transform op• Data kinds: signal → signal
• Call: import fullseye as fs; fs.ledger.abel_inverse(A, dr: 'float' = 1.0, *, method: 'str' = 'derivative', n_quad: 'int' = 64) -> 'np.ndarray' (to call the implementation directly, import mathtransforms; mathtransforms.abel_inverse(A, dr: 'float' = 1.0, *, method: 'str' = 'derivative', n_quad: 'int' = 64) -> 'np.ndarray'; from the registry, opsmath.get("abel_inverse"))
> This operator's description has not been translated yet. The original text follows as it is.
投影 A(y) から軸対称な分布 f(r) を戻す(逆 Abel 変換 = 軸対称物体の断層化)。
• `"derivative"` —— f(r) = −(1/π) ∫_r^R A'(y) / √(y² − r²) dy。y = √(r² + u²) と置いて特異点を消し、
A を 3 次スプラインで微分する。滑らかな投影には精度が高いが、微分が雑音を増幅する:
雑音の誤差は標本間隔 dr に反比例して増える(2026-10-03 実測、雑音 1%: 標本数 80 → 640 で
平均誤差 0.027 → 0.237、両対数の傾き ≈ 1)。粗い格子ではこちらが良い。
• `"onion"` —— 殻を剥く: f を半径 dr の殻ごとに一定とみなし、弦の長さの上三角行列を外側から解く。
解像度は殻の幅で決まり、微分を取らないぶん雑音の増え方が緩い(同じ実測で 0.036 → 0.100、傾き ≈ 1/2)。
細かい格子の実測ではこちら(n=640 で微分の 2.4 分の 1)、粗い格子(n≈80)では逆転する。
2 つは独立な離散化なので、同じ投影に両方を当てて食い違いを見るのが検算になる。
門: A(y) = √π σ exp(−y²/σ²) から f(r) = exp(−r²/σ²)。
Every mathops op validates its input before computing (nothing slips through silently):
• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• transforms_tour — py -3.11 examples/transforms_tour.py
signal as input)mat_solve · mat_lstsq · stat_describe · stat_histogram · stat_zscore · interp_linear · interp_cubic · interp_scattered
transform)abel_transform · abel_inverse_image · abel_revolve · hankel_transform · tf_poles_zeros · tf_freq_response · tf_impulse_response · tf_step_response
*Provenance: mathtransforms.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.