Compact astrometry and beam deconvolution

The astrometry boundary transforms a valid compact Gaussian fit from global pixel coordinates to the internal ICRS catalogue model. It is kept separate from fitting: the optimizer is a pure pixel-space operation, and this boundary owns WCS, angular geometry and restoring-beam semantics.

Coordinate transformation

Hebog rebuilds an Astropy celestial WCS from the serializable image metadata and uses zero-based continuous (x, y) pixel coordinates. A centred finite difference at the fitted centroid, with a step of 2 arcseconds on the sky converted to pixels for each axis, gives a local two-by-two Jacobian from pixel offsets to east/north tangent-plane offsets in degrees. It is within about 10⁻¹⁰ of the local scale at any pixel scale and on every platform, wherever the projection and any distortion are close to linear over 2 arcseconds; the error grows near the horizon of a SIN image (about 10⁻⁷ at 82 degrees from its reference) and with strong distortion. The Jacobian transforms the centroid to ICRS right ascension and declination, the pixel covariance to a celestial ellipse, the centroid covariance to position errors, and the local pixel area used by fitted and island fluxes. Right ascension is canonical in [0, 360) and position angle is degrees east of north modulo 180. The local Jacobian, rather than one header pixel scale, preserves signed and unequal scales, rotation, projection and wraparound.

Beam deconvolution

Fitted and restoring-beam ellipses are two-by-two east/north covariance matrices. Hebog subtracts the beam covariance and classifies the eigenvalues: two significant positive intrinsic axes give a resolved deconvolved ellipse, none gives unresolved, and one gives major-axis-only with no minor axis or position angle.

Positive geometric deconvolution is necessary but not sufficient evidence of extension. Hebog applies the one-sided ATLAS DR3 statistic, ln(S_integrated / S_peak) over the relative uncertainty of the two fluxes, combined in quadrature less their covariance when the fit gives it, at a conservative five sigma rather than ATLAS's two, because a false resolved shape is a material catalogue error and the equivalence contract requires Hebog to be no worse than released PyBDSF. A geometrically resolved fit that fails the test is unresolved with extension-not-significant; a fit with no flux uncertainty has an unavailable classification (deconvolution-uncertainty-unavailable); an exact analytic fit keeps its geometric result, so noiseless contract cases stay exact. For a free noisy fit, a finite-difference delta method propagates the covariance of major sigma, minor sigma and angle through the WCS and beam subtraction, and each deconvolved eigenvalue must exceed zero at the same five-sigma level independently: a significant major with an insignificant minor is major-axis-only, which keeps the DC_Maj that Rapthor consumes whenever it is identifiable. A noisy fit with flux uncertainty but no usable shape covariance has an unavailable deconvolution rather than a falsely precise ellipse.

The threshold is explicit configuration. The equivalence contract gates point-source specificity and recall for clearly resolved truth, selected from injected truth before fitting as a fitted-to-beam area ratio of at least 3 at SNR of at least 25; less decisive extension is a predeclared report-only population. On the paired regression population of 1,600 point sources and 200 clear extensions, point-source statistics span −2.08 to 3.38 sigma and every clear extension 17.92 to 23.83, so the five-sigma decision separates them with a wide margin; this is regression evidence, not qualification.

An unresolved internal shape is null, and a major-axis-only result stores one positive major FWHM; the zero DC_Maj of the PyBDSF/Rapthor view is produced only for an explicitly unresolved source and never re-enters a calculation. The radio_beam package was evaluated and not adopted: it solves the same small covariance problem with its own failure conventions, and Hebog needs the explicit three-state policy above.

Uncertainty status

The selected fit's nonsingular centroid covariance gives one-sigma position errors. Both are great-circle angles, because the Jacobian is east/north: E_RA is not divided by cos(dec), matching PyBDSF and the fixed angle Rapthor compares it with. Flux errors use the same covariance, flagged correlated-noise-sandwich-errors with a declared beam correlation and formal-independent-pixel-errors without one. Fitted-shape uncertainties come from the same covariance for a free fit whose shape block is usable; for a beam-constrained or circular fit, or an unusable block, they are null with shape-uncertainty-unavailable. When the fit covariance is unavailable, position and flux errors are null with position-flux-uncertainty-unavailable. Zero never means unknown. A component's published integrated flux is its infinite-plane fitted integral, resolved or not, so a source's summed flux follows PyBDSF's definition; the propagated uncertainty of a resolved component's integral is report-only.