A passive optical imaging system recovered the test target only when its decoder used the same value of a tuning parameter, called gamma, as the encoder, while naive decoding produced gamma-dependent, structured leakage.
A decoder built to match
The paper calls the design non-Hermitian biorthogonal encoding. In plain terms, it uses one set of mathematical patterns to encode an image and a paired, gamma-dependent set to decode it. With the matched pair, the channel is exactly the identity; same-side Hermitian decoding applies a different transformation instead.
The numerical workflow evaluated the forward coefficient matrix once, then changed only the decoding metric or coordinate. It used no learned prior, regularizer or iterative inverse solver.
Experimental validation used a reflective Chinese-character target in a conventional passive, intensity-only single-pixel imaging chain with programmable DMD masks, a single-pixel detector and a data-acquisition system. No non-Hermitian material was placed in the optical path. Each experimental complex coefficient was synthesized from four non-negative intensity measurements, and retaining K modes required 4K projected DMD patterns.
What the image tests found
Numerical validation used a finite 64-by-64 operator space and a grayscale 64-by-64 university-emblem target. On that target, matched reconstruction improved as ordered modes were added: at 5% sampling, PSNR was 12.31 dB, SSIM was 0.306 and Pearson correlation was 0.788; at 50%, those figures were 20.95 dB, 0.883 and 0.987. MAE fell from 0.189 to 0.077, while the naive branch had a PSNR of 6.46 dB even at 100% sampling.
A separate Cameraman simulation showed the same pattern. Matched PSNR, SSIM and Pearson correlation rose from 18.28 dB, 0.398 and 0.901 at 5% sampling to 28.04 dB, 0.913 and 0.996 at 50%. With complete sampling, the naive branch had an SSIM of 0.080 and Pearson correlation of -0.282.
In the passive single-pixel experiment, authorized decoding preserved the topology of the Chinese-character target across gamma values from 0.1 to 1.0. At a fixed encoding value of gamma = 0.6, the target was recovered only when the decoding value matched it; naive decoding generated gamma-dependent wrong-metric leakage.
Complete sampling did not overcome a mild decoding-coordinate mismatch in the emblem simulation. With decoding gamma = 0.65, the reconstruction still produced only a PSNR of 10.56 dB and an SSIM of 0.361.
The work also identified a numerical warning near an exceptional point, where a tracked pair of modes approaches coalescence. For the finite N = 64 operator, the pair approached this point at gamma approximately equal to 0.303; eigenvalue splitting and phase rigidity collapsed, and the authorized channel became more sensitive to experimental noise.
Where the evidence stops
The demonstrations do not establish unconditional security: the paper says a formal proof is beyond its scope. The partial-sampling panels were also retrospective prefix reconstructions from complete experimental acquisitions, not a real-time early-stop or dynamic-scene benchmark.
The reported image scores are target-specific and were not accompanied by confidence intervals or independent-run uncertainty.
The exceptional-point result is tied to the finite N = 64 discretization and the tracked eigenvalue pair described in the study.
Paper data and sources
Original title: A Non-Hermitian Biorthogonal Encoding Paradigm for Physical-Layer Secure Computational Imaging
Authors: Xi-Hao Chen, Kan-Xu Jia, En-Rui Zhang et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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