Mirror Lens Central Obstruction Detected as Persistent Homology Signature in 2D Photographs

A mirror lens (catadioptric) has a physical hole in its optical path, a central obstruction that projects out-of-focus highlights as donut-shaped rings rather than solid circles. We asked whether SCOTT (Structural and Coherent Order in Topological Transforms), a multi- metric image analysis instrument spanning fractal, geometric, and topological measures, could detect this three-dimensional optical feature from a flat two-dimensional photograph, with no knowledge of optics or lens type.

The prediction was locked before any image was sourced: donut bokeh should produce higher Betti-1 counts and higher H1 persistence entropy in the edge point cloud, because the physical hole creates genuine topological loops that solid bokeh does not. A pilot run (n = 5 per category, three categories) confirmed the predicted metric ordering. A replication on independently sourced specimens (n = 15 solid, n = 20 donut) confirmed both primary metrics with large effects: H1 persistence entropy at Cohen’s d = 1.47, 95% CI [0.72, 2.22], p < 0.001, and Betti-1 at d = 1.09, 95% CI [0.37, 1.81], p = 0.002. A systematic brightness difference between categories (d = 0.82) was
identified and tested directly via subsample matching, where both primary metrics survived and strengthened: H1 persistence entropy rose to d = 2.09, 95% CI [1.18, 3.00], and Betti-1 to d = 1.19, 95% CI [0.40, 1.98]. The instrument reads the topological shadow of a three-dimensional physical feature from two-dimensional image data.