Analysis of the weighted conical Radon transform

In this article, we consider the weighted conical Radon transform—the transform is motivated by Compton camera imaging as well as optical tomography. Our contribution involves introducing new inversion formulas for the weighted conical Radon transform, including explicit formulas and properties associated with convolution frames. Furthermore, we propose reconstruction formulas that solve for variety weighted parameters in the two-dimensional space.


Introduction
Single-photon emission computed tomography (SPECT) is an imaging technique in nuclear medicine that utilizes gamma rays.In the SPECT process, patients receive weakly radioactive tracers, and the emitted gamma ray photons offer valuable insights into biochemical processes.A gamma camera is then employed to detect photons entering the detector surface at a right angle.Essentially, the camera measures the integrals of the tracer distribution along straight lines that are orthogonal to its surface.
This approach results in the removal of the majority of photons, with only a few being recorded.To overcome this limitation, a novel type of camera for SPECT, proposed by Everett [1] and Singh [2], utilizes Compton scattering.This Compton camera captures the entire emission distribution on conical surfaces, with the vertices of these cones located on the detector.The main advantage of this camera design is its ability to gather information about the emission distribution more comprehensively.The mathematical problem involves determining the spatial distribution of the emitted radiation from the integrated measurements recorded by the Compton camera.
However, it is crucial to take into account the attenuation effect of photons in SPECT, as some gamma rays are absorbed during their passage through the patient's body.This attenuation leads to a degradation in the accuracy of the reconstructed image.Similar occurrences arise when Compton cameras are applied in domestic security, industrial imaging, and gamma-ray astronomy.In [10,21], the weighted parameter is equal to some power of the distance to the vertex.In [22,23], the weight has the form e − μ r where the attenuation parameter μ 0 is a known constant and r is the distance of propagating of photons.In the two-dimensional space, such transform is simplified to attenuated V-lines transform.A reconstruction formula defined on cones with vertices located on a circle is derived in [24].
According these studies, we consider a family of circular cones C(u, v) parameterized by a vertex ( , a vertical central axis, and a fixed half opening angle ( ) The attenuated conical Radon transform C μ f (u, v) of a given function f to be the exponential weighted surface integrals along circular cones C(u, v) is defined as follows In a more generalized perspective, we examine the attenuated parameter, represented by the function , positive function that decays sufficiently fast as r tends to infinity.Then, the weighted conical Radon transform of f is described by In this article, we focus on studying the weighted conical Radon transform with a vertical central axis, a fixed half opening angle, and vertices belonging to the whole space.Section 3 derives the attenuated conical transform in R n+1 and presents an inversion formula when n is an even integer.In section 4, we investigate relations between the weighted Radon cone and Fourier transforms Following this, we deduce an inversion formula based on convolution frames, thereby enabling the derivation of further inversion formulas for the latter.Section 5 explores two-dimensional space, presenting two explicit inversion formulas, each leading to a variety of weighted parameters.

Preliminaries
In this section, we will provide some necessary background knowledge that will be used in the subsequent section of our article.Let g(x) ä L 2 (R n ), we define the Fourier transform of the function g as follows Then, its inverse Fourier transform is .
Therefore, by inverse property, we have Let ( ) ( ) q j j j = cos , sin be the unit vector in R 2 , for f ä C ∞ (R 2 ), s ä R. The Radon transform of f maps f into all its line integral as , cos sin , sin cos .
A function f can be constructed from its Radon transform employing the following formula Next, we will present the definitions and related properties of convolution frames that were studied in [25].Let Λ be an at most countable index set and consider a family ( ) . Definition 2.1.Define a family of functions be a convolution frame on R n if there exit two constant positive number A B , such that, for all be two convolution frames in R n .The following statement are equivalent .
(ii) The formula holds for any If v satisfies both mentioned conditions, it is referred to as a dual convolution frame of u.We proceed to define ≔ ( ) under the following condition Therefore, both u and u + satisfy statement (i) in theorem 2.2, concluding that u + serves as the dual convolution frame of u.

Inversion formulas of the attenuated conical Radon transform with vertices on whole space
In this section, we investigate a specialized configuration of the attenuated conical Radon transform, as denoted by equation (1).Furthermore, a previously introduced inversion formula in [22] is applicable specifically when n is an odd integer.The formula is articulated as follows Next, we will complete the problem by proposing an explicit inversion formula for n, where n is an even integer.For ξ ä S n−1 , s ä R, and a fixed real number y, the Radon transform of the function ( Proof.For all x Î - S n 1 , let A denote the rotation matrix within the orthogonal group ( ) O R n responsible for rotating the unit vector For simplicity, we define Now, changing the variable ω to • w A T and considering the Jacobian of this transformation, which equals 1, it follows  Subsequently, we represent the vector ξ as ( ) By integrating both sides of this equation over the plane = x s n , we obtain When taking the 1D Fourier transform with respect to the final variable, the above equation yields p n q q q p n q q Hence, substituting (5) into (4) results in or equivalently The equation (6) can be expressed as a linear integral equation .
With reference to the findings in [27] (p.p. 90), the solution to this equation is characterized by Now, we apply the inverse formula of the Radon transform to determine f.Therefore, the proof is complete.,

The weighted conical Radon transform and convolution frames
In this section, we will explore the series inversion based on convolution frames of weighted conical Radon transform, defined in (2).To achieve this, we first provide some properties of the weighted conical Radon transform.
where Proof.Applying the nD Fourier transform to C f U yields .Now, let's perform the calculation Utilizing formula (5), the result is obtained , Next, we introduce a property of the weighted conical Radon transform.
where # x denotes the n-dimensional convolution in R n .
Proof.Combining the results of propositions 4.1 and 4.2, we infer the relationship By applying the convolution theorem, we consequently deduce that , Now, we employ the convolution frames to reconstruct the function f from its weighted conical Radon transform as Theorem 4.4.For all ( ) be the convolution frame on R n and ( ) Here,  x denotes the spatial convolution in R n .
Proof.Since + v is the dual convolution frame of v, we can express Using theorem 4.3, the factor By substituting (9) into (8), the proof is finished.,

Some explicit inversion formulas using convolution frames in 2D
In this section, we focus on the two-dimensional scenario where the initial data is supported on a bounded domain .In the two-dimensional setting, a circular cone becomes a v-line which is represented by two halflines that share a common vertex at coordinates (x ν , y ν ), see figure 2.
For simplicity, we replace the weighted conical Radon transform C U of the function with V U and characterize it as follows ⎞ ⎠ In order to corroborate the validity of our 2D reconstruction formulas, we will present practical cases with the weights having an exponential form depending on the attenuated coefficient μ.
and  m 0 be a known constant.Assume that ( ) = m -U r e r , f can be recovered from V f U as .
In above equation, the symbol erf represents the error function and is defined as x t Proof.To construct the inversion algorithm utilizing convolution frames, we have selected v k k , to embody the Gaussian form, as defined by k k , 1 2 The Fourier transform of v k k , is ( ) Examining condition (3), the dual convolution frame of v k k , is presented by Therefore, In conclusion, we determine the value of u k k , such that Next, let's recall the result for the nonattenuated v-line Radon transform, i.e., in the equation (10), the attenuated parameter U(r) equals 1.
Theorem 5.2.[28][29][30][31][32][33] For ( ) 2,max .Let ( ) ≔ ( ) , , the exact solution of the inversion problem for the v-line Radon transform is given by the formula .The formula inversion is given by According to theorem 4.4, the inversion formula for equation (12) is derived., With the given above setup of u k,k and v k,k , we propose an inversion algorithm for the v-line Radon transform based on convolution as follows.
Algorithm: Recoverf: R 2 → R from the attenuated V-line Radon transform V U f for U(r) = e − μ r .
• Step 1:For all positive integer k, compute .
In theorem (5.1), due to the specialness of the attenuated coefficient U(r), the inversion formula is derived through a one-line reduction of the 'attenuated' transform to a 'non-attenuated' transform, achieved by a change of variables.Nevertheless, under different situation , this method proves unfeasible.
Theorem 5.4.Let  m 0 and Proof.We have also chosen v k k , in Gaussian form as k k , 1 2 For any arbitrary constant m 1 , we denote As m 1 remains constant and independent of z, we choose for m = - Then, Hence, we obtain the inversion formula of equation (14)., Considering the previously mentioned arrangement of w k,k and v k,k , we present an inversion algorithm for the v-line Radon transform relying on convolution frames.
Algorithm: Recoverf: R 2 → R from the weighted V-line Radon transform V U f for ( ) = m -U r e r 2 .
• Step 1:For all positive integer k, compute f k = w k,k # x V U f. .

Figure 1 .
Figure 1.A cone is defined by its vertex (x, ν), a vertical central axis, and a fixed half opening angle ψ.

cos
Moving forward, we recall Poisson's formula for the Bessel function, (see p.p 224 of[26]), can apply the inverse Fourier transform with respect to λ to obtain Inside the integral above, we utilize the following identity ( )

Figure 2 .
Figure 2. A V-line is defined by its vertex (x ν , y ν ), a vertical central axis, and a fixed half opening angle ψ.

cos
Based on the findings presented in lemma 5.3, we conclude that Based on lemma (5.3), it can be deduced that p represents the v-line Radon transform of f * on the non-attenuated case.According theorem 5.2, we get the following results. *