Weighted V-Line Transforms in Imaging

• Weighted V-line transforms are integral transforms that assign data along two rays from a common vertex with diverse weight models such as branch, distance, and attenuation weights.
• They relate closely to Radon and exponential Radon transforms, allowing reconstruction of scalar and tensor fields through Abel-type inversion methods.
• Applications span attenuated SPECT, single-scattering models, and tensor tomography, with each weighting mechanism influencing stability and kernel structure.

Weighted V-line transforms are integral transforms that assign data to a pair of rays sharing a common vertex, with weighting introduced either by different branch coefficients, by explicit factors along the ray parameter, by attenuation, by angular branch weights in star configurations, or by tensorial contractions with the ray directions. In the recent literature, these transforms appear in scalar single-scattering models, attenuated SPECT with Compton cameras, and tensor tomography, where longitudinal, transverse, and mixed V-line transforms encode different directional components of vector or tensor fields (Haltmeier et al., 2016, Ambartsoumian et al., 31 Aug 2025, Ambartsoumian et al., 2024, Ambartsoumian et al., 2023).

1. Geometric definition and principal weight models

The unweighted baseline is the V-line transform with two rays emanating from a common vertex and integrated with respect to arc length. In the circular setup with vertex $r\theta(\varphi)$, half-opening angle $\psi$, and symmetric axis through the origin, the transform is

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$

with the two branches

$$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$

This is the reference model from which weighted variants are developed (Nguyen et al., 2020).

Weighted versions replace the unit branch coefficients or the arc-length measure by more general factors. In the formally determined swinging-branch setting, the scalar weighted V-line transform is

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$

where $u(x)$ and $v(x)$ are linearly independent unit vector fields whose integral curves are straight line segments (Ambartsoumian et al., 31 Aug 2025). In the circular-vertex setting, the weighted scalar transform is written

$$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$

for non-zero constants $c_1,c_2$ (Mishra et al., 2024). In attenuated SPECT, the weight is exponential,

$$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$

with constant attenuation coefficient $\psi$0 (Haltmeier et al., 2016).

A second major class consists of moment weights. For a scalar field, the first moment divergent beam transform is

$\psi$1

and the corresponding first moment longitudinal, transverse, and mixed V-line transforms of symmetric $\psi$2-tensor fields use the same factor $\psi$3 on each branch (Ambartsoumian et al., 2024). In tensor tomography, the directional contractions themselves also act as geometric weights: longitudinal transforms use $\psi$4, transverse transforms use $\psi$5, and mixed transforms use $\psi$6 or their higher-order analogues (Bhardwaj, 7 Feb 2025).

Weight model	Representative form	Representative source
Branch weights	$\psi$7	(Ambartsoumian et al., 31 Aug 2025)
Constant branch coefficients on circle	$\psi$8	(Mishra et al., 2024)
Distance weights	$\psi$9	(Ambartsoumian et al., 2024)
Attenuation weights	$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$0	(Haltmeier et al., 2016)
Angular branch weights	$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$1 in star transforms	(Ambartsoumian et al., 2023)

2. Relations to Radon and exponential Radon transforms

A defining structural feature of weighted V-line transforms is that many of them reduce to linear combinations of Radon data. In the unweighted circle-vertex geometry,

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$2

so each V-line datum is the sum of two line integrals at specific directions and offsets (Nguyen et al., 2020). In the branch-weighted scalar model on the unit disk,

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$3

which makes the branch weights explicit at the Radon level (Mishra et al., 2024).

For attenuation, the relevant analogue is the exponential Radon transform

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$4

and the attenuated V-line transform satisfies

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$5

This identity converts attenuation on the V-line branches into exponential weighting on Radon lines (Haltmeier et al., 2016).

The Fourier analysis of these relations leads to Abel-type inversion problems. In the attenuated case, angular Fourier coefficients of the data satisfy a generalized Abel equation with kernel

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$6

and, after reparameterization, the inversion problem becomes

$$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$7

The diagonal $$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$8 may vanish, so the problem falls outside classical Abel theory and requires a uniqueness theorem for generalized Abel equations with zeros on the diagonal (Haltmeier et al., 2016).

Star transforms provide another Radon reduction. For symmetric $$Vf(\varphi,\psi) = \sum_{\sigma=\pm 1}\int_0^{+\infty} f\bigl(r\theta(\varphi)-t\theta(\varphi+\sigma\psi)\bigr)\,dt,$$9-tensors with ray directions $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$0 and angular weights $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$1,

$$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$2

and there is a $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$3 matrix $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$4 such that, away from singular directions,

$$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$5

This converts the weighted star data into ordinary Radon data of the tensor components (Ambartsoumian et al., 2023).

3. Tensor-field formulations of weighted V-line transforms

In tensor tomography, weighting is inseparable from the field type. For symmetric $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$6-tensors $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$7, the basic contractions are

$$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$8

and these define the longitudinal, transverse, and mixed V-line transforms (Ambartsoumian et al., 2024). With two branch directions $$\ell_\sigma(\varphi,\psi)=\{r\theta(\varphi)-t\theta(\varphi+\sigma\psi): t\ge 0\}.$$9,

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$0

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$1

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$2

Their first moments,

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$3

insert the distance weight $$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$4 in each divergent beam integral and are the most immediate weighted V-line transforms in the tensor setting (Ambartsoumian et al., 2024).

For higher-order symmetric $$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$5-tensor fields supported in a disk, the mixed V-line transform takes the form

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$6

for $$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$7 (Bhardwaj et al., 30 Jun 2026). Here the weight is again geometric: the integrand is a contraction against a tensor built from the branch direction and its orthogonal complement.

This interpretation is explicit in generalized tensor V-line tomography: once one introduces tensor contractions with direction vectors, every V-line integral naturally becomes a weighted transform. Longitudinal, mixed, and transverse V-line transforms are therefore weighted V-line transforms in a tensorial sense even when no additional scalar attenuation is present (Bhardwaj, 7 Feb 2025). In the circular geometry with vertices on a circle, this viewpoint leads to generalized transforms

$$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$8

whose inversion reduces to weighted scalar V-line transforms of the scalar potentials arising in tensor decomposition (Mishra et al., 2024).

4. Injectivity, kernels, and inversion theory

The scalar weighted theory now has several distinct inversion paradigms. In the swinging-branch geometry, $$V_\alpha h(x) = \int_0^\infty h(x+t\,u(x))\,dt +\alpha \int_0^\infty h(x+t\,v(x))\,dt,$$9 is injective for any $u(x)$0. The proof differentiates $u(x)$1 along the branch fields and derives a first-order linear transport PDE for $u(x)$2, solved by the method of characteristics along the vector field $u(x)$3 (Ambartsoumian et al., 31 Aug 2025). In the attenuated circular setting, circular harmonics reduce the problem to a generalized Abel integral equation, and the transform is injective on $u(x)$4 under the condition $u(x)$5 (Haltmeier et al., 2016).

For vector fields, weighted branch coefficients alter the kernel but do not destroy explicit inversion in the constant-branch case. With symmetric directions $u(x)$6, $u(x)$7 and $u(x)$8, the weighted longitudinal and transverse transforms satisfy

$u(x)$9

in transformed coordinates $v(x)$0. The kernels are

$v(x)$1

and the pair $v(x)$2 determines $v(x)$3 (Ambartsoumian et al., 31 Aug 2025). In the unweighted swinging-branch case, the analogous statement is

$v(x)$4

and divergence and curl are recovered from $v(x)$5 and $v(x)$6 by explicit differential formulas (Ambartsoumian et al., 31 Aug 2025).

Moment data supply another route to injectivity. For symmetric $v(x)$7-tensors, combinations such as

$v(x)$8

allow full reconstruction, and the explicit formulas involve directional derivatives, divergent beam operators, and, in one regime, an elliptic boundary value problem for $v(x)$9 (Ambartsoumian et al., 2024). More generally, in the circular geometry for symmetric $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$0-tensor fields, the first $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$1 moment longitudinal transforms $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$2 or the first $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$3 moment transverse transforms $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$4 determine the field uniquely (Mishra et al., 2024).

Kernel structure is especially transparent in the tensor decomposition framework. If

$$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$5

then, for fixed $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$6, the mixed V-line transform $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$7 vanishes exactly on those tensors whose decomposition has no $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$8 term: $$V_w h(\phi,\psi) = c_1\int_0^\infty h(\Phi(\phi)+t\,u(\phi,\psi))\,dt + c_2\int_0^\infty h(\Phi(\phi)+t\,v(\phi,\psi))\,dt,$$9 Thus each $c_1,c_2$0 sees only one decomposition component, while the full family $c_1,c_2$1 is jointly injective under the support assumptions used in the disk geometry (Bhardwaj et al., 30 Jun 2026).

For star transforms of symmetric $c_1,c_2$2-tensors, injectivity is governed by the angular weight configuration. A symmetric star transform, in which the directions occur in opposite pairs with equal weights, is not invertible; if the configuration is not symmetric, then the transform is invertible (Ambartsoumian et al., 2023).

5. Numerical reconstruction, conditioning, and sampling

Numerical work on weighted V-line transforms is concentrated in attenuation- and tensor-based models. In the attenuated scalar problem, the inversion pipeline is: FFT in the vertex angle, discretization of the generalized Abel equation by product integration, Tikhonov regularization of the resulting lower-triangular systems, inverse FFT, and interpolation from a polar grid to a Cartesian grid. The paper states FFT complexity $c_1,c_2$3, linear-system cost $c_1,c_2$4 with Cholesky, and overall $c_1,c_2$5 complexity for $c_1,c_2$6 unknowns under the quoted scaling assumptions. In the reported noise-free simulations, the relative error exhibits semi-convergence and reaches an optimal $c_1,c_2$7; in photon-limited data, the optimal regularization is larger, and assuming $c_1,c_2$8 on attenuated data yields poor reconstruction quality (Haltmeier et al., 2016).

For tensor fields, the numerical implementations use direct ray marching on a pixel grid. The domain is discretized as $c_1,c_2$9 with $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$0 pixels, typically $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$1 for most experiments and $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$2 when solving PDEs. Divergent beam integrals are approximated by summing pixel values times segment lengths, and first moments are computed by multiplying additionally by the distance from the vertex to the pixel center. This implementation readily generalizes to other weights $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$3 by modifying the per-pixel factor in the summation. Reconstructions are very good for smooth phantoms, especially for $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$4 without moments and for opening angle near $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$5; moment-based inversions are ill-conditioned, and star-transform inversion is comparatively robust, with artifacts mostly outside the known support disc (Ambartsoumian et al., 2024).

Sampling theory is presently most explicit in the unweighted circular-vertex setting. The sampling paper with vertices on a circle derives standard and interlaced angular lattices for the unweighted transform and proves the supremum-norm error estimate

$$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$6

For the standard scheme, the simplified sampling conditions are

$$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$7

while the interlaced scheme yields

$$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$8

using three quarters of the leading-order sample count of the standard scheme. That work explicitly considers only the unweighted transform and does not define or analyze weighted variants (Nguyen et al., 2020). This suggests that weighted sampling theory requires separate control of the Fourier coefficients of the weighted data.

6. Scope, misconceptions, and open directions

A recurring misconception is to identify weighted V-line transforms solely with attenuation. The literature is broader. Weighting may mean unequal branch coefficients $$V_\mu f(\theta,\psi) = \sum_{\sigma=\pm 1}\int_0^\infty f\bigl(R\omega(\theta)-r\,\omega(\theta-\sigma\psi)\bigr)e^{-\mu r}\,dr,$$9 or $\psi$00, polynomial distance weights $\psi$01, exponential factors $\psi$02, angular branch weights $\psi$03 in star transforms, or the geometric tensor contractions that distinguish longitudinal, transverse, and mixed transforms (Haltmeier et al., 2016, Ambartsoumian et al., 31 Aug 2025, Ambartsoumian et al., 2024, Bhardwaj, 7 Feb 2025). Another misconception is that every V-line result automatically extends to weighted data. The sampling theory with vertices on a circle, for example, treats only the unweighted transform, and its discussion of weighted extensions is speculative rather than proved (Nguyen et al., 2020).

Several open directions are explicit. In the attenuated scalar problem, the theory addresses constant attenuation, while spatially varying attenuation

$\psi$04

is identified as an open mathematical problem (Haltmeier et al., 2016). In the swinging-branch framework, the scalar weighted transform is solved for arbitrary $\psi$05, but the fully general vector-field case with both arbitrary $\psi$06 and non-constant branch fields $\psi$07 is left open (Ambartsoumian et al., 31 Aug 2025). For tensor fields, the unweighted disk theory provides a decomposition, kernel characterization, and inversion template that points toward weighted analogues, but the paper itself does not develop a weighted tensor theory (Bhardwaj et al., 30 Jun 2026).

A further caution comes from weighted ray transforms. There exist strictly positive rotation-invariant continuous weights $\psi$08 for which the weighted straight-ray transform $\psi$09 has a non-trivial kernel in $\psi$10, and there are weights with $\psi$11 for arbitrary $\psi$12 (Goncharov et al., 2017). This suggests that weighted V-line uniqueness cannot be inferred from positivity alone; regularity and geometry of the weight are likely decisive.

Taken together, the modern theory presents weighted V-line transforms as a family rather than a single operator class. The branch-weighted, moment-weighted, attenuated, angularly weighted, and tensorially weighted formulations are linked by reductions to Radon-type operators, transport equations, or decomposition theorems, but each weighting mechanism alters the kernel structure, the inversion formula, and the stability profile in a different way.