---
title: 'Fan Beam Model: CT & Pulsar Emission'
url: https://www.emergentmind.com/topics/fan-beam-model
type: topic
---

# Fan Beam Model: CT & Pulsar Emission

In the literature represented here, the term **fan beam model** has two technically distinct meanings. In tomography, inverse problems, and detector design, it denotes a **2D divergent-beam acquisition or operator model** in which a point source emits rays that fan out toward a detector, and it functions both as a stand-alone geometry and as the principal 2D precursor to cone-beam CT extensions [1906.06472][1304.7701][2310.09567]. In pulsar radio astronomy, the same term denotes a **fan-shaped emission geometry** composed of radially extended, azimuthally organized sub-beams or streams, proposed as an alternative to nested conal beams [1405.6825][1507.08454][1701.06374].

## 1. Geometric formulations in tomography

In CT-related work, the fan-beam model is not a single parameterization but a family of closely related 2D divergent-beam geometries. One common formulation is the **equidangular fan-beam geometry**: a point x-ray source rotates on a circular trajectory of radius \(R_o\), the scan view angle is \(\phi\), and a ray in the fan is indexed by fan angle \(\gamma\). In that setting, a ray is denoted \(L(\phi,\gamma)\), with projection \(P(\phi,\gamma)\), bowtie absorbance \(B(\phi,\gamma)\), source flux \(N_i(\phi)\), and detected photons \(N_o(\phi,\gamma)\) [1304.7701]. A different but equally standard formulation is the **equidistant-detector fan-beam transform** \(Q\) with circular source trajectory radius \(r\), source angle \(\beta\), and detector coordinate \(s\):
\[
Q f(s, \beta) = Df\!\left(r \beta, \frac{s\beta^\perp-r\beta}{\|s\beta^\perp-r\beta\|_2}\right),
\]
with \(\beta^\perp = (-\sin\varphi,\cos\varphi)\) if \(\beta=(\cos\varphi,\sin\varphi)\) [2310.09567].

Other formulations are explicitly noncircular. In the **2D fan-beam geometry with sources on a line**, the source trajectory is
\[
\vec{s}_\lambda = (D,\lambda)^T,\qquad \lambda\in\mathbb{R},
\]
the detector is the vertical line \(x_1=0\), and the transform is
\[
\mathcal{D}f(\lambda,y) = \int_0^{+\infty} f(D-lD,\; \lambda+ly-l\lambda)\,dl.
\]
This linogram-style parametrization is used in a calibration setting based on range conditions on distributions [2505.08805].

A more geometric boundary-based formulation appears in tensor tomography on the Euclidean unit disk. There the incoming boundary \(\partial_+SM\) is parameterized by **fan-beam coordinates** \((\beta,\alpha)\), where
\[
x(\beta)=(\cos\beta,\sin\beta),\qquad
v=(\cos(\beta+\pi+\alpha),\sin(\beta+\pi+\alpha)),
\]
with \(\alpha\in(-\pi/2,\pi/2)\), and the straight-ray geodesic is
\[
\gamma_{\beta,\alpha}(t)=e^{i\beta}+t\,e^{i(\beta+\pi+\alpha)},\qquad 0\le t\le 2\cos\alpha.
\]
The same framework yields explicit scattering relations such as
\[
\mathcal S(\beta,\alpha)=(\beta+\pi+2\alpha,\ \pi-\alpha)
\]
and the antipodal relation
\[
\mathcal S_A(\beta,\alpha)=(\beta+\pi+2\alpha,\ -\alpha)
\]
[1510.05132][1704.08294].

In Fourier-based CT reconstruction, fan-beam geometry is also linked to Radon and linogram coordinates rather than an explicit detector model. In that context, the 2D discrete Radon transform is written using line equations \(v=qu+p\) and \(u=qv+p\), which are then converted to polar line parameters \(s\) and \(a\) via
\[
s = |p|/\sqrt{1 + q^2},\qquad a = \tan^{-1} q
\]
for basically horizontal lines, and
\[
s = |p|/\sqrt{1 + q^2},\qquad a = \tan^{-1}(-q)
\]
for basically vertical lines [1906.06472]. This suggests that, within tomography, “fan beam model” refers at once to acquisition geometry, coordinate conventions, and the operator domain in which reconstruction is carried out.

## 2. Forward models, inversion formulas, and reconstruction operators

The common analytical foundation of fan-beam CT reconstruction is the 2D Fourier/Radon relation. One paper states the Fourier slice theorem in the form
\[
F1PI = S_1 F2,
\]
meaning that the 1D Fourier transform of a projection corresponds to a slice of the 2D Fourier transform of the object [1906.06472]. The same work connects the discrete Radon transform to pseudo-polar Fourier samples through
\[
Ril = F110 PPil \quad (i = 1,2),
\]
and explicitly situates fan-beam CT as the 2D setting in which **FIRM** had already rendered possible “achieving high-quality two-dimensional (2D) images from a fan beam CT with a limited number of projections.” It also notes that **Averbuch et al.** reconstructed 2D images by applying 2D inverse DRT on 2D Radon data obtained from fan-beam projections [1906.06472].

Several later works reformulate the fan-beam model to improve computational efficiency or reduce interpolation error. In **precision learning**, the parallel-to-fan-beam conversion problem is written as
\[
\mathbf{A}_{f}\mathbf{A}_{p}^{\top}(\mathbf{A}_{p}\mathbf{A}_{p}^{\top})^{-1}\mathbf{p}_{p} = \mathbf{p}_{f},
\]
and then approximated by a Fourier-domain filter
\[
\mathbf{A}_{f}\mathbf{A}_{p}^{\top}\mathbf{F}^{H}\mathbf{K}\mathbf{F}\mathbf{p}_{p} = \hat{\mathbf{p}_{f}}.
\]
The resulting network uses known operators \(\mathbf{A}_p\), \(\mathbf{A}_p^\top\), \(\mathbf{A}_f\), \(\mathbf{F}\), and \(\mathbf{F}^H\), while learning only \(\mathbf{K}\), thereby avoiding the interpolation inherent in geometric rebinning. The paper compares this construction to the interpolation-based method of **Syben et al.** and reports sharper images from the learned method [1807.03057].

An exact-geometry reconstruction line proceeds from **Katsevich’s helical cone-beam formula**. For fan-beam CT, the resulting arc-based reconstruction is
\[
f(\underline{x})=-\frac{1}{2\pi}\int_{\text{chord}(\underline{x})} \frac{1}{\|\underline{x}-\underline{a}(\lambda)\|} \,g^F\!\left(\lambda,\frac{\underline{x}-\underline{a}(\lambda)}{\|\underline{x}-\underline{a}(\lambda)\|}\right)\,d\lambda,
\]
with filtered data obtained from differentiation in \(\lambda\) and Hilbert-transform filtering. That work then introduces a new pixel-dependent redundancy weighting \(\varpi(\underline{x},\lambda)\), in contrast to Parker weighting, and reports higher PSNR and SSIM, especially for super-short-scan trajectories [2101.01886].

A separate analytical route derives alternative fan-beam backprojection and adjoint operators from a parallel backprojection theorem. In the standard equiangular geometry, the Fourier-domain adjoint takes the Bessel–Neumann form
\[
\widehat{{}^*w}(\sigma\boldsymbol\xi_\theta) = \frac{2}{\sigma}\sum_{n=0}^\infty b_n(\theta)\,J_n(D\sigma),
\]
while the conventional backprojection becomes
\[
\widehat{w}(\sigma\boldsymbol\xi_\theta) = \frac{1}{\sigma}\sum_{n=0}^\infty \dot b_n(\theta)\,J_n(D\sigma).
\]
The paper states that these formulations can be implemented as an \(O(N^{2.3729})\) matrix multiplication and reports greater robustness in highly noisy data than conventional \(O(N^3)\) representations [1812.06519].

At the projector level, the **convolutional non-separable footprint (CNSF)** model rewrites fan-beam forward and back-projection in terms of box-spline convolutions. For a box-spline basis \(M_{\boldsymbol\Xi}\), the fan-beam projection becomes a variable-direction box spline,
\[
P\{M_{\boldsymbol\Xi}\}(s)=M_{\boldsymbol Z(s)}(s'),
\]
and for the pixel basis the practical blurred projector is expressed with an effective blur \(\tau'\) through a closed-form convolution/difference formula. That work reports improved accuracy and efficiency over LTRI and SF, while retaining memory-less on-the-fly computation suited to iterative reconstruction [1907.10526].

## 3. Alignment, calibration, and differentiable geometry learning

Because the inverse problem depends sensitively on geometry, several works treat the fan-beam model as an **additional inverse problem**. For a circular-source-trajectory fan-beam scanner with linear detector, one alignment model writes misalignment as a detector-coordinate translation,
\[
\tilde g(s,\beta)=g(s-h,\beta)\eqqcolon \tau_h g(s,\beta),
\]
where \(h\) is an unknown detector shift or center-of-rotation offset [2310.09567]. The same paper exploits the fan-beam symmetry condition
\[
g(s, \beta)=g(-s,\beta+\pi+2\arctan\tfrac{s}{r}),
\]
which is necessary but not sufficient for \(g\) to lie in the range of \(Q\). On that basis it proposes two low-cost strategies: **2D sinogram registration (2DR)** and a **fixed point method (FP)**, alongside **Linear Yang (LY)** as a refinement of the earlier averaging-based method of **Yang et al.** The reported validation shows that **FP and 2DR significantly outperform Yang and LY** on both low-noise and high-noise industrial scans, with 2DR generally best overall [2310.09567].

A complementary line of work makes the fan-beam reconstruction operator differentiable with respect to geometry itself. There each projection is represented by a \(2\times 3\) projection matrix
\[
\bm{P}_i = \bm{K}_i \cdot [\bm{R}_i \mid \bm{t}_i],
\]
and the reconstructed image value is
\[
\bm{I}(x, y) = \sum_{i=1}^{N} d_i \left( g \left( \bm{P}_i \cdot \begin{pmatrix} x\\ y\\ 1 \end{pmatrix} \right) \right).
\]
The paper derives the analytic Jacobian of \(\bm{I}(x,y)\) with respect to the six entries of \(\bm{P}_j\), implements the operator as a custom `torch.autograd.Function`, and applies it to rigid motion compensation. In the autofocus setting, the reported gain over the motion-affected reconstruction is a **35.5 % reduction in MSE** and a **12.6 % improvement in SSIM** [2212.02177].

A third calibration approach uses **range conditions on distributions** rather than on ordinary functions. In the 2D fan-beam geometry with sources on a line, the transform is extended to distributions by duality,
\[
(\mathcal{D}_\lambda f,\phi)=\langle f,\mathcal{D}_\lambda^*\phi\rangle,\qquad
\mathcal{D}_{\lambda}^{*}\phi(\vec{x}) = \frac{1}{D-x_1}\, \phi\!\left(\frac{x_2D-x_1\lambda}{D-x_1}\right),
\]
and moments satisfy
\[
\left(g_\lambda(y),y^k\right)=\mathscr{P}_k(\lambda),
\]
with \(\mathscr{P}_k(\lambda)\) polynomial of degree at most \(k\). Modeling markers as Dirac distributions then yields closed-form calibration formulas for the source positions \(\lambda_i\), detector shifts \(y_{\lambda_i}\), and marker-set parameters \(C_a,C_b,p_a,p_b\), even when the full object projections are truncated, provided the marker projections remain non-truncated [2505.08805].

Taken together, these works show that the fan-beam model is not only a forward geometry but also a calibrated object of estimation, regularization, and gradient-based optimization.

## 4. Beam shaping, detector architectures, and extended imaging modalities

In hardware and acquisition design, the fan-beam model often determines how fluence, dynamic range, and detector physics are managed. A clear example is the **dynamic bowtie** for fan-beam CT. For an elliptical object
\[
\frac{x^2}{A^2}+\frac{y^2}{B^2}\le 1,
\]
the detected photons are modeled by Beer–Lambert attenuation,
\[
N_o(\phi,\gamma)=N_i(\phi)e^{-B(\phi,\gamma)}e^{-P(\phi,\gamma)},
\]
and equalization of detected counts leads to the optimal bowtie condition
\[
B(\phi,\gamma)=P(\phi,0)-P(\phi,\gamma)+B(\phi,0)+\text{constant}.
\]
The bowtie is rotated mechanically in synchrony with source rotation, while \(N_i(\phi)\) is adaptively modulated. In the ideal elliptical phantom, the expected numbers of detected photons can be made “the same at each detector element,” and in practical head-like cross-sections the method reduces the detector dynamic range burden and peripheral overflow relative to no bowtie or a fixed circular design [1304.7701].

The fan-beam model also appears in non-x-ray tomography. A **fast-neutron transmission tomography** concept uses many point-like sources arranged around a stationary specimen and a ring-shaped detector. The detector is a THGEM-based multilayer polyethylene converter in which elastic \(n\)–\(p\) scattering produces recoil protons, gas ionization, electron drift, and THGEM multiplication. Simulations estimate a **detection efficiency of about 5–8%**, reaching approximately **8%** for a multilayer design near **300 layers**, with expected reconstructed spatial resolution of about **1 mm** and an intrinsic point-spread peak of about **500 \(\mu\)m FWHM** for the unscattered component [1203.0181].

In **fan beam coded aperture x-ray coherent scatter imaging**, the source lies at the origin, the fan beam is in the \(z=0\) plane, and a secondary coded aperture between object and detector disambiguates scatter events. The continuous forward model is
\[
g(\mathbf{r}')=\int\!\!\int H(\mathbf{r}',\mathbf{r},q)\, f(\mathbf{r},q)\, d\mathbf{r}\,dq,
\]
with geometry factors, mask transmission, scatter-angle spread, and spectral term bundled into \(H\). The paper emphasizes joint system-algorithm design and reports computational-time speedups of approximately **146** and **32** in the forward and backward models, respectively [1603.06400].

A related but spectrally richer setting is **Compton scattering tomography (CST)** in a joint CST–CT fan-beam scanner. There the spectral data are decomposed as
\[
\Spec(\mathbf{s},\mathbf{d},E)=\sum_{i=0}^{\infty} g_i(\mathbf{s},\mathbf{d},E),
\]
with \(g_0\) ballistic, \(g_1\) singly scattered, and \(g_2\) doubly scattered photons. The second-order term is modeled explicitly, while higher-order scattering is treated by conjectured smoothing behavior. The reconstruction strategy combines a sparse-view CT prior from \(g_0\) with differentiation in energy,
\[
\mathcal{D}_E^\gamma g^{\mathbf{s},\mathbf{d}} = \mathcal{F}^{-1}\!\left( i\zeta\,F_\gamma(\zeta)\,\mathcal{F}(g^{\mathbf{s},\mathbf{d}})(\zeta) \right),
\]
to suppress the smoother higher-order contributions. The simulations use **16 angular views**, **32 detectors** per view, and **256 energy bins**, and report substantially better reconstructions after energy differentiation than with a first-order-only model applied directly to \(g_1+g_2\) data [2008.06699].

## 5. Fan-beam coordinates in integral geometry and tensor tomography

A distinct mathematical tradition uses the fan-beam model as the natural boundary parametrization for X-ray and attenuated X-ray transforms on the Euclidean disk. In that setting, the X-ray transform is
\[
If(\beta,\alpha)=\int_0^{2\cos\alpha} f\!\left(e^{i\beta}+t e^{i(\beta+\pi+\alpha)},\,\beta+\pi+\alpha\right)\,dt,
\]
and the paper proves the unweighted continuity
\[
I:L^2(SM)\to L^2(\partial_+SM)
\]
with operator norm \(2\) [1510.05132]. The same work develops Fourier- and scattering-based range decompositions for tensor fields, showing that the classical moment conditions in parallel geometry are equivalent, in this fan-beam setting, to a Pestov–Uhlmann range characterization.

The tensor-tomographic significance of fan-beam coordinates is that they make the incoming/outgoing boundary, the scattering symmetry, and the fiberwise Hilbert-transform machinery explicit. The data space \(L^2(\partial_+SM)\) is decomposed into eigenspaces of the antipodal scattering symmetry, and the operators
\[
P:=A_-^* H A_+
\]
and its even/odd parts \(P_\pm\) provide range characterizations for \(I_0\) and \(I_\perp\) [1510.05132]. This is the analytical basis of the explicit Cauchy-type inversion formulas given there.

The attenuated extension keeps the same geometry but adds a position-dependent complex attenuation \(a\in C^0(M,\mathbb C)\). The attenuated X-ray transform is
\[
I_a f(x,v) = \int_0^{\tau(x,v)} f(\varphi_t(x,v))\, \exp\!\left(\int_0^t a(\gamma_{x,v}(s))\,ds\right)\,dt,
\]
and the corresponding transport equation is
\[
Xu + a u = -f \quad\text{on } SM, \qquad u|_{\partial_-SM}=0.
\]
A basic gauge obstruction is
\[
I_a[(X+a)h]=0,
\]
so injectivity is only modulo gauge. The constructive result is that every finite-order tensor field has an equivalent representative
\[
g = g_0 + X_\perp g_s + \sum_{k=1}^m g_k
\]
with the same attenuated transform, and that this representative can be uniquely and stably reconstructed [1704.08294].

In this literature, the fan-beam model is therefore not merely a scanner geometry. It is the coordinate system in which range conditions, Hilbert-transform identities, gauge structure, and explicit inversion become tractable.

## 6. Fan-beam model in pulsar radio-beam theory

In pulsar studies, the fan-beam model refers to a radio-emission geometry fundamentally different from nested cone beams. The basic picture is that broadband and coherent emission from secondary relativistic particles moving along dipolar magnetic flux tubes forms **radially extended sub-beams**. Several such sub-beams produce a fan-shaped overall beam; if only one or a few flux tubes are active, the beam becomes highly patchy [1405.6825]. A later population-synthesis paper attributes the model originally to **Michel** and further development to **Dyks et al.** and **Wang et al. (W14)**, while preserving the same central premise of radially elongated flux-tube emission [2010.13127].

A central consequence is a radial intensity law. Under the assumptions of coherent emission, free flow of secondary plasma, and single-particle power law \(p_{\rm e}\propto r^q\), the outer-beam intensity is
\[
I=AP^{q-4}\dot{P}\cos^2\alpha \left(f_{1}^{q-3}-f_{2}^{q-3}\right)\rho^{2q-6},
\]
so that
\[
I \propto \rho^{2q-6}.
\]
For \(q<3\), this produces **limb darkening** in the outer beam [1405.6825]. This differs from conal models, which assume one or more circular or elliptical rings with a global beam edge. The same work emphasizes that in the fan-beam picture there is “no global beam edge” in the same sense, because broadband emission can arise over a broad altitude range along a given flux tube [1405.6825].

The geometry also changes width predictions. For the outer beam, one derived relation is
\[
\cos\!\left(\frac{W}{2}+C\right)=\frac{\sin\alpha}{\tan(\alpha+\beta)\left(\cos^2\alpha+\tan^{-2}\varphi\right)^{1/2}},
\]
with \(C=\arctan(\sec\alpha/\tan\varphi)\), implying that pulse width \(W\) generally increases with \(|\beta|\) in the outer fan beam [1405.6825]. This is the opposite of the standard conal expectation highlighted in the same paper.

For the precessional pulse evolution of **PSR J0737−3039B**, the fan-beam geometry is specialized to two elongated beams at fixed magnetic azimuths \(\phi_{m,1}\) and \(\phi_{m,2}\), with pulse width
\[
W(t) = |\phi_2(\phi_{m2}) - \phi_1(\phi_{m1})|.
\]
The viewing angle evolves as
\[
\cos \zeta (t) = \cos \lambda \cos i + \sin \lambda \sin i \cos \Phi(t),\qquad
\Phi(t) = \Omega_p (T_0 - t),
\]
and the beam-cut geometry is mapped to pulse longitude by
\[
\cos \phi(t) = \frac{\cos \theta_m - \cos \alpha \cos \zeta (t)}{\sin \alpha \sin \zeta (t)}.
\]
Near the dipole axis, the paper derives the approximation
\[
W \approx (\zeta(t)-\alpha)\tan\phi_{m2}/\sin\alpha,
\]
which explains why the pulse width can vary nearly linearly over part of the precession cycle [1701.06374].

## 7. Observational tests, frequency evolution, and population implications in pulsars

Several empirical studies use the fan-beam model as a discriminant against traditional conal interpretations. A statistical test based on four- and five-component pulsar profiles defines
\[
R_W=\frac{W_{\rm in}}{W_{\rm out}},
\]
where \(W_{\rm in}\) and \(W_{\rm out}\) are the inner and outer peak separations. For the observed sample of 30 Q/M pulsars, the histogram is broad and centered roughly at \(R_W\approx 0.4\), whereas a nested-cone model with \(R_\rho=0.75\) predicts a sharp peak near \(R_W\approx 0.75\). The reported KS probability is about \(10^{-12}\) for the raw conal distribution versus the observed one, compared with **0.002** for the raw fan-beam model. The conal model can be reconciled only by invoking strong selection effects, including the claim that about **80% of Q and M profiles would need to be undetected** because of blending [1507.08454].

The precessional evolution of **PSR J0737−3039B** provides a different test. Using GBT width measurements from **Perera et al. (2010)**, the fan-beam model reproduces the transition from a single-peaked to a double-peaked profile, the growth in component separation, and the disappearance in March 2008 over a broad low-\(\chi^2\) region of parameter space. A nominal asymmetric outer-traverse fit gives
\[
\alpha = 4.2^\circ,\quad \lambda = 82.8^\circ,\quad \phi_{m2} = 13.0^\circ,\quad T_0 = 2006\ {\rm yr},\quad \chi^2/{\rm dof}=0.9,
\]
but the paper stresses that this solution is not unique and that acceptable reappearance times span approximately **2017–2078** [1701.06374]. This suggests that the fan-beam geometry is flexible enough to match the observed secular evolution, but not sufficiently constrained to yield a unique beam reconstruction.

Wideband frequency-evolution measurements also favor a stream- or patch-based interpretation over a simple radius-to-frequency mapping. Using Murriyang/Parkes observations of **157 pulsars** over **704–4032 MHz**, one study finds that Gaussian component peak locations vary little with frequency, with median
\[
\mu_{\rm comp.sep} = -0.04 \pm 0.48,
\]
whereas component widths narrow more clearly, with median
\[
\mu_{\rm comp.width} = -0.10 \pm 0.39,
\]
and outer-envelope widths show
\[
\mu_{\rm comp.edge} = -0.26 \pm 0.53.
\]
The paper concludes that a classic picture of a single emission height decreasing with frequency cannot explain the diversity of behavior, and that fan-beam or patchy multi-altitude structures are more consistent with the data [2509.11945].

At the Galactic-population level, the fan-beam model changes both beaming geometry and luminosity law. In a modified **PSRPOPPY/EVOLVE** framework calibrated to **1214 isolated pulsars** from the Parkes multibeam and Swinburne surveys, the fan-beam evolution model reproduces the observed distributions of Galactic longitude, latitude, spin period, period derivative, dispersion measure, and 1.4-GHz flux density. The inferred underlying population of radio-loud isolated pulsars is
\[
2.27\times10^6,
\]
and the same model predicts approximately **2700** and **240** new isolated pulsars for FAST inner- and outer-Galactic-plane surveys, respectively [2010.13127]. A plausible implication is that the fan-beam geometry creates a large hidden reservoir of weak pulsars: many objects are geometrically visible in principle but faint because the model’s luminosity depends strongly on impact angle.

Across these studies, the fan-beam model in pulsar astronomy functions less as a single closed theory than as a competing geometric paradigm. Its distinctive signatures are radially extended, azimuthally organized emission, width behavior different from conal beams, and empirical performance that is often better than the nested-cone picture in profile statistics, precessional evolution, and wideband profile phenomenology [1405.6825][1507.08454][2509.11945].

Source: https://www.emergentmind.com/topics/fan-beam-model