21≤p<2 and for almost all x∈ℝ2,(16)X∗Aμfx=fμ,x∗Kx,where(17)fμ,xz=fμz,x=2fzCμz,z−x′,with y′ = y/|y|, and(18)Ky=1y.Proof.
By the definitions ofAμ and X∗, (9), and (10), we have(19)X∗Aμfx=∫S1∫lθ,x⋅θfyCμy,θ⊥dyldθ,where Cμ (y, θ) is defined by (12). Through the change of variables in the right-hand side, (19) gives(20)X∗Aμfx=∫S1∫lθ,0fx+yCμx+y,θ⊥dyldθ=∫S1∫lθ,0y⋅θ⊥≥0fx+yCμx+y,y′dyldθ+∫S1∫lθ,0y⋅θ⊥≥0fx+yCμx+y,y′dyldθ.
By Lemma1, for f∈Lpℝ21≤p<2 and for almost all x∈ℝ2, the following integral(21)∫ℝ2fx+yydy,exists and is finite. And more, Cμ is bounded by (14). Thus, from Lemma 3 and (20), it follows that(22)X∗Aμfx=∫ℝ2fx+yCμx+y,y′ydy+∫ℝ2fx+yCμx+y,−y′ydy,for almost all x∈ℝ2. By (13), we have(23)Cμx+y,−y′=Cμx+y,y′.
Plugging (23) into (22) gives(24)X∗Aμfx=2∫ℝ2fx+yCμx+y,y′ydy=2∫ℝ2fx−yCμx−y,−y′ydy=fμ,x∗Kx,where fμ,x is defined by (17), and K is defined by (18). This completes the proof.Remark 1.
In Theorem1, the kernel function K (y) is the Riesz kernel |y|−1 (ignoring the constant). Thus, X∗Aμ is an operator similar to the Riesz potential I21, neglecting the difference between fμ,x and f.


## 3. Derivation of the Inversion Formula

In this section, we pursue the inversion of the operatorX∗Aμ in the above section. Then, the inversion of the attenuated X-ray transforms Aμ can be consequently obtained. Indeed, if we define the operator Bμ = X∗Aμ and denote the left inverse operator of Bμ by Bμ−1, then Bμ−1Bμ=Bμ−1X∗Aμ=I, where I is the identity transform. The operator Bμ−1X∗ can be viewed as the left inverse operator of Aμ.By Remark1, we know that the operator Bμ is similar to the Riesz potential I21. Then, a natural idea is whether we can invert the operator Bμ in a similar way to the Riesz potential I21 ([18], Sec. 26). This is the goal of this section. First, we need to introduce the truncated hypersingular integral operator Dϵ (ϵ > 0) ([18], Sec. 26):(25)Dεfx=∫y≥εfx−fx−yy3dy,where f (x) − f (x − y) is the finite difference of order 1 of function f with a step y and with center at the point x ([18], Sec. 25). Set φx=Bμfx. Then, Dϵφ can be written as a form of convolution, stated as follows.Theorem 2.
Letφx=Bμfx. Then, for f∈Lpℝ21≤p<2 and for almost all x∈ℝ2,(26)Dεφx=∫ℝ2fμ,xx−εξK˜ξdξ,where(27)K˜ξ=1ξ2∫z|≤|ξKz−Kz−e1dz,and K is defined by (18).Proof.
We begin the proof with the difference ofφ. Applying Theorem 1 and then changing the variables of integral, we have(28)Bμfx−Bμfx−y=fμ,x∗Kx−fμ,x∗Kx−y,(29)=∫ℝ2fμ,xx−ξΔξ,ydξ,where(30)Δξ,y=Kξ−Kξ−y.
Thus, combination of (25) and (29) and then changing the order of integration give(31)Dεφx=∫y≥ε1y3∫ℝ2fμ,xx−ξΔξ,ydξdy,(32)=∫ℝ2fμ,xx−ξ∫y≥ε1y3Δξ,ydydξ.
Lete1 = (1, 0) and ωx:e1⟶x/x=x′ be a rotation converting e1 into x′. It is easy to check that ωy′−1ξ′ and ωξ′−1y′ are symmetric with respect to the horizontal axis in ℝ2. Hence,(33)ξ−ky=yωy−1ξy−ke1=yξy2ωξ−1y−ke1.
Putξ/y2ωξ−1y=z in the right-hand side of (32). Then, ξ/y=z, y=y2/ξωξz=ξ/zωξz′, and dy=ξ2/z4dz. Thus, from (32) and (33), it follows that(34)Dεφx=∫ℝ2fμ,xx−ξ1ξ2∫z≤ξ/εΔzdzdξ,where(35)Δz=Kz−Kz−e1.
Through the change of variablesξ ⟶ ϵξ in the right-hand side, (34) turns into (26).
Finally, we verify the reasonability of interchanging the order of integration in the right-hand side of (31) for almost all x∈ℝ2. By Fubini’s theorem, the boundedness of Cμ (14), and the definition of Δ (ξ, y) (30), it suffices to prove that the following integrals exist and are both finite:(36)I1x=∫y≥ε1y3∫ℝ2fx−ξξdξdy,I2x=∫y≥ε1y3∫ℝ2fx−ξξ−ydξdy.
Easy computation gives(37)I1x=2πε∫ℝ2fx−ξξdξ,which exists and is finite for f∈Lpℝ21≤p<2 for almost all x∈ℝ2 and each ϵ > 0 following from Lemma 1.
For |ξ − y| ≥ 1, by Hölder’s inequality, we have for all x∈ℝ2,(38)∫ξ−y≥1fx−ξξ−ydξ≲fp,where “≲” means less than up to a constant, which implies(39)∫y≥ε1y3∫ξ−y≥1fx−ξξ−ydξdy<+∞.
For |ξ − y| ≤ 1, by the generalized Minkowski inequality, we have(40)∫ξ−y≤1fx−ξξ−ydξp,x≤2πfp,where ‖·‖p,x denotes the Lp-norm of function with respect to variable x. Combination of (40) and the generalized Minkowski inequality yields for almost all x∈ℝ2 and for each ϵ >  0 that(41)∫y≥ε1y3∫ξ−y≤1fx−ξξ−ydξdy<+∞.
Hence, from (39) and (41), it follows that I2 (x) exists and is finite for almost all x∈ℝ2 and for each ϵ > 0, which completes the verification.
Now, we consider the inversion ofBμ. For l∈ℙ2, let(42)L1l=μ:∫lμydyl<+∞.Theorem 3.
The operatorDf=limε⟶0Dεf is the left inverse to Bμ within the frames of the spaces Lpℝ21≤p<2 up to a bounded function mathcalK (x), that is,(43)DBμfx=Kxfx,for f∈Lpℝ21≤p<2 in the Lp-norm sense, where(44)Kx=2∫ℝ2K˜ξCμx,ξ′dξ,with K˜ defined by (27) and μ ∈ L1 (l) for all l∈ℙ2.Proof.
We use the Lebesgue dominated convergence theorem to prove this theorem. By Theorem2 and (44), we have(45)DεBμfx−Kxfxp,x=∫ℝ2fμ,xx−εξK˜ξdξ−2fx∫ℝ2K˜ξCμx,ξ′dξp,x=∫ℝ2fμ,xx−εξ−2fxCμx,ξ′K˜ξdξp,x.
Due to the generalized Minkowski inequality, we have(46)∫ℝ2fμ,xx−εξ−2fxCμx,ξ′K˜ξdξp,x≤∫ℝ2fμ,xx−εξ−2fxCμx,ξ′p,xK˜ξdξ.
Next, we investigate ∥fμ,x (x − ϵξ) − 2f (x) Cμ (x, ξ′)∥p,x. By the definition of fμ,x (17) and (13), we have(47)fμ,xx−εξ−2fxCμx,ξ′p,x=2fx−εξCμx−εξ,ξ′−2fxCμx,ξ′p,x≤I+II,where(48)I=2fx−εξ−fxCμx−εξ,ξ′p,x,(49)II=2fxCμx−εξ,ξ′−Cμx,ξ′p,x.
Set(50)ωpf,h=∫ℝ2fx−h−fxpdx1/p.
Then, from (14) and (48), it follows that(51)I≤2ωpf,εξ⟶0,ε⟶0,for f∈Lpℝ21≤p<2.
It follows from (12) that(52)Cμx−εξ,ξ′−Cμx,ξ′≤12exp−Dμx−εξ,ξ′−exp−Dμx,ξ′+12exp−Dμx−εξ,−ξ′−exp−Dμx,−ξ′.
For allh∈ℝ2 and all θ∈S1,(53)Dμx+εh,θ−Dμx,θ≤∫lθ⊥,x⋅θ⊥μy+εh−μydyl.
Becauseμ ∈ L1 (l) for all l∈ℙ2,(54)∫lθ⊥,x⋅θ⊥μy+εh−μydyl⟶0,ε⟶0.
Thus, from (53) and (54), it follows that(55)Dμx+εh,θ−Dμx,θ⟶0,ε⟶0.
Then, from (52) and (55),) it follows that(56)Cμx−εξ,ξ′−Cμx,ξ′⟶0,ε⟶0.
By (14), (49), (56), and the Lebesgue dominated convergence theorem, we have the right-hand side of (49) which tends to 0 as ϵ tends to 0. Thus, from (49), it follows that(57)II⟶0,ε⟶0.
Combination of (47), (51), and (57) yields(58)fμ,xx−εξ−2fxCμx,ξ′p,x⟶0,ε⟶0.
Due to (14) and (17), it follows that(59)fμ,xx−εξ−2fxCμx,ξ′p,x≤fμ,xx−εξp,x+2fxCμx,ξ′p,x≤4fp.
The functionK˜ defined by (27) plays the role of the approximate kernel in the proof of Theorem 3. Thus, the integrability of K˜ on ℝ2 is a necessary condition, which can be obtained by Lemma 26.4 in [18]. Actually, the function K˜ can be denoted by Kl,α⋅ (neglecting the constant cl,α,n), defined by formula (26.30) in [18]:(60)Kl,αx=cl,α,nxn∫y<x∑k=0l−1llky−ke1α−ndy,with l = 1, α = 1, n = 2, and lk=l⋅l−1⋅⋅⋅l−k+1/1⋅2⋅⋅⋅k, e1 = (1, 0). From the conclusion that Kl,α⋅∈Lqℝn,1−α/n<1/q≤1,α>0,q>0,l=1,2,3,… in Lemma 26.4 in [18], we know that K1,1⋅∈Lℝ2 (letting l = 1, α = 1, and n = 2), namely,(61)∫ℝ2K˜ξdξ<+∞.
Thus, from (58), (59), (61) and the Lebesgue dominated convergence theorem, it follows that(62)∫ℝ2fμ,xx−εξ−2fxCμx,ξ′p,xK˜ξdξ⟶0,ε⟶0.
Finally, from (55), (56), and (62), it follows that(63)DεBμfx−Kfxp,x⟶0,ε⟶0,and the boundedness of Kx follows from (14), (44), and (61).


## 4. Main Results and Discussion

Based on the inversion of the operatorX∗Aμ in Section 3, we have the following inversion of the attenuated X-ray transforms Xμ on Lpℝ21≤p<2 space. By Theorem 3 and the definition of operator Bμ, we have the following.Theorem 4.
For the attenuated X-ray transformsXμ with μ ∈ L1 (l) for all l∈ℙ2, the following formula holds:(64)DX∗Aμfx=Kxfx, for f∈Lpℝ21≤p<2 in the senses of Lp-norm and almost everywhere on ℝ2, where Dφ=limε⟶0Dεφ, K is a bounded function on ℝ2 defined by (44), and Aμ is the symmetric attenuated X-ray transforms defined by (8).The correctness of Theorem4 ensures that the mapping Aμ f ⟶ f exists when f∈Lpℝ21≤p<2, that is, the symmetric attenuated X-ray transforms Aμ, and thus the attenuated X-ray transforms Xμ, are invertible when f∈Lpℝ21≤p<2. The uniqueness of f is then in the almost everywhere sense due to the metric in the Lp space.Finally, we give some comparisons between some classic results and our inversion formula (64). Novikov’s inversion formula [12] is little complicated, where some smoothing conditions and decaying conditions at infinity are needed for μ. Natterer’s inversion formula [13] is derived but more succinct in form than Novikov’s, where the class of μ is not definitely determined. The divergence operator div and the compound Hehg of Hilbert transform H, h=1/2I+iHXμ and g = Xμ f (I, the identity operator; i, the imaginary number), are all simultaneously involved in Natterer’s inversion formula, which makes the formula not simple. Our inversion formula is relatively concise, which mainly contains the operators X∗ and Dϵ, where only the integrability condition of μ is needed, namely, μ ∈ L1 (l) for all l∈ℙ2. The operator X∗ is an ordinary integral operator on S1, and the operator Dϵ is a truncated hypersingular integral operator on ℝ2, see [18], Sec. 26 for more details. The appearance of the bounded function Kx may be a defect of our inversion formula. But, it seems inevitable for the attenuated X-ray transforms by our model. The ideal one is that Kx is identically equivalent to a nonzero constant, which can be attained for μ (x) ≡ 0 when the attenuated X-ray transforms become the X-ray transforms and Cμ ≡ 1, whereas which may never be attained for general attenuated X-ray transforms due to the nonvanishing attenuation functions μ.



---

*Source: 5896328-2020-03-23.xml*