--- title: 'Star Transform: A Generalized Radon Model' url: https://www.emergentmind.com/topics/star-transform type: topic --- # Star Transform: A Generalized Radon Model 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 [2005.01918, 1401.7655]. ## 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 $c_i=1$, the transform is unweighted; otherwise it is weighted [2005.01918]. A physically motivated formulation uses a strip \[ \mathbb S=\{(Y,Z):02$ produces a genuine star of rays and improves invertibility [1401.7655]. 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 $\psi\in\mathbb S^1$. Writing $\langle \psi,\gamma_i\rangle$ for the Euclidean dot product, the Type-1 singularities are \[ \mathcal Z_1=\bigcup_{i=1}^k\{\psi:\langle \psi,\gamma_i\rangle=0\}, \] where individual beams become tangent to the Radon lines, and the Type-2 singularities are \[ \mathcal Z_2=\left\{\psi:\sum_{i=1}^k c_i\prod_{j\neq i}\langle \psi,\gamma_j\rangle=0\right\}, \] where the overall weighting degenerates [2005.01918]. The necessary and sufficient condition for $\mathcal S_k$ to be injective on $C_c(\mathbb R^2)$ is that the set of directions and weights be non-symmetric in the specific sense that there is no partition of the $k$ rays into matched pairs $(\gamma,-\gamma)$ carrying equal weights. Equivalently, the Type-2 zero-locus is not all of $\mathbb S^1$. 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 [2005.01918]. The operator has several elementary symmetries. It is translation invariant: \[ \mathcal S_k[f(\cdot-a_0)](a)=\mathcal S_k f(a-a_0). \] It is equivariant under joint rotation of the domain and ray directions: if $R\in SO(2)$, then \[ \mathcal S_k[f\circ R](a)=\mathcal S_k[f](R^T a) \] when each $\gamma_i$ is replaced by $R\gamma_i$. 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 [2005.01918]. 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 $g(a)=\mathcal S_k f(a)$ be known for all $a$ in a sufficiently large disk. Reconstruction proceeds through the standard Radon transform $\mathcal R$. For each $\psi\in\mathbb S^1\setminus(\mathcal Z_1\cup\mathcal Z_2)$, define \[ w(\psi)=\sum_{i=1}^k \frac{c_i}{\langle \psi,\gamma_i\rangle}, \qquad q(\psi)=-\frac{1}{w(\psi)}. \] Then \[ \mathcal R f(\psi,t)=q(\psi)\,\partial_t[\mathcal R g](\psi,t), \] and hence \[ f=\mathcal R^{-1}\big[q(\psi)\,\partial_t\,\mathcal R(\mathcal S_k f)(\psi,t)\big]. \] The derivation passes through half-plane integrals: each divergent-beam integral is rewritten as an integral of $f$ over a half-plane weighted by $1/\langle \psi,\gamma_i\rangle$, the weighted sum is differentiated in $t$ to convert half-plane integrals into line integrals, and the resulting algebraic relation is solved for $\mathcal R f$ before applying $\mathcal R^{-1}$, for example by filtered backprojection [2005.01918]. In the strip geometry, inversion is expressed in Fourier coordinates. Writing \[ \mu(y,z)=\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,e^{iqy}\,\tilde\mu(q,z), \qquad \tilde\mu(q,z)=\frac1L\sum_{n\in\mathbb Z}\mu_n(q)e^{i\kappa_n z}, \qquad \kappa_n=\frac{2\pi n}{L}, \] one obtains, for each $q$, an infinite system \[ \Phi_n = d_n\,\mu_n + \sum_{k=1}^K \frac{s_k\,\alpha_k}{\beta_k+\kappa_n} \sum_m\frac{\mu_m}{\beta_k+\kappa_m}, \] with \[ d_n=\sum_{k=1}^K\frac{i\,s_k}{u_{kZ}(q u_{kY}+\kappa_n u_{kZ})}, \qquad \beta_k(q)=q\,u_{kY}/u_{kZ}. \] Direct inversion is obtained by defining \[ x_k=\sum_n \mu_n/(\beta_k+\kappa_n), \] solving a finite $K\times K$ system for the $x_k$, and then recovering the Fourier coefficients $\mu_n$. The same framework also yields a low-frequency closed form when $\Sigma_0\neq0$ and $\Sigma_1\neq0$, and a local Katsevich–Krylov-type formula \[ \mu({\bf R})=-\frac1{\sum_k \sigma_k}\,\nabla\!\cdot\boldsymbol\Phi({\bf R}) \] for a suitably defined vector-valued data function $\boldsymbol\Phi$ satisfying $\sum_k \sigma_k\hat{\bf u}_k=0$ [1401.7655]. 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 $q(\psi)$. Near Type-2 singular directions $\psi_0$ where $w(\psi_0)=0$, one has $|q(\psi)|\to\infty$, and the inversion becomes ill-conditioned. The error propagation is described by the bound \[ \|\delta f\| \le C\cdot \sup_{\psi\in\mathbb S^1}|q(\psi)|\cdot \|\delta g\|, \] so the global condition number is controlled by $\max |q(\psi)|$ away from $\mathcal Z_1\cup\mathcal Z_2$. For an odd regular star with uniform weights at the vertices of a regular $(2m+1)$-gon, $\mathcal Z_2=\varnothing$ and $w(\psi)$ never vanishes, giving uniformly stable inversion. By contrast, even-ray stars always have infinitely many Type-2 singularities on $\mathbb S^1$, which produces severe artifacts along specific directions in numerical reconstructions [2005.01918]. In the strip formulation, the low-frequency regime $qL\ll1$ requires $\Sigma_0\neq0$ and $\Sigma_1\neq0$; if $\Sigma_1=0$, the transform is singular at $q=0$. In the high-frequency regime $qL\gg1$, boundary terms are negligible and invertibility is controlled by the diagonal entries $d_n(q)$, equivalently by the function \[ f(\theta)=\sum_{k=1}^K \frac{s_k}{\cos(\theta-\theta_k)}. \] Stable inversion requires that $f(\theta)$ have no real zeros. A simple necessary condition is that the number of rays $K$ be odd, since for even $K$ the function $f$ always crosses zero; a more refined necessary condition is that the weighted directions $s_k\hat{\bf u}_k$ not all lie in a single half-plane [1401.7655]. Several computational strategies follow from this structure. Fourier-based block-diagonalization treats each $q$ independently. An iterative Tikhonov pseudo-inverse writes a truncated matrix as $A=D+V$ with $V$ a sum of $K$ separable terms, leading to complexity per $q$ of $O(KN^2)$ and overall $O(KN^3)$ for an $N\times N$ slice. A direct separable inversion solves the small $K\times K$ system in $O(K^3+N)$ operations per $q$, giving overall $O(N^2)$ for the full slice. These costs compare favorably with a naive pixel-domain method using an $N^2\times N^2$ matrix, which would cost $O(N^6)$ or $O(N^4)$ per iteration [1401.7655]. The reported numerical experiments used a uniform square inclusion and a Shepp–Logan-type phantom on a $125\times125$ grid with Poisson fluctuations added to each $\phi_{jk}$ at $\mathcal N=4\times10^4,10^4,2.5\times10^3$. The $K=2$ case showed severe high-frequency artifacts unless heavily regularized; a $K=3$ case with all $s_k=1$ was stable with good reconstructions and $\lambda=0$; and a second $K=3$ family designed for simultaneous $\mu_a,\mu_s$ recovery separated stable and unstable geometries according to the presence or absence of zeros of $f$. The full $125\times125$ 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 [1401.7655]. ## 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 $Z=0$ in direction $\hat{\bf u}_j$ and an angularly resolved detector at $Z=L$ in direction $\hat{\bf u}_k$ measure power \[ W_{jk}({\bf R}) = W_0\,S_{jk}\,\mu_s({\bf R})\, \exp\!\bigl[-(I_j({\bf R})+I_k({\bf R}))\bigr], \] where $\mu_s$ is the local scattering coefficient and $S_{jk}$ is a known geometric factor. Defining \[ \phi_{jk}({\bf R})=\ln\!\bigl[W_{jk}/(W_0\,S_{jk}\,\bar\mu_s)\bigr], \] one obtains \[ \phi_{jk}({\bf R})=I_j({\bf R})+I_k({\bf R})+\eta({\bf R}), \qquad \eta({\bf R})=\ln[\mu_s({\bf R})/\bar\mu_s]. \] Appropriate linear combinations of the $\phi_{jk}$ eliminate $\eta({\bf R})$ while retaining all ray integrals. For symmetric coefficients $c_{jk}$ with $\sum_{j,k}c_{jk}=0$, $c_{jk}=c_{kj}$, and $s_k=\sum_j c_{jk}\neq0$, \[ \Phi=\tfrac12\sum_{j,k}c_{jk}\phi_{jk}=\sum_{k=1}^K s_k I_k. \] This is the star transform of the attenuation field [1401.7655]. A related Euclidean model is \[ \phi_{ij}(x)=\mathcal X_{\gamma_i}f(x)+k_{ij}\mathcal X_{\gamma_j}f(x)+\eta(x), \] where $f$ is the spatial attenuation, $\eta$ is a background term, and $k_{ij}$ is known physics. By taking suitable linear combinations $\sum_{i,j}\omega_{ij}\phi_{ij}$, one eliminates $\eta$ and obtains a weighted star transform of $f$. Inverse star reconstruction then yields the attenuation map, while a secondary pass recovers $\eta$ [2005.01918]. 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 [1401.7655]. Similar transforms also arise in emission tomography with Compton cameras, where cone transforms appear, and in single-scattering ultrasonic imaging [2005.01918]. ## 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 \[ e_{k-1}(y_1,\dots,y_k)=\sum_{i=1}^k \prod_{j\neq i} y_j \] cannot contain a $(k-1)$-dimensional real linear subspace when $k$ is odd. Equivalently, for $m=2n+1$, the set $\{y:e_{m-1}(y)=0\}$ has no $m-1$-planes. The argument constructs a concrete two-dimensional plane $T$ in $\mathbb R^m$ spanned by the aperture vectors of a regular $(2n+1)$-star, shows that $e_{m-1}(y)\neq0$ on the unit circle in $T$, and uses homogeneity to deduce \[ T\cap\{e_{m-1}=0\}=\{0\}. \] This resolves the conjecture of A. Conflitti in the odd-$m$ case and underpins the completeness of the inversion theory [2005.01918]. 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 \[ \phi(x)\coloneqq F_Y(F_X(x)) \] be star-shaped on $(0,\infty)$, equivalently that $\phi(x)/x$ be increasing for $x>0$. In the two-component heterogeneous exponential parallel-system setting, if $(\lambda_1,\lambda_2)\prec(\theta_1,\theta_2)$, then \[ \max\{X_1,X_2\}\leq_*\max\{Y_1,Y_2\}, \] while comparison via the convex transform order fails in general [1901.03238]. 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 $\sharp$ constructed in one framework by braided Fourier transform on a braided exterior algebra, with \[ \sharp=g\circ F|_{\Lambda^m}:\Lambda^m\to\Lambda^{n-m}. \] That usage concerns bicovariant differential calculi on Hopf algebras and is unrelated to the tomographic star transform described above [1511.00190]. Source: https://www.emergentmind.com/topics/star-transform