README.md

Pose-Free Weighted Anisotropic Inverse Experiment

Purpose

This experiment combines the two nuisance variables that were separated in the previous inverse extensions:

• unknown rotation of the observed boundary
• unknown medium anisotropy alpha

The forward family is still

[ w_1 d_\alpha(x, p_1) + w_2 d_\alpha(x, p_2) + w_3 d_\alpha(x, p_3) = S ]

with positive normalized weights and the axis-aligned anisotropic metric

[ d_\alpha((x, y), (u, v)) = \sqrt{(x-u)^2 + \alpha^2 (y-v)^2}. ]

The recovery target remains the joint latent object:

• the normalized source triangle relative to budget
• the normalized weight vector in the simplex
• the anisotropy parameter alpha

This is the first inverse test where the boundary is pose-free and the medium is hidden at the same time.

Research Question

Does the weighted multi-source control object remain recoverable when unknown rotation and unknown medium anisotropy appear together, and does an anisotropy-aware inverse still beat a Euclidean weighted shortcut in that combined-nuisance setting?

Scope Of This Test

This is a pose-free anisotropic inverse, but it is still not the most general warped-medium problem.

The assumptions are:

• three sources
• weighted constant-sum boundary
• translation removed from the boundary observation
• scale removed from the boundary observation
• one unknown axis-aligned anisotropy parameter alpha
• rotation treated as an observational pose nuisance on the radial-signature grid

So this experiment does not solve unknown anisotropy-axis orientation.

It solves the combined problem where:

• the observed signature can be rotated
• the medium strength parameter alpha is unknown
• geometry and weights are also unknown

Inverse Method

The experiment script is run_pose_free_weighted_anisotropic_inverse_experiment.py.

The method stays deliberately simple:

1. build an anisotropy-aware reference bank over geometry, weights, and alpha
2. encode each forward boundary as a centroid-centered, mean-radius-normalized radial signature
3. randomly rotate the observed signature on the 64-bin grid
4. recover the best bank element under cyclic-shift-aware masked L2 matching

So the inverse is not handed either the orientation or the medium parameter.

Baseline

The baseline is a Euclidean weighted reference bank with the same geometric and weight ranges but fixed alpha = 1, evaluated under the same pose-free cyclic-shift matching.

This keeps the comparison clean:

• if medium anisotropy is truly part of the latent state, the anisotropy-aware bank should still beat a Euclidean weighted shortcut even when rotation is unknown

Parameter Sweep

Reference banks:

• anisotropy-aware bank size: 300
• Euclidean weighted baseline bank size: 150

Test set:

• 40 weighted anisotropic test cases per observation regime

Anisotropy range:

• alpha in [0.60, 1.80]

Observation regimes:

• full_clean
• full_noisy
• partial_arc_noisy
• sparse_full_noisy
• sparse_partial_high_noise

Main Result

The combined-nuisance inverse works, but unevenly across the latent variables.

In the pose-free boundary-only setting, the weighted three-source latent object remains operational when unknown rotation and unknown anisotropy are present together: geometry and weights stay recoverable to a useful degree, and the anisotropy-aware inverse still decisively outperforms a Euclidean weighted shortcut, but alpha becomes much more weakly identified.

The summary file is pose_free_weighted_anisotropic_inverse_summary.json.

Global summary:

• anisotropy-aware bank size: 300
• Euclidean baseline bank size: 150
• trials per regime: 40
• best mean geometry MAE: 7.1066e-02
• worst mean geometry MAE: 1.0414e-01
• best mean weight MAE: 1.3793e-01
• worst mean weight MAE: 1.6562e-01
• best mean alpha absolute error: 1.4557e-01
• worst mean alpha absolute error: 3.0421e-01
• best mean anisotropy-aware fit RMSE: 7.8338e-03
• worst mean anisotropy-aware fit RMSE: 4.6687e-02
• smallest mean improvement factor over the Euclidean baseline: 3.7804
• largest mean improvement factor: 1.5925e+01

By regime:

• full_clean

  • geometry MAE mean: 7.1066e-02
  • weight MAE mean: 1.6163e-01
  • alpha absolute error mean: 1.9084e-01
  • anisotropy-aware fit RMSE mean: 8.0384e-03
  • Euclidean baseline fit RMSE mean: 9.8257e-02
  • mean improvement factor: 1.4075e+01

• full_noisy

  • geometry MAE mean: 7.8980e-02
  • weight MAE mean: 1.3793e-01
  • alpha absolute error mean: 2.6749e-01
  • anisotropy-aware fit RMSE mean: 7.8338e-03
  • Euclidean baseline fit RMSE mean: 1.0664e-01
  • mean improvement factor: 1.5925e+01

• partial_arc_noisy

  • geometry MAE mean: 8.7021e-02
  • weight MAE mean: 1.5521e-01
  • alpha absolute error mean: 2.3414e-01
  • anisotropy-aware fit RMSE mean: 1.5793e-02
  • Euclidean baseline fit RMSE mean: 1.2223e-01
  • mean improvement factor: 1.2703e+01

• sparse_full_noisy

  • geometry MAE mean: 8.6516e-02
  • weight MAE mean: 1.5850e-01
  • alpha absolute error mean: 1.4557e-01
  • anisotropy-aware fit RMSE mean: 1.5624e-02
  • Euclidean baseline fit RMSE mean: 1.1404e-01
  • mean improvement factor: 9.4438

• sparse_partial_high_noise

  • geometry MAE mean: 1.0414e-01
  • weight MAE mean: 1.6562e-01
  • alpha absolute error mean: 3.0421e-01
  • anisotropy-aware fit RMSE mean: 4.6687e-02
  • Euclidean baseline fit RMSE mean: 1.1085e-01
  • mean improvement factor: 3.7804

These numbers say something important:

• the anisotropy-aware latent model still carries much more explanatory power than the Euclidean shortcut
• geometry recovery remains fairly stable even under the combined nuisance
• weight recovery degrades, but not catastrophically
• alpha recovery degrades sharply once rotation is hidden too

Comparison To The Canonical Anisotropic Inverse

This experiment also records direct penalties relative to the earlier canonical-pose anisotropic inverse.

The comparison is in pose_free_weighted_anisotropic_inverse_penalties.csv.

The high-level picture is:

• geometry MAE stays in roughly the same range, with penalty factors from 0.917 to 1.238
• weight MAE worsens by about 1.24x to 1.70x
• alpha MAE worsens much more sharply, by about 4.77x to 14.39x
• anisotropy-aware fit RMSE is comparable to canonical in some regimes and modestly worse in the harder ones

Because both experiments use finite reference banks and nearest-neighbor recovery, penalty factors below 1 in a few geometry and fit entries should be read as finite-bank variation, not as evidence that unknown rotation makes anisotropic inversion easier.

The structural takeaway is:

• unknown rotation does not destroy the latent-object result
• it specifically damages anisotropy identifiability much more than geometry identifiability
• the combined nuisance creates a real conditioning problem for alpha

Interpretation

This is still a strong result, but it is strong in a more specific way than the previous inverse extensions.

The project now has evidence that:

• geometry and participation remain operational latent variables under combined pose and medium nuisance
• medium-aware modeling still matters a great deal for fit quality
• alpha is not equally well-conditioned under the current signature representation

So the right reading is not “everything survived equally well.”

The right reading is:

the latent-object framework survives the combined nuisance well enough to stay operational, but the medium parameter becomes the weak link under pose-free observation.

That is scientifically useful. It tells us where the framework is strong and where the inverse needs a better representation or better matching strategy.

One plausible mechanism is partial aliasing between cyclic pose variation and anisotropic deformation in the current radial-signature encoding.

Scope Of The Result

This experiment does show:

• pose-free boundary-only recovery of geometry, weights, and alpha
• continued strong advantage for an anisotropy-aware latent model over a Euclidean weighted shortcut
• a clear conditioning map showing that alpha is much more fragile than geometry under the combined nuisance

This experiment does not address:

• good alpha identifiability under pose-free observation
• recovery when the anisotropy axis itself is unknown
• recovery under a full positive-definite metric matrix rather than a single scalar alpha
• recovery in non-quadratic or spatially varying media

Figures

• pose_free_weighted_anisotropic_inverse_heatmap.png
• pose_free_weighted_anisotropic_inverse_baseline_and_penalty.png
• pose_free_weighted_anisotropic_inverse_examples.png

The clearest figures are:

• pose_free_weighted_anisotropic_inverse_heatmap.png for the uneven recovery profile across geometry, weights, and alpha
• pose_free_weighted_anisotropic_inverse_baseline_and_penalty.png for the continued baseline gap and the rotation penalty relative to canonical anisotropic inversion

Artifacts

Data:

• pose_free_weighted_anisotropic_inverse_trials.csv
• pose_free_weighted_anisotropic_inverse_summary.csv
• pose_free_weighted_anisotropic_inverse_penalties.csv
• pose_free_weighted_anisotropic_inverse_summary.json

Code:

• run_pose_free_weighted_anisotropic_inverse_experiment.py

Recommended Next Step

The cleanest next step is not a broader medium yet.

It is to improve alpha identifiability under the same combined nuisance.

At this point the project has shown that the latent-object result survives, but it has also revealed a solver challenge:

• geometry and weights are relatively robust
• alpha is much more weakly identified once pose is hidden

So the highest-leverage next experiment is to test a richer inverse representation or alignment scheme that tries to separate anisotropy from pose rather than broadening immediately to unknown anisotropy axes or richer warped media.