transform op• Data kinds: signal → signal
• Call: import fullseye as fs; fs.ledger.abel_transform(f, dr: 'float' = 1.0, *, n_quad: 'int' = 64) -> 'np.ndarray' (to call the implementation directly, import mathtransforms; mathtransforms.abel_transform(f, dr: 'float' = 1.0, *, n_quad: 'int' = 64) -> 'np.ndarray'; from the registry, opsmath.get("abel_transform"))
Maps an axisymmetric distribution f(r) (samples at r = 0, dr, 2dr, …) to its projection A(y) (forward Abel transform).
A(y) = 2 ∫_y^R f(r) r / √(r² − y²) dr. Substituting r = √(y² + u²) gives A(y) = 2 ∫_0^{√(R²−y²)} f(√(y²+u²)) du, and the singularity at y = r disappears (the integrand is smooth). f is interpolated with a cubic spline, and the integral over u uses Gauss–Legendre with n_quad points. Outside R, f = 0 is assumed.
Gate: if f = exp(−r²/σ²), then A(y) = √π σ exp(−y²/σ²) (when R is large enough). The return value has the same length and the same grid as f (y = 0, dr, 2dr, …).
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_inverse · 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.