Efficient tensor tomography in fan-beam coordinates. II: Attenuated transforms

This article extends the author's past work [Inv. Probl. Imaging, 10:2 (2016), 433--459] to attenuated X-ray transforms, where the attenuation is complex-valued and only depends on position. We give a positive and constructive answer to the attenuated tensor tomography problem on the Euclidean unit disc in fan-beam coordinates. For a tensor of arbitrary order, we propose an equivalent tensor of the same order which can be uniquely and stably reconstructed from its attenuated transform, as well as an explicit and efficient procedure to do so.


Introduction
We present a sequel to [11], concerned with the reconstruction of tensor fields from their X-ray transform, to the case of transforms with attenuation. Let M = {x = (x, y) ∈ R 2 , x 2 + y 2 ≤ 1} the Euclidean unit disc, a ∈ C 0 (M, C) and SM = M × S 1 the unit circle bundle of M . The variable in S 1 will be referred to as "angular". For f ∈ L 2 (SM ), we define the attenuated X-ray transform of f by where we denote ϕ t (x, v) = (x + tv, v) = (γ x,v (t), v) the Euclidean geodesic flow and denotes the ingoing boundary (data space where the transform is defined). Considering an integer m, we denote by L 2 (m) (SM ) the subspace of L 2 (SM ) consisting of elements with harmonic content in the angular variable contained in {−m, . . . , m}.
Upon restricting it to certain subspaces L 2 (m) (SM ), the transform (1) encompasses several problems: when f (x, v) = f (x) (m = 0), this is the mathematical formulation of SPECT [2,6,7,14,16]; when f is a vector field (i.e., linear in v in the above form, or m = 1), this is the mathematical formulation of Doppler Tomography [30,8,10,26]; for general m, this corresponds to symmetric m-tensors and the tensor tomography problem [29,25,19]; finally, for integrands with non-polynomial dependence (or infinite harmonic content in the angular variable), this problem has applications in the study of the Boltzmann transport equation, with applications to Bioluminescence Imaging [5,31], as will be illustrated in forthcoming work. While the first two cases have now been studied for a few decades, the present setting allows for a comprehensive understanding of the problem for integrands with general angular dependence; moreover, the fan-beam viewpoint adopted (traditionally rebinned into parallel geometry in the Euclidean case, see [15]) allows to elucidate certain questions related to X-ray transforms on manifolds, which continue to receive serious attention [3,12,18,19,21].
The attenuated tensor tomography problem we consider may be formulated as follows: given m and f ∈ L 2 (m) (SM ), what is reconstructible of f from I a f , and how to reconstruct it ? Except for isolated cases, the operator I a restricted to L 2 (m) (SM ) has a non-trivial kernel given by the following: for any h ∈ L 2 (m−1) (SM ) with spatial components in H 1 0 (M ), I a [(X + a)h] = 0 (with X = cos θ∂ x + sin θ∂ y the geodesic vector field). It is natural to ask whether these integrands are the only elements in the kernel, and we provide a positive answer to this question. Further, the size of the kernel of I a restricted to such tensors increases, and for reconstruction purposes, we present a candidate to be reconstructed modulo this kernel. As can be expected heuristically from [11], when considering tensors of order m ≥ 1, the form of the reconstructed candidate only differs from the case m = 1 by residual terms. By "residual" here we mean that such terms are harmonic in position, and therefore represent very little relevant information compared to the main "bulk" made up of two full functions in L 2 (M ) and H 1 0 (M ). In particular, these residual terms contain no singularity inside the domain. The reconstruction procedure then consists in reconstructing the residual terms first, then the bulk.
The reconstruction of the residual terms requires the explicit construction of invariant distributions, that is, distributional solutions of Xw = 0 on SM , with conditions on their moments (e.g., fiberwise holomorphic with prescribed fiberwise average). The quest for such invariant distributions has been rather active [20,21], as injectivity statements of X-ray transforms have been proven to be equivalent to the existence of certain invariant distributions [22], which can also be formulated as surjectivity results for backprojection operators (e.g., I * 0 ). A salient feature here is the explicit construction of such distributions, whose existence is usually based on ellipticity arguments [24], or series of iterated Beurling transforms in higher dimensions [21]. Such invariant distributions, via appropriate integrations by parts on SM , allow to obtain reconstruction formulas for the residual terms mentioned above.
We then carry out the reconstruction of the main bulk (g 0 and g s below). In a recent work with Assylbekov and Uhlmann [3], the author provided range characterizations and reconstruction formulas for the attenuated ray transform on surfaces, restricted to the case m = 1 (i.e., sums "function + vector field") in smooth topologies, in particular generalizing the approach in [10] to complex-valued attenuations and non-Euclidean geometries following [28,12]. We revisit these results here, and the Euclidean case allows for more precise statements. The main tools involved are a holomorphization operator (as introduced in [3]) and holomorphic integrating factors first introduced in [28] which are so crucial in two-dimensional tomography problems [3,19,18].
It should be noted that this reconstructed representative is also valid for vector fields alone (Doppler transform), though unlike the reconstruction formulas provided in [10,12], this formula provides a partial reconstruction of a vector field (namely, its solenoidal part) even where the attenuation vanishes. This is similar in spirit to [34], see also Remark 3. We restrict this article to the case of attenuations which depends on position only. Other types are considered in other settings, that is, linearly-dependent in angle [18,13]. Such "attenuations" have a different physical meaning (that of a connection, see [17,18,35]), and their inversion can sometimes be tackled without the use of holomorphic integrating factors, see the recent work [13].
Other approaches for tackling Euclidean attenuated transforms have been A-analytic function theoryà la Bukhgeim [2,10,34], leading in particular to recent range characterizations of the attenuated transform over functions, one-forms and second-order tensors [25,27,26], and Riemann-Hilbert problems, leading in particular to an efficient reconstruction of functions from their attenuated transform [6,14,16] and studies of more general integrands and partial data problems in [4].
We finally point out a few differences with the previous work [11] on unattenuated tensor tomography: • The decompositions modulo kernel presented no longer split according to the parity of the tensor order m, as attenuated transport equations now mix all even and odd angular modes.
• In order to recover the residual elements, the approach in [11] was to compute their forward transform, whose frequency content in data space could easily be described, and an inversion formula easily derived. In the attenuated case, this is no longer the case, hence the necessity of constructing special invariant distributions, combined with integrations by parts on SM .
• Unlike the attenuated case, residual elements must be reconstructed in a specific order, that is, from higher to lower.
We now state the main resuts.

Main results
Spaces and notation. Denote by C P the Poincaré constant 1 of the unit disc, that is, The spaces L 2 (SM ) and L 2 (M ) are endowed with their usual inner products denoted ·, · SM and ·, · M and corresponding norms We decompose L 2 (SM ) into circular harmonics where H 0 is isometric to L 2 (M ) and for , We also denote L 2 (m) (SM ) := m k=0 H k , such a space can be viewed as the restriction to SM of sums of symmetric tensor fields of order up to m. Denote L 2 (ker ∂) and L 2 (ker ∂) the subspaces of L 2 (M ) made of complex-analytic and antianalytic functions (where ∂ = 1 2 (∂ x − i∂ y ) and ∂ = 1 2 (∂ x + i∂ y )). Such spaces are closed in L 2 (M ) (see e.g. [32, Ex. 6 p254]), and for k ≥ 1, we then define By the observations above, H sol k is a closed subspace of H k (hence Hilbert). Remark 1. The space H sol k corresponds to restrictions to SM of trace-free (if k ≥ 2), divergencefree symmetric k-tensors. To see this, decompose a k-tensor into f = k p=0 f p σ(dz p ⊗ dz k−p ), whose trace is computed as 1 Via Rayleigh quotient, Cp = λ −1 1 , where λ1 > 0 is the smallest eigenvalue of the Dirichlet Laplacian on M , see [33,Ch. 11] Then tr σ(dz p ⊗ dz k−p ) is nonzero proportional to σ(dz p−1 ⊗ dz k−1−p ) if 0 < p < k, and zero otherwise. In particular, a trace-free k-tensor takes the form f 0 dz k + f k dz k , and its divergence is then given by div f = (∂f 0 )dz k−1 + (∂f k )dz k−1 , see, e.g., [21, Appendix B]. Hence the claim.
Remark 2. For the sake of brevity in Theorem 1 and its proof, we are changing notation slightly from [11], keeping circular harmonics of same magnitude ±k in the same subspace H k . In this correspondence and recalling the definitions η + = e iθ ∂ and η − = e −iθ ∂, H sol k,+ corresponds to L 2 (ker k η − ) in [11], and H sol k,− corresponds to L 2 (ker −k η + ).
For a ∈ C 0 (M, C), the attenuated ray transform extends into a bounded operator Proof. This is a direct consequence of [11,Lemma 4.1] and the obvious pointwise estimate |I a f (β, α)| ≤ e 2a∞ I|f |(β, α), where the factor 2 is the diameter of M and I denotes the unattenuated transform.
Gauge representatives of the attenuated transform.
We denote X = cos θ∂ x + sin θ∂ y the geodesic vector field and X ⊥ = sin θ∂ x − cos θ∂ y the "transverse derivative". Theorem 1. Let a ∈ C 0 (M, C) with supremum a ∞ and let m a natural integer. For any f ∈ L 2 (m) (SM ), there exists g ∈ L 2 (m) (SM ), linear in f , satisfying I a f = I a g, of the form Moreover, with C := 4 + 8a 2 ∞ C P , we have the following stability estimates Theorem 1 was established in the unattenuated case [11], though the dependence of the continuity estimate on m is now made explicit. Note that such an estimate for general f ∈ L 2 (SM ) could not be possible, unless we assume some decay on the angular moments of f . To this end, let us define, for κ ≥ 1 Theorem 2. Suppose f ∈ L 2,C (SM ) where C is the constant in Theorem 1. Then there exists g ∈ L 2 (SM ), linear in f , satisfying I a f = I a g, of the form Moreover, we have the estimate Uniqueness and reconstruction. We now explain the reconstruction procedure, which requires introducing additional kernels and concepts. We define the kernel constructed in (32) G(x, y; β, α) := 1 4π 2 Also introduce the so-called holomorphization operator (see (39)) for h ∈ L 2 (SM ) with A ± , A * ± defined in Sec. 3.1 and P † in Lemma 3. We finally define a holomorphic integrating factor for a (see Prop. 7) by with P * − defined in (21). The reconstruction procedure then goes as follows. Theorem 3. Considering (a, f, g, m) as in Theorem 1, the representative g is unique and is reconstructed as follows. For k = m down to 1, reconstruct g k = e ikθ g k,+ + e −ikθ g k,− ∈ H sol k via where I k := I a (g − m =k+1 g ). Then defining I := I a (g − m =1 g ) = I a (g 0 + X ⊥ g s ), the two functions g 0 , g s are reconstructed via where we have defined D := e wa ( B(Ie −ρa )) ψ , D := e w a ( B(Ie −ρ a )) ψ , and where ∂g + = ∂g − = 0, so that g ± are expressed as Cauchy formulas in terms of their known boundary conditions Remark 3 (Connection with the Doppler transform). We briefly explain how Theorem 3 applies to the Doppler transform, i.e., the attenuated transform over a vector field: a vector field V ∈ L 2 , restricted to SM , takes the form Applying a Helmholtz decomposition to V , we may write V = Xf + X ⊥ g for f ∈ H 1 0 (M ) and g ∈Ḣ 1 (M ). Then a direct calculation shows that I a V = I a (−af + X ⊥ g), out of which Theorem 3 implies that −af and g can both be reconstructed uniquely and stably in all cases. This allows to recover the following facts: • As previously established in [9], if a vanishes nowhere, then both f and g, and thus V , can be reconstructed stably throughout the domain. In general, this also clarifies why f (or the potential part of V ) can only be recovered on the support of a.
• As previously established in [34], curl V = ∆g can be reconstructed everywhere regardless of whether the attenuation vanishes.
Theorems 1 and 3 motivate the following result. Such a result was proved before in the case of smooth tensors for the magnetic ray transform in [1].
Outline and additional results. The rest of the article is organized as follows. In Section 3, we recall preliminaries on the geometry of SM and its boundary ( §3.1), and on transport equations ( §3.2). We prove Theorems 1, 2 and 4 in Section 4. The reconstruction aspects are covered in Section 5. We first describe important boundary operators in §5.1, namely P ± (defined in (20)) and provide explicit pseudo inverses P † ± for them in Lemma 3. We then show in Proposition 5 that P † ± are the "filters" in the (unattenuated) filtered-backprojection formulas for functions and solenoidal vector fields in fan-beam coordinates, allowing as a novelty for vector fields to be supported up to the boundary. This then allows us to construct holomorphic integrating factors ( §5.2) for functions and vector fields in Propositions 6 and 7. Next in Section 5.3, we cover in Theorem 9 the explicit construction of invariant distributions with one-sided harmonic content and prescribed fiberwise average. Using such distributions via integration by parts on SM , we then show in §5.4 how to reconstruct the residual terms g m ∈ H sol m . Finally, we cover the reconstruction of the last two functions (g 0 , g s ) in §5.5 in Theorem 11, defining and using along the way the holomorphization operator in Proposition 10.

Geometry of SM and its boundary
The generator of all lines in the unit disc is the geodesic vector field X = cos θ∂ x + sin θ∂ y . X is completed into a global frame of SM using X ⊥ := sin θ∂ x − cos θ∂ y and V := ∂ θ . In the harmonic decomposition (3), we define the fiberwise Hilbert transform H : We also denote H : L 2 (∂SM ) → L 2 (∂SM ) the operator defined by the same formula on the circles sitting above points at the boundary ∂M . The following commutator was derived in [24]: where we define fiberwise average u 0 (x) := 1 2π 2π 0 u(x, θ) dθ. H splits into H = H + + H − where H +/− denotes H restricted to even/odd harmonics. A function u ∈ L 2 (SM ) is called fiberwise holomorphic (resp antiholomorphic) if (Id + iH)u = 0 (resp. (Id − iH)u = 0). Fiberwise holomorphic functions are preserved under product and exponentiation (when they are defined in appropriate topologies).
We parameterize the boundary ∂SM with fan-beam coordinates (β, α) ∈ S 1 × S 1 , where x(β) = cos β sin β parameterizes the point at the boundary and v = cos(β+π+α) sin(β+π+α) is a vector at the basepoint x(β). The inward boundary ∂ + SM is the subset of ∂SM for which −π 2 ≤ α ≤ π 2 (making v inward-pointing) and the outward boundary ∂ − SM is the subset of ∂SM for which π 2 ≤ α ≤ 3π 2 (making v outward-pointing). The scattering relation S : ∂ ± SM → ∂ ∓ SM maps an ingoing/outgoing point to the outgoing/ingoing other end of the unique geodesic passing through it, and the antipodal scattering relation S A : ∂ + SM → ∂ + SM maps an ingoing point to that at the other of the unique geodesic passing through it. These maps are given explicitly by We define A ± : L 2 (∂ + SM ) → L 2 (∂SM ) the operators of even/odd extension via scattering relation, i.e. Their adjoints are given by

Transport equations and integration by parts on SM
The transform (1) may be realized as the influx trace u| ∂ + SM of the unique solution u = u f a to the transport problem For h ∈ C 0 (∂ + SM ), we denote h ψ = v ∈ C 0 (SM ) the unique solution to Using (11), we can derive the following integration by parts formula, true for any u, v ∈ C 1 (SM ): In particular, if u solves a transport equation of the form and if w is a solution of Xw = −a with ρ := w| ∂ + SM , then the function u = ue −w satisfies In addition, if v = h ψ is as in (12), then the integration by parts above applied to u and v yields: 4 Gauges and uniqueness. Proofs of Theorems 1, 2 and 4.
Preliminaries. Recall the notation η + = e iθ ∂ and η − = e −iθ ∂, and note that By virtue of the fact that [η + , η − ] = 0 and η * + = −η − , we have the following property: By density, we can extend this property to integrands of the form u = e ikθ v(x), with v ∈ H 1 0 (M ). In addition, for any such integrand, we have where the last term vanishes via Green's formula. In particular, using the Poincaré constant C P defined in (2), we have for any u = e ikθ v(x) with v ∈ H 1 0 (M ) and k integer, and similarly u 2 . These estimates will be useful below to control the growth of constants.
Moreover, we have the continuity estimate v k−1 2 ≤ 4C P f k 2 and Proof. Let f k ∈ H k and write f k = f k,+ e ikθ + f k,− e −ikθ . Using the elliptic decompositions associated with ∂, ∂ operators, write together with estimates If k > 2, the sum is orthogonal and we have The case k = 2 is a direct consequence of the inequality (a + b) 2 ≤ 2(a 2 + b 2 ). On to the stability hence the proof.
We now prove Theorem 1.
Proof of Theorem 1. The case m = 0 is trivial, with g 0 = f 0 and continuity constant C = 1. For the case m = 1, writing f = f 0 + f 1 , we decompose where v 0,± ∈ H 1 0 (M ) and g 1,± ∈ L 2 (M ) with ∂g 1,− = ∂g 1,+ = 0, with stability estimates With the identity Then the transport equation Xu + au = −f can be rewritten as Upon defining g 0 := f 0 − ag p , and since g p vanishes on ∂SM , we have Now for the estimation, we have by orthogonality By definition, Regarding X ⊥ g s , we have, using (14) so that, using (15), Combining all estimates, we obtain as advertised in (6). We now prove all cases m ≥ 2 by induction.
Then the transport equation can be rewritten in the form Since v m−1 vanishes at the boundary, h satisfies We now build the estimate, separating the cases m = 2 and m > 2. For m = 2, we have, by orthogonality of Fourier modes, Bounding each term separately and using the estimates from Lemma 2: At this point, the component h 0 + h 1 does not have the desired form, and we therefore apply the case m = 1 to it, to obtain the existence of g = g 0 + X ⊥ g ⊥ + g 1 , such that I a g = I a [h 0 + h 1 ] and, by (16), Now defining g := g +g 2 , g has the desired form and we clearly have I a g = I a h = I a f . Moreover, combining the last two estimates displayed, This in particular satisfies the base case m = 2 for estimate (6). Suppose now that m ≥ 3 and that we already decomposed f m and defined h as in the case m = 2. We have, by orthogonality where, using Lemma 2, the following estimates hold and for k ≤ m − 3, h k = f k . At this point, the term h = m−1 k=0 h k does not have the desired form, thus we apply the induction step to it, so that there exists g ∈ L 2 (m−1) (SM ) of the desired form, such that I a g = I a h with estimate Thus upon defining g = g + g m , we clearly have I a g = I a h = I a f , with estimate so that, since C = 4 + 8a 2 ∞ C P , the last term is bounded by C m f m 2 , and the hypothesis is reconducted. The proof of Theorem 1 is complete.
The generalization of Theorem 1 to integrands with infinite harmonic content with sufficient decay is then immediate.
By linearity, we have Since the right-hand-side converges to zero as m → ∞ regardless of n, the sequence g (m) is Cauchy in L 2 (SM ), thus converges to some g ∈ L 2 (SM ). By continuity of I a : L 2 (SM ) → L 2 (∂ + SM ), we have The form (7) of g is inherited from the form (5) of each g (m) and the fact that each summand belongs to a closed subspace of L 2 (SM ). Estimate (8) follows from sending m → ∞ in (19).
Assuming Theorem 3, we now provide a brief proof of Theorem 4.
Proof of Theorem 4. Let f ∈ L 2 (m) (SM ) and define u such that Examining the proof of Theorem 1 closer, we prove that there exists v of degree m − 1 with components in H 1 0 (M ), vanishing at ∂SM such that with g as in (5)  where we have defined φ p,q (β, α) := 1 π √ 2 e i(pβ+2qα) . For further use, we define is the pullback of φ by the antipodal scattering relation. In particular, we have the splitting L 2 (∂ + SM ) = V + ⊕ V − , where V + := ker(Id − S * A ) is spanned by either {u p,q } or {u p,q }, and V − := ker(Id + S * A ) is spanned by either {v p,q } or {v p,q }. Recall that splits into P = P + + P − upon defining P ± = A * − H ± A + : V ± → V ∓ , and that from [23], Range I 0 = Range P − and Range I ⊥ = Range P + in smooth topologies. It was then proved in [11] that in the L 2 (∂ + SM ) → L 2 (∂ + SM ) setting, the singular value decompositions of P ± makes them roughly L 2 → L 2 isometries onto their respective ranges. To be more specific, let us define: These subspaces of L 2 (∂ + SM ) capture exactly the modes achieved by I 0 and I ⊥ , and describing the range of these operators there would only require describing rates of decay in the Fourier coefficients. It is also immediate to find that, in the functional setting of (20), We now write an explicit right-inverse for the operator P on V +,0 ⊕ V −,⊥ . In what follows, we recall the definition of the operator C := 1 2 A * − HA − , which upon splitting H = H + + H − , splits accordingly into C + : V − → V − and C − : V + → V + . Lemma 3. The operators P † − := 1 4 P * − , P † + := 1 4 P * + (Id − 12C 2 + ) and P † := P † + + P † − are such that P − P † − , P + P † + and P P † are the L 2 (∂ + SM )-orthogonal projections onto V +,0 , V −,⊥ and V +,0 ⊕ V −,⊥ , respectively.
In particular, P † is a right inverse for P on the ranges of I 0 and I ⊥ .
as well as Considering the adjoints similar calculations allow to establish that In particular we obtain otherwise, which means that 1 4 P * − inverts P − on V +,0 . For P + , because of the appearing half spectral values, we need to modify slightly using C + , and arrive at the following if q > 0 and p < q, v p,q if (q > 0 and p = q) or (q = 0 and p < 0), 0 otherwise.
Adding everything together, we see that the operator As recorded in the proposition below, these pseudo-inverses are, in fact, the "filters" in the filtered-backprojection formulas inverting I 0 over L 2 (M ) and I ⊥ oveṙ The main novelty here is that the inversion formula for solenoidal one-forms allows solenoidal potentials to be supported up to the boundary.
Proposition 5. With the pseudo-inverses P † + and P † − defined in Lemma 3, we have the reconstruction formulas Proof of Proposition 5. Equation (23) is a direct consequence of the formula f = 1 8π I ⊥ A * + HA − I 0 f (see [11]) and the definition of P † − . On to proving (24), by [11,Lemma 4 We already know that for h 0 ∈ H 1 0 (M ), the following reconstruction formula holds (see, e.g., [12,Prop. 2

.2])
Surprisingly, the same is only true up to a factor 1 4 for the term h ∂ , as we now show that Proof of (25). It is enough to prove it for h ∂ = z k for any integer k ≥ 1 and (25) follows by linearity and complex conjugation. Applying the operators one at a time, we first have I ⊥ z k = −iπ √ 2(−1) k v k,k (see, e.g., [11,Prop. 3]). Then by (22), we have P * + v k,k = iu k,k so that P * + I ⊥ z k = π √ 2(−1) k u k,k . Finally by (30), we have I 0 u k,k = 2π ((u k,k ) ψ ) 0 = (−1) k √ 2 z k , so that 1 2π With (25) proved, we finally return to proving (24). By virtue of [11,Prop. 3], we have the relations C 2 + I ⊥ h 0 = 0 and C 2 Combining this with both reconstruction formulas, we deduce that hence the proof.

Holomorphic integrating factors
As a direct consequence of Proposition 5, we can construct so-called holomorphic integrating factors for functions and solenoidal one-forms explicitly. Proposition 6. Let P † defined in Lemma 3. For any f 0 ∈ L 2 (M ) and f s ∈Ḣ 1 (M ), the function fiberwise holomorphic by construction, satisfies Proof of Proposition 6. Proving claim (i) amounts to computing

Proving claim (ii) amounts to computing
hence the result.
While these will be used to construct a holomorphization operator of transport solutions in Section 5.5, integrating factors for attenuation a will be used at several places throughout.
Proposition 7. For a function a ∈ C 0 (M, C), the function defined on SM by is a fiberwise odd, holomorphic solution of Xw a = −a, whose restrictions to ∂ + SM and ∂SM are given by Proof. That w a is holomorphic is immediate and w a solves Xw a = −a as a consequence of Proposition 6, and since n ∈ V − , n ψ is odd. We now compute, using that (n ψ )| ∂SM = A + n, Symmetry considerations give that P * − I 0 a ∈ V − . The sum above splits into four terms, whose leftmost factors are A * − A + ≡ 0, A * + A + = 2Id and A * − HA + = P − (when acting on V − ) and A * + HA + , leading up to By Lemma 3, we see that the second term in the right-hand side equals 1 2 I 0 a. On to the last term, with the fact that we deduce that hence the formula for ρ a in (27) holds. In addition, since w a is fiberwise odd, then w a | ∂SM is the fiberwise odd extension of ρ a to ∂SM , and since I 0 a ∈ V + and P * − I 0 a ∈ V − , this is equivalent to writing The proof is complete.

Invariant distributions with prescribed harmonic moments
The present section aims at producing fiberwise holomorphic invariant distributions h ψ (as in (12)) with fiberwise average (h ψ ) 0 ∈ L 2 (ker ∂). Since h → (h ψ ) 0 is the adjoint of the ray transform in the L 2 (M ) → L 2 (∂ + SM, cos α) setting, this can also be formulated as a surjectivity statement for this adjoint, as was initially done in [24,Theorem 1.4] for smooth topologies and simple Riemannian surfaces. The main difference here is that the target space in the ray transform is a different one and, while leading to more explicit constructions (see Theorem 9 below), it would not be amenable to the argument in [24] since in the present setting, the normal operator I * 0 I 0 associated with the restriction I 0 : H 0 → L 2 (∂ + SM ) is not an elliptic pseudodifferential operator after being extended to a slightly larger domain. In the present setting, a direct calculation using Santaló's formula leads to the expression Here we first aim at finding explicit preimages by I * 0 of elements in L 2 (ker ∂), and in the case of the Euclidean unit disc, this is again rather explicit, by directly exhibiting the singular value decomposition of I 0 • ι : L 2 (ker ∂) → L 2 (∂ + SM ), where ι : L 2 (ker ∂) → H 0 is the inclusion map.
Singular value decomposition (SVD) of I 0 • ι. We first define normalized so that Z k L 2 (SM ) = 1. In particular, by construction, In addition, it is computed easily (see e.g., [11, pp449-450]), that Moreover, since u k,k , u n,n ∂ + SM = 2δ kn , we obtain directly that the SVD of I 0 • ι : , is given by This implies in particular that the SVD of (I 0 • ι) * = ι * I * 0 : 2 u k,k ∞ k=0 → L 2 (ker ∂) is nothing but or, in other words where ι * is the L 2 (SM )-orthogonal projection onto L 2 (ker ∂). For the statement above to express the existence of invariant distributions with prescribed average, it is now absolutely necessary to remove ι * from the equalities above, and make it a surjectivity result for I * 0 and not ι * I 0 . To this end, we must go through the following direct calculation, whose proof is relegated to the Appendix.
Proposition 8. For any integer k ≥ 0, we have Now defining and with Z k as in (29), Proposition 8 indeed implies This motivates the definition of the kernel G, based on the series ∞ k=0 (k + 1)ζ k = (1 − ζ) −2 , convergent for |ζ| < 1: Based on the property that I * 0 W k = Z k , we are able to fomulate the following given by Moreover, the distribution where the 2π factor comes from the fact that the initial integral is over SM . Proof of Theorem 9. The first statement is a direct consequence of Proposition 8, and we now prove (i) and (ii). Proof of (i). In light of Lemma 4 below, it is enough to show that A + W k cos α is holomorphic on the fibers of ∂SM , which means that its harmonic content in α only consists of nonnegative harmonics. In order to check, it should be noted that W k cos α , initially defined on ∂ + SM , belongs to V + , which means that extending it to ∂ − SM by evenness is the same as extending by evenness w.r.t. α → α + π, and judging by the given expression, this simply consists of extending the expression we already have to ∂SM . Then the calculation (44) applies to A + u k,k cos α on the whole of ∂SM , i.e. Proof of (ii). It suffices to show that for every p ≥ 0, For m odd, this is already clear because then e imθ Z k is fiberwise odd, while, since W k cos α ∈ V + , W k cos α ψ is fiberwise even. Therefore it remains to check orthogonality for m = 2q even. In this case, the integration by parts formula (13) with a = 0 reads . Now using [11,Proposition 4], I[e i2qθ Z k ] is proportional to u 2q+k,q+k , and since W p is a multiple of u p,p , their inner product always vanishes. (ii) is proved.
Then h ψ is fiber-holomorphic on SM .
Reconstruction of e imθ g m,+ ∈ H sol m,+ . Since H sol m, (29). By Parseval's, we then have We now explain how to recover the inner products above from known data. Choose as integrating factor w a , defined via Proposition 7, that is, w a is a fiberwise antiholomorphic, odd solution of Xw = −a, with w a | ∂ + SM = ρ a . Then the integration by parts formula (13)  The result is still true when m = 1 and g = g 0 + X ⊥ g s = g 0 − 1 i e −iθ ∂g s + 1 i e iθ ∂g s , in which case the only potentially troublesome term e iθ ∂g s , e iθ Z k SM is still zero, as can be seen by integrating by parts on M .
With the notation of the current paper, we then recover, via a different way, the formulas presented in [11,Theorem 2.4].
Reconstruction of e −imθ g m,− ∈ H sol m,− . We have the following obvious identity where g ∈ L 2 (m−1) (SM ) and g m ∈ H sol m (M ), with decomposition g m = e −imθ g m,+ + e imθ g m,− .
Using the formulas above, we can then reconstruct g m,− from I a (g + g m ) via (37) mutatis mutandis: Complex-conjugating, we arrive at: Remark 7 (Speed-up of formula (37)). As it stands, formula (37) can be sped up, noticing that e −ρ a I a (g + g m )e i(−m+1)α dα dβ, (similarly for (38)), where the integral in α does not depend on x. This is significantly faster than (37), which requires integrating against the non-separable kernel G(z; β, α).

Reconstruction of g 0 and g s
The approach above showed that we can reconstruct the "residual terms" first, from highest order to lowest. After their forward transform is successively removed from the data, we are them left with reconstructing (g 0 , g s ) from I a [g 0 + X ⊥ g s ], which is the purpose of this section. Such an inversion method was first proposed in the context of simple surfaces in [3], though the present Euclidean case allows for even more expliciteness, and we will repeat the arguments here for completeness. With the right inverse P † of P constructed in Section 5.1, we first construct a so-called holomorphization operator, adapting [3, Proposition 6.1]. Define the operator B : Then we have the following 3 It is also proved in [3] to be a mapping B : C ∞ (SM ) → C ∞ (∂+SM ).
Applying the Hodge decomposition to the one-form 2f −1 , there exists g ∈ H 1 0 (M ) and h ∈ H 1 (M ) such that 2f −1 = Xg + X ⊥ h, in which case the previous equation can be rewritten as Upon integrating along each line, we make appear This motivate that we define u is holomorphic and by virtue of Proposition 6, u also solves and u 0 = −ih − u 0 . We then rewrite u as where the first summand u := 1 2 (u (+) − g + u ) is holomorphic, and where the second summand satisfies X 1 2 (u (−) + g − u ) = 0, so that it is equal to some h ψ , where h = 1 2 (u (−) + g − u )| ∂ + SM = B(u| ∂SM ) by construction. Therefore, Claim 1 is proved. As for Claim 2, if f −1 = 0, then the Hodge decomposition above becomes h = g = 0, and we read Thus Proposition 10 is proved.
Out of the holomorphization operator, we are able to derive reconstruction formulas for g 0 and g s . Such formulas were derived in [3] and we repeat the proof here for completeness.
Theorem 11 (Reconstruction of (g 0 , g s )). Let a ∈ C ∞ (M ). Define w a and w a following Eq. (26), ρ a := w a | ∂ + SM and ρ a := w a | ∂ + SM , and let B and B as above. Then the functions (g s , g 0 ) ∈ H 1 0 (M ) × L 2 (M ) can be reconstructed from data I := I a (g 0 + X ⊥ g s ) (extended by zero on ∂ − SM ) via the following formulas: where we have defined D := e wa ( B(Ie −ρa )) ψ , D := e w a ( B(Ie −ρ a )) ψ , and where ∂g + = ∂g − = 0, so that g ± are expressed as Cauchy formulas in terms of their boundary conditions Proof of Theorem 11. Since e −wa is a holomorphic solution of Xu − au = 0, we have Thanks to Proposition 10, defining v := ue −wa , the function v = v−( B(v| ∂SM )) ψ is holomorphic and satisfies X v = −b. Then defining u := e wa v = u − e wa ( B(ue −wa | ∂SM )) ψ = u − D, u solves the equation Similarly using e −w a , an antiholomorphic solution of Xu − au = 0, the function u = u − e w a ( B(ue −w a | ∂SM )) ψ = u − D is antiholomorphic and solves Projecting (41) onto H −1 and (42) onto H 1 , we obtain This implies the relations: which, since g s vanishes at the boundary, completely determines g ± from their boundary values, which are in turn determined from the boundary values of u and u (via a Cauchy formula), which are known from data. Taking the half-sum, we obtain where the right-hand side is completely determined by data. On to the determination of g 0 , we project the equation Xu + au = −g 0 − X ⊥ g s onto H 0 to make appear and show how to determine each term from the data. Since u is holomorphic, then u −1 = 0 = u −1 − D −1 , so u −1 = D −1 . Since u is antiholomorphic, u 1 = 0 = u 1 − D 1 , so u 1 = D 1 . Finally, We arrive at the following formula for g 0 Theorem 11 is proved.

A Proof of Proposition 8
Proof of Proposition 8. A first calculation shows that thus upon defining we have The proof will then be complete once we prove that J k,0 (x) = (−1) k J k,k (x) = 1 2 (x + iy) k , and J k,p (x) = 0, p / ∈ {0, k}.
Expanding the first factor in the right hand side using the binomial formula, we obtain a series in powers of sin θ, where the only term with a nonzero c −k coefficient is (−ρ 2 sin 2 θ) s (iρ sin θ) k−2s = (iρ sin θ) k = ρ k (−1) k 2 −k e −ikθ + =k c e i θ .