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

'e attenuated X-ray transform arises from the image reconstruction in single-photon emission computed tomography. 'e theory of attenuated X-ray transforms is so far incomplete, and many questions remain open. 'is paper is devoted to the inversion of the attenuated X-ray transforms with nonnegative varying attenuation functions μ, integrable on any straight line of the plane. By constructing the symmetric attenuated X-ray transformAμ on the plane and using the method of Riesz potentials, we obtain the inversion formula of the attenuated X-ray transforms on Lp(R2)(1≤p< 2) space, with nonnegative attenuation functions μ, integrable on any straight line in R2. 'ese results are succinct and may be used in the type of computerized tomography with attenuation.


Introduction
Computerized tomography (CT) means the reconstruction of a function from its line or plane integrals. e Radon transforms are the bases of the mathematics of computerized tomography [1]. Denote the hyperplane in R n with normal vector θ ∈ S n− 1 and with distance |t|(t ∈ R) from the origin by where "·" is the standard inner product in R n . en, the Radon transform [2][3][4][5] of a function f on R n is defined by where dy H 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: where dx l 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 R n . 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. e 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. ere 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].
e attenuated X-ray transforms arise from the singlephoton emission computed tomography (SPECT) [1]. e 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 X μ is invertible on R 2 . 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 X μ with general attenuation μ. And the injectivity of X μ on R 2 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 R 2 is given first, and the attenuated X-ray transform (see [2], Sec.8.8; [15]; [16], p.432; [13], p.113) is defined by for (θ, s) ∈ S 1 × R 1 , where Dμ is the divergent beam transform of μ, f is the density function, dx l is the Lebesgue measure on the straight line l(θ, s) � x ∈ R 2 : x · θ � s , and θ ⊥ � (− θ 2 , θ 1 ) for θ � (θ 1 , θ 2 ) ∈ S 1 . When μ ≡ 0, the attenuated X-ray transform X μ becomes the X-ray transform X, namely, the Radon transform R on R 2 . Obviously, 0 ≤ exp(− (Dμ)(x, θ ⊥ )) ≤ 1 for all x and θ. From the existence of the Radon transforms ( [17], eorem 4.28), we have (X μ f)(θ, s) which exists and is finite for almost all (θ, s) ∈ S 1 × R 1 when f ∈ L p (R 2 )(1 ≤ p < 2). From the mapping properties of the Radon transforms ( [17], eorem 4.34), we know that , with c p,q,r a constant depending only on p, q, r. is paper is devoted to the inversion of the attenuated X-ray transforms on L p (R 2 )(1 ≤ p < 2) space, with varying nonnegative attenuation functions μ, integrable on any is, X μ f has two different values on the same straight line l (θ, s). is differs from that of the X-ray transforms. e 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) ∈ S 1 × R 1 . erefore, Xf can be regarded as a function on the set of straight lines in the plane, denoted by P 2 , whereas the attenuated X-ray transform X μ f cannot be seen as a function on P 2 , but on S 1 × R 1 instead. In account of these facts, we construct the symmetric attenuated X-ray transform A μ , and then by the method of Riesz potentials [1,7], we obtain the inversion formula of the attenuated X-ray transform X μ on L p (R 2 )(1 ≤ p < 2) space, with nonnegative attenuation function μ, integrable on any straight line in R 2 . ese results are succinct and may be used in the type of computerized tomography with attenuation.

Preliminaries: Symmetric Attenuated X-Ray Transforms and Riesz Potentials
In this section, we construct the symmetric attenuated X-ray transform A μ on the plane and then convert it into an operator similar to the Riesz potentials I 1 2 ( [18], Sec. 25) on with the aid of the dual X-ray transform X with k α (y) � (c n (α)) − 1 |y| α− n and x, y ∈ R n , see [18], Sec. 25. ese processes are the preliminaries for the inversion of A μ in the next section. e symmetric attenuated X-ray transform A μ on the plane is defined by for (θ, s) ∈ S 1 × R 1 and nonnegative function μ on R 2 . roughout this paper, we suppose that the attenuation function μ is nonnegative on R 2 . It is easily verified that, like the X-ray transforms, the symmetric attenuated X-ray transforms A μ also have the symmetry property ( which is crucial to our inversion of the attenuated X-ray transforms. Due to this property, A μ f 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 transforms A μ 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: We attempt to represent X * A μ as some convolution operator with the Riesz kernel on R 2 , namely, an operator similar to the Riesz potentials. First, some preliminaries are needed for our derivation.
for x ∈ R 2 , θ ∈ S 1 . en, Proof. e proof is simple. □ Lemma 3. e following equalities hold: 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. For f ∈ L p (R 2 )(1 ≤ p < 2) and for almost all where with y′ � y/|y|, and Proof. By the definitions of A μ and X * , (9), and (10), we have where Cμ (y, θ) is defined by (12). rough the change of variables in the right-hand side, (19) gives f(x + y)Cμ x + y, y ′ dy l dθ f(x + y)Cμ x + y, y ′ dy l dθ. (20) By Lemma 1, for f ∈ L p (R 2 )(1 ≤ p < 2) and for almost all x ∈ R 2 , the following integral exists and is finite. And more, Cμ is bounded by (14). us, from Lemma 3 and (20), it follows that for almost all x ∈ R 2 . By (13), we have Plugging (23) into (22) gives where f μ,x is defined by (17), and K is defined by (18). is completes the proof.
□ Remark 1. In eorem 1, the kernel function K (y) is the Riesz kernel |y| − 1 (ignoring the constant). us, X * A μ is an operator similar to the Riesz potential I 1 2 , neglecting the difference between f μ,x and f.

Derivation of the Inversion Formula
In this section, we pursue the inversion of the operator X * A μ in the above section. en, the inversion of the attenuated X-ray transforms A μ can be consequently obtained. Indeed, if we define the operator B μ � X * A μ and denote the left inverse operator of B μ by B − 1 μ , then 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 φ(x) � (B μ f)(x). en, D ϵ φ can be written as a form of convolution, stated as follows. where and K is defined by (18).
Finally, we verify the reasonability of interchanging the order of integration in the right-hand side of (31) for almost all x ∈ R 2 . 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: Easy computation gives which exists and is finite for f ∈ L p (R 2 )(1 ≤ p < 2) for almost all x ∈ R 2 and each ϵ > 0 following from Lemma 1. For |ξ − y| ≥ 1, by Hölder's inequality, we have for all where "≲" means less than up to a constant, which implies For |ξ − y| ≤ 1, by the generalized Minkowski inequality, we have where ‖·‖ p,x denotes the L p -norm of function with respect to variable x. Combination of (40) and the generalized Minkowski inequality yields for almost all x ∈ R 2 and for each ϵ > 0 that Hence, from (39) and (41), it follows that I 2 (x) exists and is finite for almost all x ∈ R 2 and for each ϵ > 0, which completes the verification. Now, we consider the inversion of B μ . For l ∈ P 2 , let □ Theorem 3. e operator Df � lim ε⟶0 D ε f is the left inverse to B μ within the frames of the spaces L p (R 2 )(1 ≤ p < 2) up to a bounded function mathcalK (x), that is, for f ∈ L p (R 2 )(1 ≤ p < 2) in the L p -norm sense, where us, from (58), (59), (61) and the Lebesgue dominated convergence theorem, it follows that Finally, from (55), (56), and (62), it follows that and the boundedness of K(x) follows from (14)

Main Results and Discussion
Based on the inversion of the operator X * A μ in Section 3, we have the following inversion of the attenuated X-ray transforms X μ on L p (R 2 )(1 ≤ p < 2) space. By eorem 3 and the definition of operator B μ , we have the following.

Theorem 4.
For the attenuated X-ray transforms X μ with μ ∈ L 1 (l) for all l ∈ P 2 , the following formula holds: for f ∈ L p (R 2 )(1 ≤ p < 2) in the senses of L p -norm and almost everywhere on R 2 , where Dφ � lim ε⟶0 D ε φ, K is a bounded function on R 2 defined by (44), and A μ is the symmetric attenuated X-ray transforms defined by (8).
e correctness of eorem 4 ensures that the mapping A μ f ⟶ f exists when f ∈ L p (R 2 )(1 ≤ p < 2), that is, the symmetric attenuated X-ray transforms A μ , and thus the attenuated X-ray transforms X μ , are invertible when f ∈ L p (R 2 )(1 ≤ p < 2). e uniqueness of f is then in the almost everywhere sense due to the metric in the L p 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. e divergence operator div and the compound He h g of Hilbert transform H, h � (1/2)(I + iH)(Xμ) and g � X μ 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 D ϵ , where only the integrability condition of μ is needed, namely, μ ∈ L 1 (l) for all l ∈ P 2 . e operator X * is an ordinary integral operator on S 1 , and the operator D ϵ is a truncated hypersingular integral operator on R 2 , see [18], Sec. 26 for more details. e appearance of the bounded function K(x) may be a defect of our inversion formula. But, it seems inevitable for the attenuated X-ray transforms by our model. e ideal one is that K(x) 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest.