Sampling for the V-line Transform with Vertex in a Circle

The V-line transform arises in Compton camera imaging. Let us first recall the classical setup single-photon emission computed tomography SPECT, which is a nuclear medicine tomographic imaging technique using gamma rays. In SPECT, weakly radioactive tracers are given to the patient. The radioactive tracers can be detected through the emission of gamma ray photons revealing the information about biochemical processes. Then one uses a gamma camera to record photons that enter the detector surface perpendicularly. That is, the camera measures the integrals of the tracer distribution over straight lines that are orthogonal to its surface, see Fig 1.

This technique removes most photons and only a few photons are recorded. Therefore, a new type of camera for SPECT, which makes use of Compton scattering, was proposed by Everett [6] and Singh [27]. It uses electronic collimation as an alternative to mechanical collimation, which provides both high efficiency and multiple projections of the object. The camera consists of two plane gamma detectors positioned one behind the other. An emitted photon undergoes Compton scattering in the first detector surface $D_{1}$ and is absorbed by the second detector surface $D_{2}$. In each detector surface, the position and the energy of the photon are measured. The scattering angle at $D_{1}$ is determined via the Compton scattering formula $\cos\psi=1-\dfrac{mc^{2}(E_{1}–E_{2})}{E_{1}E_{2}}$, where $m$ is the electron mass, $c$ the speed of light, $E_{1}$ the photon energy at $D_{1}$, and $E_{2}$ the energy of the photon measured at $D_{2}$. So a photon observed at $x_{1}\in D_{1}$ and $x_{2}\in D_{2}$ with the energy $E_{1},E_{2}$ respectively must have been emitted on the surface of the circular cone, whose vertex is at $x_{1}$, central axis points from $x_{2}$ to $x_{1}$, and the half-opening angle is given by $\psi$ (see Fig. 2).

As a result, a Compton camera gives us the integrals of emission distribution on conical surfaces whose vertices are on the $D_{1}$. The mathematical problem of Compton camera imaging is to reconstruct the emission distribution from such integrals.

There are quite a few works on the mathematical problems of Compton camera. Specially in the two dimensional space, the cone become V-line and there exist some inversion formulas (e.g. [4, 23, 30]). In the three dimensional space, the space of conical surfaces whose vertices are on a detector surface is a five dimensional manifold. One is in the situation of redundant data. Taking advantage of such redundancy was the topic of several works (see, e.g.,[16]). One, however, may wish to restrict themselves into the lower dimensional data. In this article, we are interested in the sampling theory for Compton camera imaging in two dimensional space (i.e., the $V$-line transform).

Namely, let $f$ be a $b$-essentially band-limited function supported in the circle of radius $r_{0}\leq 1$ centered at the origin. We consider the V-line transform $V(f)$ of $f$ on all the V-lines whose vertex is on the circle of radius $r>1$ centered at the origin and the symmetric axis passing through the origin, see Fig. 3.

We consider the problem of recovering function $Vf$ from its discrete measures $Vf\left(\varphi_{k},\alpha_{m}\right)$. That is, we consider efficient sampling schemes to recover the V-line transform. Using Shannon sampling series and classical Fourier analysis, several authors [7, 8, 10, 18] study the reconstruction of the standard Radon and circular Radon transforms from its discrete measurement. More recently, Stefanov [28] employed semi-classical analysis to study the same problem for generalized Radon transform.

In this work, we introduce two sampling schemes. The first one is the standard sampling scheme, which is $(\varphi,\psi)=(\varphi_{k},\psi_{m})$, $0\leq k\leq N_{\varphi}-1,0\leq m\leq N_{\psi}-1$. Here, $\varphi_{k}$ and $\psi_{m}$ are evenly spaced in their domains. By restricting on the cones that intersect with the support of $f$, we only consider

	$\displaystyle\varphi_{k}=$	$\displaystyle\dfrac{k2\pi}{N_{\varphi}},\hskip 28.45274pt\text{for}\hskip 28.45274pt0\leq k\leq N_{\varphi}-1,$
	$\displaystyle\psi_{m}=$	$\displaystyle\dfrac{m2\pi}{N_{\psi}},\hskip 28.45274pt\text{for}\hskip 28.45274pt0\leq m<\dfrac{N_{\psi}\arcsin(r_{0}/r)}{2\pi}.$

To well recover $g$ from the discrete data we need

$\displaystyle N_{\varphi}\geq 2\pi rb,\quad N_{\psi}\geq 4rb.$

The total number $M_{0}$ of needed samples has to satisfy

$$M_{0}\geq 4(rb)^{2}\arcsin(r_{0}/r).$$

The second, more efficient, sampling scheme is given by

	$\displaystyle\varphi_{k}=$	$\displaystyle\dfrac{k2\pi}{N_{\varphi}}\hskip 28.45274pt\text{for}\hskip 28.45274pt0\leq k\leq N_{\varphi}-1,$
	$\displaystyle\psi_{k,m}=$	$\displaystyle\dfrac{\pi m}{N_{\psi}}\hskip 28.45274pt\text{for k, m are parity and}\hskip 28.45274pt0\leq m<\dfrac{N_{\psi}\arcsin(r_{0}/r)}{\pi},$

with the sampling conditions

$\displaystyle N_{\varphi}\geq 2\pi rb,\quad N_{\psi}\geq 3rb.$

The total number of samples satisfies

$$M_{0}\geq 3(rb)^{2}\arcsin(r_{0}/r),$$

which is three quarters of the standard scheme.

The article is organized as follows. In Section 2, we introduce some preliminaries.

The 2D Radon transform. We recall the 2D Radon transform that maps a function $f$ on $\mathbb{R}^{2}$ into the set of its integrals over the lines on $\mathbb{R}^{2}$. If $\theta(\varphi)=\left(\cos\varphi,\sin\varphi\right)\in S^{1}$ is a unit vector and $s$ is a real number. The Radon transform $Rf(\varphi,s)$ is defined as

$$Rf(\varphi,s)=\int_{x\cdot\theta=s}f(x)\,dl=\int^{+\infty}_{-\infty}{f\left(s\cos\varphi-t\sin\varphi;s\sin\varphi+t\cos\varphi\right)dt}.$$

In this notation, $Rf(\varphi,s)$ is the integral of $f$ over the line with a normal direction $\theta$ and of the distance $s$ from the origin. Let $R_{\varphi}f(\cdot)=Rf\left(\varphi,\cdot\right)$, a useful relation between the 1D Fourier transform of $R_{\varphi}f$ and the 2D Fourier transform of $f$ is as follows

$$\left(R_{\varphi}f\right)^{\wedge}(\sigma)=(2\pi)^{1/2}\hat{f}\left(\sigma\cdot\theta\right).$$

Here, the Fourier transform of a function $g$ defined on $\mathbb{R}^{n}$ is given by

$$\hat{g}(\xi)=\dfrac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}{g(x)e^{-ix\cdot\xi}dx}.$$

The inversion formula of 2D Radon transform is

$$f(x,y)=\dfrac{-1}{4\pi^{2}}\int\limits_{0}^{\pi}{\int\limits_{-\infty}^{+\infty}{\dfrac{\frac{\partial}{\partial s}Rf(\varphi,s)}{s-x\cos\varphi-y\sin\varphi}ds}d\varphi}.$$

One can see the following relationship between V-line transform and Radon transform

$$Vf\left(\varphi,\psi\right)=Rf\left(\varphi+\psi-\frac{\pi}{2},r\sin\psi\right)+Rf\left(\varphi-\psi-\frac{\pi}{2},-r\sin\psi\right).$$

Sampling of periodic functions. Let $g$ be a function in $\mathbb{R}^{n}$ that is periodic with respect to $n$ vectors $p_{1},p_{2},...,p_{n}$. If the matrix $P=\left(p_{1},p_{2},\cdots,p_{n}\right)$ is nonsingular then $g$ is called a P-periodic function. One denotes $L_{P}=P\mathbb{Z}^{n}$ and its reciprocal lattice $L^{\perp}_{P}=L_{2\pi P^{-T}}$. We define the discrete Fourier transform of $g$ be the following function on $L^{\perp}_{P}$:

$$\hat{g}(\xi)=\left|\det(P)\right|^{-1}\int\limits_{P[0,1)^{n}}{g(x)e^{-ix\xi}dx},\hskip 28.45274pt\xi\in L^{\perp}_{P}.$$

Let us note that $g$ can be recovered from its Fourier coefficients by the series [29]

$$g(x)=\sum\limits_{\xi\in L^{\perp}_{P}}{\hat{g}(\xi)e^{ix\xi}}.$$

Let W be a real nonsingular $n\times n$ - matrix, our goal is study the sampling of $g$ on $L_{W}$. For this to make, we assume $L_{P}\subset L_{W}$. This case is implied if and only if $P=WM$ with an integer matrix $M$. Hence, $L^{\perp}_{W}\subset L^{\perp}_{P}$. Our study relies heavily on Poisson summation formula for a P-periodic function $g$, see [8]. It reads

$$\sum\limits_{y\in L_{W}/L_{P}}{g(x+y)}=\left|\dfrac{det(P)}{det(W)}\right|\sum\limits_{\xi\in L_{W}^{\perp}}{\hat{g}(\xi)e^{ix\xi}}.$$

Here, $L_{W}/L_{P}$ is the quotient space whose elements are of the form $y+L_{P}$. In particular, using the Poisson summation formula one obtains (see [7, 11])

Sampling of the V-Line transform. Because of $supp(f)\subset D_{1}(0)$, so that $Vf(\varphi,\psi)=0$ with $\psi\in\left[\dfrac{\pi}{2},\pi\right)$. Consequently, $Vf$ can expand to an even function in $\psi$ variable and $2\pi$-periodic in both variables. We will make use of the two dimension form of Theorem 2.1 to recover the V-line transform. For $f\in{{C}^{\infty}}\left({{D}_{1}}\left(0\right)\right)$, we denote $g\left(\varphi,\psi\right)=Vf\left(\varphi,\psi\right)$. Since, $g$ is $2\pi$-periodic in each variables, then the periodic matrix of $g$ is $P=2\pi I_{2\times 2}$ and $L_{\mathcal{P}}^{\perp}=\mathbb{Z}^{2}$. We chose matrix $W$ which satisfies the condition $L_{\mathcal{P}}\subset L_{\mathcal{W}}$ as

$$W=2\pi\left(\begin{array}[]{cc}1/P&0\\ N/(PQ)&1/Q\end{array}\right)$$

where $P,Q,N$ are three integers such that $P,Q>0$ and $0\leq N<P$. Therefor,

$$L_{W}/L_{P}=\{(s_{j},t_{jl}):s_{j}=\frac{j2\pi}{P},t_{jl}=\frac{(l+Nj/P)2\pi}{Q},j=0,...,P-1,l=0,...,Q-1\}.$$

The sampling theorem for this setting follows as

Our goal is to use the Theorem 2.2 to propose some sampling conditions and derive a corresponding sampling error estimate. For this purpose, we assume $\hat{f}(\xi)$ be negligible for $|\xi|>b$, in the sense that the integral $\epsilon_{d}(f,b):=\int_{|\xi|>b}{|\xi|^{d}\,|\hat{f}(\xi)|\,d\xi}$ is small for all real number $d$. Such a function $f$ is called essentially b-band-limited. Here is the main result of this article:

In the rest of this article, we present the proof of this theorem. We then discuss two related sampling schemes.