3D Ray-Based Motion Field Equations

• 3D ray-based motion field equations are evolution equations along rays that couple field quantities to spatial and velocity variations.
• They unify methodologies in radiative transfer, elastodynamics, and computer vision by incorporating Doppler effects, geometric spreading, and amplitude corrections.
• Efficient numerical schemes, including finite-element solvers and RANSAC-based methods, enable practical implementations across astrophysics, seismology, and real-time scene analysis.

A 3D ray-based motion field equation formally relates the evolution of physical or observed quantities—such as intensity, wavefronts, or geometric rays—along a spatially propagating ray in three-dimensional space, accounting for effects such as medium motion, velocity fields, geometric anisotropy, or camera/scene kinematics. Such equations arise as fundamental objects in domains including radiative transfer, dynamic ray tracing in moving or anisotropic media, visual odometry, and light field scene flow analysis. The defining feature is that the physical law is cast as an evolution equation along a ray parameterized by position, direction, and, when relevant, auxiliary internal variables (frequency, wavelength, slowness, etc.), with explicit coupling to velocity or motion fields.

1. General Structure and Physical Interpretation

Ray-based motion field equations universally formalize the propagation of a field quantity $I$ along a spatial ray $\mathbf{x}(s)$, parametrized by arc length $s$, direction vector (or momentum) $\mathbf{n}$, and possibly frequency or wavelength $\nu$ or $\lambda$. The canonical form of the motion field equation includes terms describing:

• The geometric propagation or "drift" along the ray.
• Modulation or coupling between internal variables and ray direction due to velocity or motion fields (e.g., Doppler effects, kinematics).
• Source ($\eta$), extinction ($\chi$), or analogous inhomogeneity/amplitude terms.

The presence of a non-zero velocity field, spatial inhomogeneity, or anisotropy introduces explicit coupling terms, usually derivatives or projections of the velocity field onto the ray direction, which account for phenomena such as Doppler shifts, geometric spreading, or convective transport.

In mathematical terms, such an equation typically reads as:

$$\frac{dI(\mathbf{r},\mathbf{n},\nu)}{ds} - \text{[velocity-induced coupling]} = \text{sources} - \text{extinction}$$

with detailed structure dependent on the application domain.

2. 3D Ray-Based Radiative Transfer with Arbitrary Velocity Fields

The classical domain for ray-based motion field equations is radiative transfer in astrophysics or atmospheric sciences. In the presence of an arbitrary velocity field $\mathbf{v}(\mathbf{r})$, the Eulerian-frame, frequency-dependent transfer equation takes the form (Seelmann et al., 2010):

$$\frac{dI(\mathbf{r},\mathbf{n},\nu)}{ds} - \left(\frac{\nu_0}{c}\right)[\mathbf{n} \cdot \nabla(\mathbf{n} \cdot \mathbf{v}(\mathbf{r}))]\,\frac{\partial I}{\partial \nu} = \eta(\mathbf{r},\mathbf{n},\nu) - \chi(\mathbf{r},\mathbf{n},\nu)I(\mathbf{r},\mathbf{n},\nu)$$

where

• $I$ is the specific intensity,
• $\eta$ the emission,
• $\chi$ the absorption,
• the $$(\mathbf{n} \cdot \nabla)(\mathbf{n} \cdot \mathbf{v})$$ term encodes the 3D, directional Doppler velocity gradient along the ray,
• and $\nu_0$ is the line-center frequency.

Key features:

• The motion field term couples the evolution of intensity at a given frequency to the spatially varying velocity field component along the ray.
• 3D, arbitrary velocity fields, including non-monotonic flows, can be treated, provided a sufficiently fine angular and spectral discretization.
• The equation is ready for discretization along rays and can be implemented in $\Lambda$-iteration or operator-splitting solvers for large-scale radiative transfer calculations (Seelmann et al., 2010, Baron et al., 2012).

This formalism is crucial for modeling line transfer in stellar winds, supernovae, or convective astrophysical environments.

3. Dynamic Ray Tracing in 3D Heterogeneous Anisotropic Media

In elastodynamics and seismology, the 3D ray-based motion field equation governs both the geometry of rays (kinematic problem) and the evolution of neighboring rays (dynamic/paraxial problem), particularly in heterogeneous and anisotropic media (Koren et al., 2020, Ravve et al., 2020).

Kinematic Equation

From variational principles, (Euler-Lagrange), the ODE for geometric ray tracing is (Koren et al., 2020):

$$\frac{d}{ds}[ V_\mathrm{ray}(\mathbf{x},\mathbf{r})\mathbf{r} ] = \nabla_\mathbf{x} V_\mathrm{ray}(\mathbf{x},\mathbf{r})$$

where $V_\mathrm{ray}$ is the group velocity depending on position and direction.

Dynamic (Jacobi/Paraxial) Equation

For the evolution of infinitesimal ray bundles (geometric spreading, amplitude), the second variation produces the 3D Jacobi equation (Ravve et al., 2020, Ravve et al., 2020):

$$\frac{d}{ds}[ L_{rr}(s)\,\dot{\mathbf{u}}(s) + L_{rx}(s)\,\mathbf{u}(s) ] - L_{xr}(s)\,\dot{\mathbf{u}}(s) - L_{xx}(s)\,\mathbf{u}(s) = 0$$

where $\mathbf{u}(s)$ is the shift vector (normal to the ray), and $L_{xx}, L_{xr}, L_{rx}, L_{rr}$ are Hessians (second derivatives) of the Lagrangian with respect to position and direction.

Key points:

• The coupling structure formalizes caustics, geometric spreading, and amplitude/phase corrections for complex multidimensional media.
• The Lagrangian and Hamiltonian formalism are equivalent via explicit relations between their Hessians (Ravve et al., 2020).
• Numerically, efficient finite-element solvers directly reuse the global traveltime Hessian, ensuring stability and accuracy across caustics (Ravve et al., 2020).

4. Generalization to Moving Anisotropic Media (Plasma and Wave Propagation)

In moving, possibly anisotropic backgrounds—e.g., magnetized plasma with flow—the ray-based motion field equation accounts for Lorentz-transformed dispersion relations (Braud et al., 19 Jul 2024): Let $D'(\mathbf{r},\mathbf{k}\,',\omega')=0$ be the rest-frame dispersion. The lab-frame evolution equations to first order in flow, for eikonal rays, are: $$\begin{aligned} \frac{d\mathbf{r}}{ds} &\approx \nabla_{\mathbf{k}'} D' - \mathbf{u}\,\frac{\partial D'}{\partial \omega'} \ \frac{d\mathbf{k}}{ds} &\approx -\nabla_\mathbf{r} D' + (\mathbf{k}\cdot\nabla\mathbf{u})\frac{\partial D'}{\partial \omega'} + \frac{\omega}{c^2}(\nabla\mathbf{u})^\top \nabla_{\mathbf{k}'} D' \end{aligned}$$ where $\mathbf{u}(\mathbf{r})=\mathbf{v}(\mathbf{r})/c$ is the dimensionless flow field.

Consequences:

• The group velocity and Fresnel drag, medium inhomogeneity, and Doppler gradients are all present.
• Structure is general: valid for arbitrary rest-frame dispersion and vectorial flow profiles (Braud et al., 19 Jul 2024).

5. Geometric and Computer Vision Motion Field Equations

The 3D ray-based motion field formalism also underpins direct visual ego-motion estimation and scene flow in computer vision.

Visual Ego-Motion (Ray-Flow Equation)

For a pinhole or generalized camera: $$\dot{\mathbf{r}} = [\mathbf{r}]_\times\,\boldsymbol{\omega} + \frac{\mathbf{r}\mathbf{r}^\top - I}{d}\,\mathbf{v}$$ where $\dot{\mathbf{r}}$ is the instantaneous time derivative of the unit ray, $\boldsymbol{\omega}$ the angular velocity, $\mathbf{v}$ the linear velocity, $d$ the tracked depth, and $[\mathbf{r}]_\times$ the cross-product matrix (Yang et al., 12 Nov 2025).

The same structure is used for efficient, camera-model-agnostic visual odometry pipelines, and has demonstrated practical capabilities on resource-constrained embedded platforms.

Light Field Scene Flow (Ray-Flow for Scene Reconstruction)

In 4D light fields, the linear ray-flow equation relates 3D scene flow $\mathbf{V}$ to spatiotemporal gradients: $L_x V_x + L_y V_y + L_z V_z + L_t = 0$ Here, the $L$'s are partial derivatives of the light field, and $V$ is the local scene motion. This is under-determined and is regularized with local or global methods (Ma et al., 2019).

6. Numerical Implementation and Solution Strategies

Across application domains, the 3D ray-based motion field equations require careful numerical handling:

• In radiative transfer, discretization along rays (characteristics), spectral upwinding, and operator splitting (e.g., $\Lambda$-iteration) are standard (Seelmann et al., 2010).
• In dynamic ray tracing, finite-element methods with Hermite interpolation and penalty terms for normalization/spacing are deployed to yield sparse, stable algebraic systems and efficient caustic handling (Koren et al., 2020, Ravve et al., 2020).
• In computer vision, stacking per-feature equations into large, overdetermined linear systems, followed by RANSAC and least-squares for robust estimation, leverages the algebraic structure for high-throughput real-time operation (Yang et al., 12 Nov 2025).
• In light field analysis, local structure tensors govern motion recoverability and under-determination, informing both direct solvers and global optimization (Ma et al., 2019).

7. Applications, Limitations, and Physical Implications

Applications of the 3D ray-based motion field equations span astrophysical radiative transfer in complex velocity fields, elastodynamic and acoustic ray tracing in Earth science, electromagnetic wave propagation in moving anisotropic plasmas, and real-time egomotion or scene flow recovery in computer vision. Fundamental limits include:

• Validity of the underlying approximations (e.g., non-relativistic, geometrical optics, linearizations).
• Numerical resolution in angular, spectral, or spatial grids to adequately capture Doppler broadening, caustic focusing, and anisotropy-induced complexity.
• In vision, rank-deficient structure tensors or ill-conditioned setups fundamentally limit extractable motion information (Ma et al., 2019).
• For highly relativistic or highly nonlinear flows, full generalizations involving relativistic transfer or Hamilton-Jacobi theory are required (Baron et al., 2012, Braud et al., 19 Jul 2024).

These equations unify geometric, kinematic, and dynamic propagation phenomena across physical and computational settings, providing the backbone for both theoretical analysis and practical modeling.

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References

• (Seelmann et al., 2010) "A 3D radiative transfer framework: VII. Arbitrary velocity fields in the Eulerian frame"
• (Baron et al., 2012) "A 3D radiative transfer framework: X. Arbitrary Velocity Fields in the Co-moving Frame"
• (Koren et al., 2020) "Eigenrays in 3D heterogeneous anisotropic media: Part I -- Kinematics, Variational formulation"
• (Ravve et al., 2020) "Eigenrays in 3D heterogeneous anisotropic media: Part VI -- Dynamics, Lagrangian vs. Hamiltonian approaches"
• (Koren et al., 2020) "Eigenrays in 3D heterogeneous anisotropic media: Part III -- Kinematics, Finite-element implementation"
• (Ravve et al., 2020) "Eigenrays in 3D heterogeneous anisotropic media: Part VII -- Dynamics, Finite-element implementation"
• (Braud et al., 19 Jul 2024) "Ray tracing methods for wave propagation in moving anisotropic media : application to magnetized plasmas"
• (Yang et al., 12 Nov 2025) "SMF-VO: Direct Ego-Motion Estimation via Sparse Motion Fields"
• (Ma et al., 2019) "Differential Scene Flow from Light Field Gradients"