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arXiv · 2610.23912

Junction Geometry Survives an Unidentifiable Blur

Sohail·Sarkar·Shakuntala Baichoo
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Abstract

Thin filaments cross in microscopy, and dispersed spectra cross on X-ray detectors. At a junction, where these structures cross or branch, how many straight arms meet, and which way do they go? Unknown blur obscures the answer. At a balanced crossing, distinct Gaussian blurs can even produce the same observation law. We show that the arms can still be recovered. For separated rays of nonvanishing weight and an unknown centered Gaussian-mixture blur, a distance combining arm count, directions, relative weights and supporting lines is bounded by a constant times the Euclidean difference in cell probabilities on a sufficiently fine fixed grid. The bound is uniform over a bounded model class, including unidentifiable blurs. From $n$ positions or photon counts, the minimax error is of order $n^{-1/2}$. This rate is proved for a global estimator. In practice we fit rays and blur by constrained local likelihood, reducing the BIC count when two detectors agree on fewer arms in the microscopy studies. Across three studies with protocols fixed before evaluation, on 360 crossings and branches assembled from measured single-molecule localizations, the fit gets the arm count and every direction within $10^\circ$ 24 to 37 percentage points more often than a tuned fixed-scale Hessian ridge detector, averaged over 25 to 400 localizations. On acute angles with correct counts, mean absolute error is $3.5^\circ$ to $4.3^\circ$, versus $10.4^\circ$ to $15.5^\circ$. On Chandra spectra, 38 of 90 fits at 800 photons find all four arms within $10^\circ$; among these, median error is $0.023^\circ$.

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Sohail, Sarkar, Shakuntala Baichoo. 2026-09-30. Junction Geometry Survives an Unidentifiable Blur. https://arxiv.org/abs/2610.23912

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