Star Transform: A Generalized Radon Model

• Star transform is a generalized Radon transform that maps two-variable functions to their integrals along star-shaped ray families.
• It provides explicit inversion formulas via both Radon and Fourier methods, ensuring practical image reconstruction in scattering-based modalities.
• Stability and injectivity depend on ray configuration, with odd-ray designs offering uniform stability compared to even-ray setups.

The star transform is a generalized Radon transform that maps a function of two variables to its integrals along star-shaped trajectories consisting of a finite number of rays emanating from a common vertex. In its weighted form, it sums divergent-beam transforms along prescribed directions with prescribed coefficients; in imaging theory it arises in mathematical models of modalities based on scattering of elementary particles, and the broken-ray or V-line transform is the two-ray special case (Ambartsoumian et al., 2020, Zhao et al., 2014).

1. Definition and geometric formulations

Let $f(x,y)$ be a compactly supported, or sufficiently rapidly decaying, function on $\mathbb R^2$. For distinct directions $\theta_1,\dots,\theta_k\in[0,2\pi)$ and optional nonzero weights $c_1,\dots,c_k\in\mathbb R$, define unit vectors

$\gamma_i=(\cos\theta_i,\sin\theta_i).$

The $i$-th divergent-beam transform at a point $a\in\mathbb R^2$ is

$$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$

and the weighted $k$-ray star transform is

$$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$

When all $\mathbb R^2$0, the transform is unweighted; otherwise it is weighted (Ambartsoumian et al., 2020).

A physically motivated formulation uses a strip

$\mathbb R^2$1

a total attenuation field $\mathbb R^2$2, and $\mathbb R^2$3 rays through each vertex $\mathbb R^2$4 with unit directions $\mathbb R^2$5, $\mathbb R^2$6. For each ray,

$\mathbb R^2$7

where $\mathbb R^2$8 is the distance from $\mathbb R^2$9 to the boundary of the strip along $\theta_1,\dots,\theta_k\in[0,2\pi)$0. The transform is then the linear combination

$\theta_1,\dots,\theta_k\in[0,2\pi)$1

Within this formulation, the broken-ray transform is recovered at $\theta_1,\dots,\theta_k\in[0,2\pi)$2 with $\theta_1,\dots,\theta_k\in[0,2\pi)$3, while $\theta_1,\dots,\theta_k\in[0,2\pi)$4 produces a genuine star of rays and improves invertibility (Zhao et al., 2014).

Both formulations encode the same structural idea: data at each vertex are obtained by summing line integrals over a finite family of half-rays sharing a common origin. This makes the transform a natural intermediate object between the classical Radon transform and the broken-ray transform.

2. Injectivity, singular sets, and symmetry structure

Invertibility is governed by two finite singular sets of directions $\theta_1,\dots,\theta_k\in[0,2\pi)$5. Writing $\theta_1,\dots,\theta_k\in[0,2\pi)$6 for the Euclidean dot product, the Type-1 singularities are

$\theta_1,\dots,\theta_k\in[0,2\pi)$7

where individual beams become tangent to the Radon lines, and the Type-2 singularities are

$\theta_1,\dots,\theta_k\in[0,2\pi)$8

where the overall weighting degenerates (Ambartsoumian et al., 2020).

The necessary and sufficient condition for $\theta_1,\dots,\theta_k\in[0,2\pi)$9 to be injective on $c_1,\dots,c_k\in\mathbb R$0 is that the set of directions and weights be non-symmetric in the specific sense that there is no partition of the $c_1,\dots,c_k\in\mathbb R$1 rays into matched pairs $c_1,\dots,c_k\in\mathbb R$2 carrying equal weights. Equivalently, the Type-2 zero-locus is not all of $c_1,\dots,c_k\in\mathbb R$3. A direct consequence is that any star with an odd number of rays is injective, while the only non-injective, hence non-invertible, stars are those that are centrally symmetric with matched weights (Ambartsoumian et al., 2020).

The operator has several elementary symmetries. It is translation invariant: $c_1,\dots,c_k\in\mathbb R$4 It is equivariant under joint rotation of the domain and ray directions: if $c_1,\dots,c_k\in\mathbb R$5, then

$c_1,\dots,c_k\in\mathbb R$6

when each $c_1,\dots,c_k\in\mathbb R$7 is replaced by $c_1,\dots,c_k\in\mathbb R$8. It is also invariant under permutation of rays together with the corresponding permutation of weights. Among these symmetries, central symmetry is exceptional because it is precisely the symmetry class that destroys invertibility (Ambartsoumian et al., 2020).

This symmetry classification is significant because it reduces the injectivity problem to a concrete combinatorial-geometric obstruction rather than a generic functional-analytic one.

3. Exact inversion and reconstruction formulas

For the Euclidean formulation, let $c_1,\dots,c_k\in\mathbb R$9 be known for all $\gamma_i=(\cos\theta_i,\sin\theta_i).$0 in a sufficiently large disk. Reconstruction proceeds through the standard Radon transform $\gamma_i=(\cos\theta_i,\sin\theta_i).$1. For each $\gamma_i=(\cos\theta_i,\sin\theta_i).$2, define

$\gamma_i=(\cos\theta_i,\sin\theta_i).$3

Then

$\gamma_i=(\cos\theta_i,\sin\theta_i).$4

and hence

$\gamma_i=(\cos\theta_i,\sin\theta_i).$5

The derivation passes through half-plane integrals: each divergent-beam integral is rewritten as an integral of $\gamma_i=(\cos\theta_i,\sin\theta_i).$6 over a half-plane weighted by $\gamma_i=(\cos\theta_i,\sin\theta_i).$7, the weighted sum is differentiated in $\gamma_i=(\cos\theta_i,\sin\theta_i).$8 to convert half-plane integrals into line integrals, and the resulting algebraic relation is solved for $\gamma_i=(\cos\theta_i,\sin\theta_i).$9 before applying $i$0, for example by filtered backprojection (Ambartsoumian et al., 2020).

In the strip geometry, inversion is expressed in Fourier coordinates. Writing

$i$1

one obtains, for each $i$2, an infinite system

$i$3

with

$i$4

Direct inversion is obtained by defining

$i$5

solving a finite $i$6 system for the $i$7, and then recovering the Fourier coefficients $i$8. The same framework also yields a low-frequency closed form when $i$9 and $a\in\mathbb R^2$0, and a local Katsevich–Krylov-type formula

$a\in\mathbb R^2$1

for a suitably defined vector-valued data function $a\in\mathbb R^2$2 satisfying $a\in\mathbb R^2$3 (Zhao et al., 2014).

These inversion formulas show that the star transform is not merely injective in favorable geometries; it admits explicit reconstruction operators with several distinct analytic realizations.

4. Stability, conditioning, and numerical inversion

For the Radon-based inversion, stability is governed by the multiplier $a\in\mathbb R^2$4. Near Type-2 singular directions $a\in\mathbb R^2$5 where $a\in\mathbb R^2$6, one has $a\in\mathbb R^2$7, and the inversion becomes ill-conditioned. The error propagation is described by the bound

$a\in\mathbb R^2$8

so the global condition number is controlled by $a\in\mathbb R^2$9 away from $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$0. For an odd regular star with uniform weights at the vertices of a regular $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$1-gon, $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$2 and $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$3 never vanishes, giving uniformly stable inversion. By contrast, even-ray stars always have infinitely many Type-2 singularities on $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$4, which produces severe artifacts along specific directions in numerical reconstructions (Ambartsoumian et al., 2020).

In the strip formulation, the low-frequency regime $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$5 requires $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$6 and $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$7; if $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$8, the transform is singular at $$\mathcal X_{\gamma_i}f(a)=\int_0^\infty f(a+t\gamma_i)\,dt,$$9. In the high-frequency regime $k$0, boundary terms are negligible and invertibility is controlled by the diagonal entries $k$1, equivalently by the function

$k$2

Stable inversion requires that $k$3 have no real zeros. A simple necessary condition is that the number of rays $k$4 be odd, since for even $k$5 the function $k$6 always crosses zero; a more refined necessary condition is that the weighted directions $k$7 not all lie in a single half-plane (Zhao et al., 2014).

Several computational strategies follow from this structure. Fourier-based block-diagonalization treats each $k$8 independently. An iterative Tikhonov pseudo-inverse writes a truncated matrix as $k$9 with $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$0 a sum of $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$1 separable terms, leading to complexity per $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$2 of $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$3 and overall $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$4 for an $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$5 slice. A direct separable inversion solves the small $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$6 system in $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$7 operations per $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$8, giving overall $$(\mathcal S_k f)(a;\theta_1,\dots,\theta_k) =\sum_{i=1}^k c_i\,\mathcal X_{\gamma_i}f(a) =\sum_{i=1}^k c_i\int_0^\infty f(a+t(\cos\theta_i,\sin\theta_i))\,dt.$$9 for the full slice. These costs compare favorably with a naive pixel-domain method using an $\mathbb R^2$00 matrix, which would cost $\mathbb R^2$01 or $\mathbb R^2$02 per iteration (Zhao et al., 2014).

The reported numerical experiments used a uniform square inclusion and a Shepp–Logan-type phantom on a $\mathbb R^2$03 grid with Poisson fluctuations added to each $\mathbb R^2$04 at $\mathbb R^2$05. The $\mathbb R^2$06 case showed severe high-frequency artifacts unless heavily regularized; a $\mathbb R^2$07 case with all $\mathbb R^2$08 was stable with good reconstructions and $\mathbb R^2$09; and a second $\mathbb R^2$10 family designed for simultaneous $\mathbb R^2$11 recovery separated stable and unstable geometries according to the presence or absence of zeros of $\mathbb R^2$12. The full $\mathbb R^2$13 reconstruction by the direct-separable method took on the order of a few seconds on a workstation, versus minutes-to-hours for pixel-domain solvers (Zhao et al., 2014).

5. Physical origin and imaging applications

The star transform is derived from first-order, or single, scattering models. In the strip geometry, an angularly resolved source at $\mathbb R^2$14 in direction $\mathbb R^2$15 and an angularly resolved detector at $\mathbb R^2$16 in direction $\mathbb R^2$17 measure power

$\mathbb R^2$18

where $\mathbb R^2$19 is the local scattering coefficient and $\mathbb R^2$20 is a known geometric factor. Defining

$\mathbb R^2$21

one obtains

$\mathbb R^2$22

Appropriate linear combinations of the $\mathbb R^2$23 eliminate $\mathbb R^2$24 while retaining all ray integrals. For symmetric coefficients $\mathbb R^2$25 with $\mathbb R^2$26, $\mathbb R^2$27, and $\mathbb R^2$28,

$\mathbb R^2$29

This is the star transform of the attenuation field (Zhao et al., 2014).

A related Euclidean model is

$\mathbb R^2$30

where $\mathbb R^2$31 is the spatial attenuation, $\mathbb R^2$32 is a background term, and $\mathbb R^2$33 is known physics. By taking suitable linear combinations $\mathbb R^2$34, one eliminates $\mathbb R^2$35 and obtains a weighted star transform of $\mathbb R^2$36. Inverse star reconstruction then yields the attenuation map, while a secondary pass recovers $\mathbb R^2$37 (Ambartsoumian et al., 2020).

The transform is therefore useful precisely because it converts scattering data into a linear tomographic operator. In the formulation of the 2014 work, its advantages include the possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously from the same data and the possibility to utilize scattered radiation that conventional X-ray tomography discards (Zhao et al., 2014). Similar transforms also arise in emission tomography with Compton cameras, where cone transforms appear, and in single-scattering ultrasonic imaging (Ambartsoumian et al., 2020).

6. Algebraic connection and related terminology

A notable by-product of the injectivity analysis is a result in algebraic geometry. A crucial step is to prove that the zero set of the elementary symmetric polynomial

$\mathbb R^2$38

cannot contain a $\mathbb R^2$39-dimensional real linear subspace when $\mathbb R^2$40 is odd. Equivalently, for $\mathbb R^2$41, the set $\mathbb R^2$42 has no $\mathbb R^2$43-planes. The argument constructs a concrete two-dimensional plane $\mathbb R^2$44 in $\mathbb R^2$45 spanned by the aperture vectors of a regular $\mathbb R^2$46-star, shows that $\mathbb R^2$47 on the unit circle in $\mathbb R^2$48, and uses homogeneity to deduce

$\mathbb R^2$49

This resolves the conjecture of A. Conflitti in the odd-$\mathbb R^2$50 case and underpins the completeness of the inversion theory (Ambartsoumian et al., 2020).

The term star transform also appears in other mathematical contexts with different meanings. In reliability theory, the star transform order is an order relation on nonnegative random variables with continuous distribution functions, defined by the requirement that

$\mathbb R^2$51

be star-shaped on $\mathbb R^2$52, equivalently that $\mathbb R^2$53 be increasing for $\mathbb R^2$54. In the two-component heterogeneous exponential parallel-system setting, if $\mathbb R^2$55, then

$\mathbb R^2$56

while comparison via the convex transform order fails in general (Arab et al., 2019). This is a stochastic order, not an integral transform on star-shaped ray families.

Likewise, in noncommutative geometry the phrase Hodge star refers to an operator $\mathbb R^2$57 constructed in one framework by braided Fourier transform on a braided exterior algebra, with

$\mathbb R^2$58

That usage concerns bicovariant differential calculi on Hopf algebras and is unrelated to the tomographic star transform described above (Majid, 2015).