Parabolic Orientation Sihti

Does the angular law of a lossless sieve decide where the objects go?

Sihti sieves an image into a core plus octave residues that add back exactly, and its G4 test asks where the objects end up. This repo splits every octave residue further by direction, under six direction laws, keeps only a budget of coefficients, and scores the rebuild with Sihti's own G4 measure on BSDS500. Live viewer: anttiluode.github.io/ParabolicOrientationSihti (your image or webcam, all six laws, both rankings, the coherence map).

Why this test

OpenAI's local-smoothing preprint (Oct 2026, §1.4–2) filters a superposition of waves by direction by translating spacetime along an imaginary light ray, −iρ(1, ω). That multiplies mode ξ by exp(−ρ|ξ|·|ω−ξ̂|²/2): a Gaussian in direction whose width shrinks as (ρ|ξ|)^(−1/2). That is the parabolic scaling law of curvelets (Candès–Donoho 2004), produced by analytic continuation. The proof also insists the modes be summed before squaring, so their interference survives.

Two questions followed for Sihti. First, does a parabolic direction law put more object structure into a budget than a fixed one? Second, does a phase-relationship score (cross-scale coherence, the local cross term) pick better coefficients than magnitude? The paper claims nothing about images. Everything below is our test, not its consequence.

Result (BSDS500 test, n = 200, pre-registered)

Rules and predictions were pushed in PREREG.md (commit 7bc172e) before the test split was touched. Primary: Sihti G4 alignment of the rebuild at a 10 % budget. Decision rule: higher mean, wins on ≥ 60 % of images, sign test p < 0.01.

	claim (a beats b)	wins / losses	mean a vs b	p	predicted	outcome
H1	parabolic > fixed 8	145 / 55	0.126 vs 0.112	1.5e-10	pass	PASS
H1k	parabolic > steerable-pyramid angular windows, sp8 (kill condition)	171 / 29	0.126 vs 0.105	1e-25	pass	PASS: not killed
H1m	parabolic > fixed 8 at equal PSNR	112 / 88	0.116 vs 0.112	0.10	uncertain	fail
H1mk	parabolic > sp8 at equal PSNR	134 / 66	0.111 vs 0.105	1.7e-6	uncertain	PASS
H2	ray (the paper's window) > parabolic steps	82 / 118	0.125 vs 0.126	0.013	fail	fail
H3	coherence ranking > magnitude (G4 alignment)	45 / 155	0.107 vs 0.126	2e-15	fail	fail
H3b	coherence ranking puts kept detail on human boundaries	148 / 52	0.090 vs 0.035	7e-12	pass	PASS
H0	parabolic > isotropic Sihti	31 / 169	0.126 vs 0.159	3e-24	fail	fail

What it means

1. The direction law matters, and parabolic is the right one. At equal redundancy (48 bands each), parabolic beats fixed 8 and sp8 at every budget from 1 % to 25 %. Anti-parabolic is worst (parabolic beats it 153/47), so it is the direction of the scaling that matters, not merely its unevenness. The kill condition did not fire.

2. Most of that is the known curvelet sparsity, and a smaller part may not be. At equal reconstruction quality the gain over fixed 8 is not significant (H1m), so against fixed 8 parabolic mainly compresses better: Candès–Donoho, now measured on objects. Over sp8 a gain survives equal PSNR (H1mk). A post-hoc check (not pre-registered) shows window shape alone does not explain it: fixed 8 vs steerable pyramid at equal PSNR is 105/95, p = 0.52. So a beyond-sparsity object advantage of the parabolic law is suggested, not established: +0.0044 against fixed 8 at p = 0.10.

3. The paper's window is a curvelet window. The continuous imaginary-ray window (ray) ties the stepwise parabolic one, nominally a hair worse. The physics produces the right law; it does not produce a better one.

4. "Relationships before squaring" finds outlines, and outlines are not what G4 calls objects. The local cross-scale coherence score moves 2.5× more of the kept energy onto human boundaries (H3b), and you can see it in the examples: sharper trunk edges and wing lines. But it lowers G4 alignment (H3). G4 rewards rebuilds that fill regions with their own colour, while coherence spends the budget drawing their edges. Both are object information; they are different kinds, and this is now measured rather than argued.

5. No direction law beats having no directions here. Plain isotropic Sihti wins (H0). The likely cause is redundancy: iso has 6 bands, the oriented arms 48, all undecimated, so each oriented coefficient carries less. The equal-PSNR comparison that was meant to separate this is not readable: iso's PSNR at 10 % lies outside parabolic's whole budget range on 166/200 images. A fair version needs a decimated (critically sampled) oriented transform.

First four test images in sorted order (not chosen by result), 10 % budget. 10 % of the image's reals is ~0.2 % of a 48-band undecimated frame's coefficients, which is why every rebuild is blurry (~17 dB). The comparison is what survives at that budget. Rightmost: cross-scale coherence, bright where adjacent octaves agree in phase.

Ledger

Verified

• Every arm is a tight frame and lossless: core + Σ residues = image to 1e-12, plus the frame synthesis (33 Gate 0 tests, tests/test_gate0.py)
• The JS viewer reproduces the Python rebuilds to 2e-15
• H1, H1k, H1mk, H3b pass; H1m, H2, H3, H0 fail; all eight came out on the side predicted, or split where the prediction was "uncertain"

Found on the way (Gate 0, before any data)

• Selecting coefficients in the original quadrant only breaks exactness. The fix: one chosen coefficient switches on its three mirror copies, and a flip maps orientation θ to π − θ, so the copies live in the partner band.
• The cross term p²/|p|·ā flips sign with edge polarity, so mirrored copies cancel. Averaging q²/|q| makes it polarity-invariant.
• Octave-0 coherence is at noise level even on a clean antialiased edge (0.22 vs 0.14), so it is excluded from coherence ranking. Octaves 1–5 separate edge from noise (0.66–0.83 vs 0.10–0.17).

Not established / open

• Whether the parabolic law adds object structure beyond sparsity (H1m p = 0.10)
• Isotropic vs oriented at equal redundancy: needs a decimated transform
• Whether G4 or boundary concentration is the right reading of "object". This repo shows they disagree.

Killed

• "The paper's window is better than curvelet steps" (H2)
• "Phase-coherence ranking makes the rebuild more object-like" in G4's sense (H3)

Correction (Oct 27, from Astra's review). sp8 is the Simoncelli–Freeman angular window (|cos|^(K−1), 8 orientations) placed inside this repo's own undecimated tight frame, with the same radial octaves as every other arm. It is not a stock complex steerable pyramid, which uses its own radial filters and recursive subsampling. So H1k and H1mk show that parabolic beats that angular-window control in this frame. They do not show that it beats established steerable-pyramid systems; that comparison was never run.

Prior art. Curvelets and the parabolic law: Candès & Donoho 2004. Steerable pyramid: Simoncelli & Freeman 1995. Phase-doubled cross-scale statistics p²/|p|: Portilla & Simoncelli 2000. Phase congruency: Kovesi 1999. Complex-shifted Gaussian beams: Deschamps 1971. The G4 measure and BSDS protocol: anttiluode/Sihti. The window formula: OpenAI, Local smoothing for the Euclidean wave equation, Oct 25 2026, §1.4–2.

Run it

pip install -r requirements.txt
python -m pytest -q tests                                   # Gate 0
git clone https://github.com/BIDS/BSDS500
python bench.py --bsds BSDS500/BSDS500/data --phase discovery   # val, 30 images, ~7 min
python bench.py --bsds BSDS500/BSDS500/data --phase test        # test, 200 images, ~45 min
python make_figures.py

posihti/frames.py holds the six frames, posihti/transform.py the decomposition, m-term selection and coherence, posihti/metrics.py the Sihti G4 measure (copied unchanged), analyze.py the frozen decision rules. Per-image numbers are in results/test_cache.jsonl.