Fan Beam Model: CT & Pulsar Emission

• Fan beam model is a versatile framework defining 2D divergent-beam geometries in tomography, enabling precise CT acquisition and calibration.
• It integrates Fourier, Radon, and backprojection techniques to derive inversion formulas and differentiable operators for improved reconstruction quality.
• In pulsar studies, the model describes radially extended, azimuthally organized emission streams that offer an alternative to conal beams, explaining pulse width and intensity variations.

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 (Teyfouri et al., 2019, Liu et al., 2013, Guerrero et al., 2023). 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 (Wang et al., 2014, Dyks et al., 2015, Saha et al., 2017).

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 $N_o(\phi,\gamma)$ (Liu et al., 2013). A different but equally standard formulation is the equidistant-detector fan-beam transform $Q$ with circular source trajectory radius $r$, source angle $\phi$0, and detector coordinate $\phi$1: $\phi$2 with $\phi$3 if $\phi$4 (Guerrero et al., 2023).

Other formulations are explicitly noncircular. In the 2D fan-beam geometry with sources on a line, the source trajectory is

$\phi$5

the detector is the vertical line $\phi$6, and the transform is

$\phi$7

This linogram-style parametrization is used in a calibration setting based on range conditions on distributions (Konik et al., 11 May 2025).

A more geometric boundary-based formulation appears in tensor tomography on the Euclidean unit disk. There the incoming boundary $\phi$8 is parameterized by fan-beam coordinates $\phi$9, where

$\gamma$0

with $\gamma$1, and the straight-ray geodesic is

$\gamma$2

The same framework yields explicit scattering relations such as

$\gamma$3

and the antipodal relation

$\gamma$4

(Monard, 2015, Monard, 2017).

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 $\gamma$5 and $\gamma$6, which are then converted to polar line parameters $\gamma$7 and $\gamma$8 via

$\gamma$9

for basically horizontal lines, and

$L(\phi,\gamma)$0

for basically vertical lines (Teyfouri et al., 2019). 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

$L(\phi,\gamma)$1

meaning that the 1D Fourier transform of a projection corresponds to a slice of the 2D Fourier transform of the object (Teyfouri et al., 2019). The same work connects the discrete Radon transform to pseudo-polar Fourier samples through

$L(\phi,\gamma)$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 (Teyfouri et al., 2019).

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

$L(\phi,\gamma)$3

and then approximated by a Fourier-domain filter

$L(\phi,\gamma)$4

The resulting network uses known operators $L(\phi,\gamma)$5, $L(\phi,\gamma)$6, $L(\phi,\gamma)$7, $L(\phi,\gamma)$8, and $L(\phi,\gamma)$9, while learning only $P(\phi,\gamma)$0, 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 (Syben et al., 2018).

An exact-geometry reconstruction line proceeds from Katsevich’s helical cone-beam formula. For fan-beam CT, the resulting arc-based reconstruction is

$P(\phi,\gamma)$1

with filtered data obtained from differentiation in $P(\phi,\gamma)$2 and Hilbert-transform filtering. That work then introduces a new pixel-dependent redundancy weighting $P(\phi,\gamma)$3, in contrast to Parker weighting, and reports higher PSNR and SSIM, especially for super-short-scan trajectories (Wang et al., 2021).

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

$P(\phi,\gamma)$4

while the conventional backprojection becomes

$P(\phi,\gamma)$5

The paper states that these formulations can be implemented as an $P(\phi,\gamma)$6 matrix multiplication and reports greater robustness in highly noisy data than conventional $P(\phi,\gamma)$7 representations (Guerrero et al., 2018).

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 $P(\phi,\gamma)$8, the fan-beam projection becomes a variable-direction box spline,

$P(\phi,\gamma)$9

and for the pixel basis the practical blurred projector is expressed with an effective blur $B(\phi,\gamma)$0 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 (Zhang et al., 2019).

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,

$B(\phi,\gamma)$1

where $B(\phi,\gamma)$2 is an unknown detector shift or center-of-rotation offset (Guerrero et al., 2023). The same paper exploits the fan-beam symmetry condition

$B(\phi,\gamma)$3

which is necessary but not sufficient for $B(\phi,\gamma)$4 to lie in the range of $B(\phi,\gamma)$5. 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 (Guerrero et al., 2023).

A complementary line of work makes the fan-beam reconstruction operator differentiable with respect to geometry itself. There each projection is represented by a $B(\phi,\gamma)$6 projection matrix

$B(\phi,\gamma)$7

and the reconstructed image value is

$B(\phi,\gamma)$8

The paper derives the analytic Jacobian of $B(\phi,\gamma)$9 with respect to the six entries of $N_i(\phi)$0, 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 (Thies et al., 2022).

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,

$N_i(\phi)$1

and moments satisfy

$N_i(\phi)$2

with $N_i(\phi)$3 polynomial of degree at most $N_i(\phi)$4. Modeling markers as Dirac distributions then yields closed-form calibration formulas for the source positions $N_i(\phi)$5, detector shifts $N_i(\phi)$6, and marker-set parameters $N_i(\phi)$7, even when the full object projections are truncated, provided the marker projections remain non-truncated (Konik et al., 11 May 2025).

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

$N_i(\phi)$8

the detected photons are modeled by Beer–Lambert attenuation,

$N_i(\phi)$9

and equalization of detected counts leads to the optimal bowtie condition

$N_o(\phi,\gamma)$0

The bowtie is rotated mechanically in synchrony with source rotation, while $N_o(\phi,\gamma)$1 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 (Liu et al., 2013).

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_o(\phi,\gamma)$2–$N_o(\phi,\gamma)$3 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 $N_o(\phi,\gamma)$4m FWHM for the unscattered component (Cortesi et al., 2012).

In fan beam coded aperture x-ray coherent scatter imaging, the source lies at the origin, the fan beam is in the $N_o(\phi,\gamma)$5 plane, and a secondary coded aperture between object and detector disambiguates scatter events. The continuous forward model is

$N_o(\phi,\gamma)$6

with geometry factors, mask transmission, scatter-angle spread, and spectral term bundled into $N_o(\phi,\gamma)$7. The paper emphasizes joint system-algorithm design and reports computational-time speedups of approximately 146 and 32 in the forward and backward models, respectively (Odinaka et al., 2016).

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

$N_o(\phi,\gamma)$8

with $N_o(\phi,\gamma)$9 ballistic, $Q$0 singly scattered, and $Q$1 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 $Q$2 with differentiation in energy,

$Q$3

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 $Q$4 data (Kuger et al., 2020).

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

$Q$5

and the paper proves the unweighted continuity

$Q$6

with operator norm $Q$7 (Monard, 2015). 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 $Q$8 is decomposed into eigenspaces of the antipodal scattering symmetry, and the operators

$Q$9

and its even/odd parts $r$0 provide range characterizations for $r$1 and $r$2 (Monard, 2015). 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 $r$3. The attenuated X-ray transform is

$r$4

and the corresponding transport equation is

$r$5

A basic gauge obstruction is

$r$6

so injectivity is only modulo gauge. The constructive result is that every finite-order tensor field has an equivalent representative

$r$7

with the same attenuated transform, and that this representative can be uniquely and stably reconstructed (Monard, 2017).

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 (Wang et al., 2014). 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 (Huang et al., 2020).

A central consequence is a radial intensity law. Under the assumptions of coherent emission, free flow of secondary plasma, and single-particle power law $r$8, the outer-beam intensity is

$r$9

so that

$\phi$00

For $\phi$01, this produces limb darkening in the outer beam (Wang et al., 2014). 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 (Wang et al., 2014).

The geometry also changes width predictions. For the outer beam, one derived relation is

$\phi$02

with $\phi$03, implying that pulse width $\phi$04 generally increases with $\phi$05 in the outer fan beam (Wang et al., 2014). 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$06 and $\phi$07, with pulse width

$\phi$08

The viewing angle evolves as

$\phi$09

and the beam-cut geometry is mapped to pulse longitude by

$\phi$10

Near the dipole axis, the paper derives the approximation

$\phi$11

which explains why the pulse width can vary nearly linearly over part of the precession cycle (Saha et al., 2017).

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

$\phi$12

where $\phi$13 and $\phi$14 are the inner and outer peak separations. For the observed sample of 30 Q/M pulsars, the histogram is broad and centered roughly at $\phi$15, whereas a nested-cone model with $\phi$16 predicts a sharp peak near $\phi$17. The reported KS probability is about $\phi$18 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 (Dyks et al., 2015).

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-$\phi$19 region of parameter space. A nominal asymmetric outer-traverse fit gives

$\phi$20

but the paper stresses that this solution is not unique and that acceptable reappearance times span approximately 2017–2078 (Saha et al., 2017). 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

$\phi$21

whereas component widths narrow more clearly, with median

$\phi$22

and outer-envelope widths show

$\phi$23

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 (Jaroenjittichai et al., 15 Sep 2025).

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

$\phi$24

and the same model predicts approximately 2700 and 240 new isolated pulsars for FAST inner- and outer-Galactic-plane surveys, respectively (Huang et al., 2020). 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 (Wang et al., 2014, Dyks et al., 2015, Jaroenjittichai et al., 15 Sep 2025).