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"))
Recovers an axisymmetric distribution f(r) from the projection A(y) (inverse Abel transform = tomography of an axisymmetric object).
• "derivative" —— f(r) = −(1/π) ∫_r^R A'(y) / √(y² − r²) dy. Substituting y = √(r² + u²) removes the singularity, and A is differentiated with a cubic spline. Accurate for smooth projections, but differentiation amplifies noise:
the noise error grows inversely with the sample spacing dr (measured 2026-10-03, 1% noise: 80 → 640 samples gives
mean error 0.027 → 0.237, log-log slope ≈ 1). On coarse grids this one is better.
• "onion" —— onion peeling: f is taken as constant per shell of radius dr, and the upper-triangular matrix of chord lengths is solved from the outside in. Resolution is set by the shell width; since no derivative is taken, noise grows more slowly (same measurement: 0.036 → 0.100, slope ≈ 1/2). In measurements on fine grids, this one wins (1/2.4 of the derivative's error at n=640); on coarse grids (n≈80) it reverses.
The 2 are independent discretizations, so applying both to the same projection and checking the discrepancy serves as a cross-check.
Gate: from A(y) = √π σ exp(−y²/σ²), 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.