Curvature-aware inverse specular geometry for camera-assisted mmWave NLoS localization: identifiability, stability, and numerical analysis

Shih-Lin Lin; Shih-Lin Lin

doi:10.3934/math.20261351

AIMS Mathematics

2026, Volume 11, Issue 10: 34661-34698. doi: 10.3934/math.20261351

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Research article

Curvature-aware inverse specular geometry for camera-assisted mmWave NLoS localization: identifiability, stability, and numerical analysis

Shih-Lin Lin ^,

Graduate Institute of Vehicle Engineering, National Changhua University of Education, Changhua City, Taiwan

Received: 31 July 2026 Revised: 19 September 2026 Accepted: 08 October 2026 Published: 23 October 2026
MSC : 65C05, 65H10, 65K10, 93B07

This paper develops a curvature-aware specular-reflection inverse-geometry model for camera-assisted millimeter-wave non-line-of-sight localization with an unknown curved reflector and a hidden target. In a calibrated two-dimensional setting, a finite-dimensional C³ regular curve, a locally selected nondegenerate Fermat branch, radar range/azimuth measurements, and camera-derived reflector constraints form a whitened augmented forward operator. Implicit differentiation yields a reduced Jacobian for analyses of local identifiability, Fisher-Schur information, and small-perturbation stability. The tangent-plane approximation induces a second-order range bias, but the azimuth and generic localization biases remain first order. A scaled Levenberg-Marquardt solver is evaluated using 400 paired synthetic trials. Under nominal noise, the curved fusion model attains a localization root-mean-square error (RMSE) of 0.145 m versus 0.199 m for the planar model; the paired 95% bootstrap improvement interval is [0.0444, 0.0625] m. Initialization-scale tests and a deterministic 12-geometry, two-level Plackett-Burman screen further characterize empirical local recovery and within-design geometric sensitivity. These synthetic results are limited to calibrated two-dimensional geometry, a finite-dimensional reflector, known single-bounce association, a nongrazing local branch, and branch-valid initialization; they do not establish global path uniqueness, convergence from arbitrary initial values, or vehicle-mounted experimental validity.
- non-line-of-sight localization,
- mmWave radar,
- camera-radar fusion,
- specular-reflection inverse problem,
- curvature-aware modeling,
- local identifiability,
- Levenberg-Marquardt method
Citation: Shih-Lin Lin. Curvature-aware inverse specular geometry for camera-assisted mmWave NLoS localization: identifiability, stability, and numerical analysis[J]. AIMS Mathematics, 2026, 11(10): 34661-34698. doi: 10.3934/math.20261351

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Abstract

This paper develops a curvature-aware specular-reflection inverse-geometry model for camera-assisted millimeter-wave non-line-of-sight localization with an unknown curved reflector and a hidden target. In a calibrated two-dimensional setting, a finite-dimensional C³ regular curve, a locally selected nondegenerate Fermat branch, radar range/azimuth measurements, and camera-derived reflector constraints form a whitened augmented forward operator. Implicit differentiation yields a reduced Jacobian for analyses of local identifiability, Fisher-Schur information, and small-perturbation stability. The tangent-plane approximation induces a second-order range bias, but the azimuth and generic localization biases remain first order. A scaled Levenberg-Marquardt solver is evaluated using 400 paired synthetic trials. Under nominal noise, the curved fusion model attains a localization root-mean-square error (RMSE) of 0.145 m versus 0.199 m for the planar model; the paired 95% bootstrap improvement interval is [0.0444, 0.0625] m. Initialization-scale tests and a deterministic 12-geometry, two-level Plackett-Burman screen further characterize empirical local recovery and within-design geometric sensitivity. These synthetic results are limited to calibrated two-dimensional geometry, a finite-dimensional reflector, known single-bounce association, a nongrazing local branch, and branch-valid initialization; they do not establish global path uniqueness, convergence from arbitrary initial values, or vehicle-mounted experimental validity.

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