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Curved Ray Transport Model Review

Purpose

This document defines a structured research review and ranking of mathematical models for curved-ray transport in xPRIMEray. The goal is to identify which approaches provide the highest efficiency per unit accuracy for field traversal, given that fixture-based validation is now operational.

Core Framing

There is no single universally optimal method. Efficiency depends on the objective:

  • Full path-accurate curved-ray transport
  • First-arrival / travel-time approximation
  • Long-term structural fidelity (Hamiltonian preservation)
  • Global probing / sampling efficiency

This review separates models by what problem they actually solve.

Model Ranking Matrix

Model Family Theoretical Efficiency Physics Fidelity Role in xPRIMEray Priority
Symplectic Hamiltonian Ray Tracing + Derivative-Aware Stepping High High Primary transport model 1
Embedded Adaptive RK (Dormand-Prince style) High Medium-High Benchmark baseline 2
Fast Marching / Fast Sweeping (Eikonal) Very High (for first-arrival) Low-Medium Planning / guidance field 3
Trajectory Optimization / Control Medium Medium-High Probe policy layer 4
Lie-Group / Manifold Integrators Medium High Structural refinement 5
Neural Adaptive Sampling Potentially High Variable Budget allocation layer 6

Key Insight

Traditional adaptive stepping reacts to curvature.

Next-generation stepping should react to curvature AND its derivatives.

This introduces a predictive rather than reactive traversal model.

1. Symplectic Hamiltonian Transport + Derivative-Aware Stepping

This is the highest-value next experiment.

Why:

  • Preserves Hamiltonian structure of optical transport
  • Aligns with GRIN / refractive index field physics
  • Allows integration of derivative-based control

Step Controller Concept

Let:

  • k = curvature proxy
  • dk = first derivative along path
  • d2k = second derivative

Define:

difficulty = a*k + b*|dk| + c*|d2k|
step_length ∝ 1 / difficulty

Interpretation:

  • High curvature, low derivative → smooth bend → moderate steps
  • Moderate curvature, high derivative → transition → reduce early
  • Low curvature, low derivative → long stride

Secondary Baseline

2. Embedded Adaptive RK

Use as a trusted comparison model:

  • Provides error-controlled stepping
  • Well understood behavior
  • Acts as validation reference

Metrics:

  • Runtime
  • Accepted steps
  • Image deviation
  • Stability

Supporting Models

3. Fast Marching / Fast Sweeping

Not a replacement for ray tracing.

Best use:

  • Compute travel-time fields
  • Predict high-difficulty regions
  • Guide scheduler / sampling priorities

4. Trajectory Optimization / Control

Interpret ray traversal as a control problem:

  • State = ray position/direction
  • Control = step size / direction updates

Use for:

  • Adaptive refinement policies
  • Probe targeting

5. Lie / Manifold Integrators

Useful when:

  • State evolves on constrained geometric spaces
  • Strong structure preservation is required

Lower priority for current fixtures.

6. Neural Adaptive Sampling

Not primary physics engine.

Potential use:

  • Sample allocation
  • Importance prediction

Experimental Plan

Experiment 1: Derivative-Aware Controller

Modify current RayBeamRenderer:

  • Add curvature history buffer
  • Compute first and second derivatives
  • Apply smoothed difficulty metric

Evaluate:

  • Runtime reduction
  • Step count reduction
  • Visual stability

Experiment 2: RK Baseline

Run identical fixtures using:

  • Embedded adaptive RK stepping

Compare:

  • Accuracy vs runtime
  • Step efficiency

Experiment 3: Symplectic Integrator

Implement Hamiltonian-consistent stepping:

Compare against RK:

  • Long-path stability
  • Energy drift
  • Visual coherence

Experiment 4: Eikonal Guidance Field

Precompute travel-time field:

Use for:

  • Scheduler hints
  • Candidate region prioritization

Architectural Synthesis

The system can be divided into three layers:

1. Transport Layer (Local Physics)

  • Symplectic / RK stepping
  • Derivative-aware control

2. Field Awareness Layer

  • Eikonal / gradient maps
  • Difficulty estimation

3. Control / Scheduling Layer

  • Probe allocation
  • Refinement strategy

Final Recommendation

The most efficient next step is:

Implement derivative-aware adaptive stepping within the current transport system, then benchmark against an embedded RK baseline, and finally evaluate a symplectic integrator for long-term structural gains.

One-Line Philosophy

Curvature tells us where the ray is.

Derivatives tell us where the field is going.

Status

  • Fixtures: Ready
  • Measurement pipeline: Ready
  • Next phase: Model experimentation

End of document