Some uniqueness theorems for a conical Radon transform

The conical Radon transform, which assigns to a given function $f$ on $\mathbb R^3$ its integrals over conical surfaces, arises in several imaging techniques, e.g. in astronomy and homeland security, especially when the so-called Compton cameras are involved. In many practical situations we know this transform only on a subset of its domain. In these situations, it is a natural question what we can say about $f$ from partial information. In this paper, we investigate some uniqueness theorems regarding a conical Radon transform.


Introduction
The conical Radon transform is the integral transform that maps a function f on R 3 to its integrals over conical surfaces.This transform is related to various imaging techniques, e.g. in optical imaging [8], primarily when the so-called Compton cameras are used.This camera was introduced for use for Single Photon Emission Computed Tomography(SPECT) [27] but it is also used in astronomy, and homeland security imaging [1].More information on Compton cameras can be found, for example, in [1,2,5,18,22,27].
Inversion formulas for various types of conical Radon transforms are provided in [2,3,4,10,11,12,18,19,15,22,21,16,25,32,29,33].However, only a few of these articles study uniqueness of a certain kind of a conical Radon transform (for example, [21]).Here, we investigate the uniqueness property of the conical Radon transform which was first introduced in [16].This conical Radon transform arises in an application of a Compton camera consisting of two linear detectors.
The conical Radon transform of when s is between −1 and 1 and β = (cos β, 0, sin β) ∈ S 2 .(As mentioned in [16, eq.(2.4)], s is the opening angle of the cone of integration and β is the unit vector which indicates the central axis of the cone.)Here δ is the Dirac delta function and dS(α) is the standard measure on the unit sphere.When s is outside of the interval [−1, 1], Cf is set to be zero.If f (x), x = (x 1 , x 2 , x 3 ) ∈ R 3 is odd with respect to x 2 , then Cf is equal to zero.We thus assume that f is compactly supported in {x = (x 1 , x 2 , x 3 ) ∈ R 3 : x 2 > 0}.
Our results can be applicable to the conical Radon transform obtained by the typical Compton camera, which consists of two planar detectors.As a formula, this is (2) Applicability of our results to this conical Radon transform C T f follows from the fact that C T f contains Cf .
The conical Radon transform can be decomposed into two following transforms: • The weighted ray transform P maps a continuous function f on R 3 with compact support into Many articles including [9,14,24,26] studied this spherical sectional transform Qφ.However only a few articles [13,16,22,20] have investigated P f , although various articles [6,7,28,30,31] did study ∞ 0 f ((u, 0, 0) + rw)dr.The next theorem follows from the definition of C, Q and P and is well known.
The next section is devoted to three uniqueness theorems for the conical Radon transform.

Uniqueness theorems
In this section, we present some uniqueness theorems for the conical Radon transform Cf .
Theorem 3. Let S ⊂ S 1 be a set such that no non-trivial homogeneous polynomial vanishes on S.
This theorem is similar to one for the regular Radon transform derived in [23,Theorem 3.4 in Chapter II].The proof follows the idea in [23].
Proof.Let u ∈ R be fixed.Taking the 1-dimensional Fourier transform of Cf with respect to s, we have for (cos β, sin β) ∈ S and (u, σ) ∈ R × R. Notice that a j is a homogeneous polynomial of degree k.Since no non-trivial homogeneous polynomial vanishes on S, we have for a fixed u ∈ R and any j = 0, 1, 2, • • • , a j = 0. Therefore, F s (Cf ) is equal to zero and by the inversion formula [16, Theorem 5], f is equal to zero.
The following theorem is similar to one for the regular divergent beam transform derived in [23,Theorem 3.5].
We introduce two notations.For any set Proof.For the moment, let u ∈ R be fixed.For From Theorem 2, we have Cf = R 2 Φ where By the hole theorem [23, Theorem 3.1 in Chapter II] for the regular Radon transform, Φ(u, •) is supported in S ′ (u), since S ′ (u) is a convex and compact set by the compactness of S(u), and thus P f (u, α) = 0 for {(u, 0, 0) + rα : r ≥ 0, α ∈ S(u)} not meeting U.
The following lemma completes our proof.This lemma is very similar to Theorem 3.3 in Chapter II [23].
Proof.Similar to the proof of Theorem 3.3 in Chapter II [23], we can show ∞ 0 r k f (a + rα)dr = 0 for (a, α) ∈ A × S and k ≥ 1.
Note that by the Stone-Weierstrass theorem, the space spanned by {r k : k ≥ 1} is dense in the set {g ∈ C ∞ (R) : g(0) = 0, g has compact support}.

Lemma 6 .
Let S be an open set on S 2 , and let A be a continuously differentiable curve outside U. Let U ⊂ R 3 be bounded and open.Assume that for each α ∈ S there is an a ∈ A such that the half-line a + rα, r ≥ 0 misses U.If f ∈ C ∞ (R 3 ) has compact support in U and ∞ 0 f (a + rα)rdr = 0 for a ∈ A and α ∈ S, then f = 0 in {a + rα : a ∈ A, α ∈ S, r ≥ 0}.