Mathematical theory explains super-resolution from curved wave sensing
A Mathematical Theory of Near-Field Super-Resolution
Information Theory
Summary
When sensors look very closely at objects using waves, the way waves bend and curve changes how well details can be seen. The authors developed a new math theory that explains how to tell apart very close objects using these curved waves, which is different from older methods that assumed waves travel straight. They show a new rule to decide when objects can be fully distinguished and prove a method that perfectly recovers the objects’ positions under this rule. Their math also connects back to standard techniques when objects are far away and waves don't curve as much.
What this means in practice
- •For ultrasound imaging engineers: Design systems that distinguish closely spaced scatterers when wavefront curvature is non-negligible, improving near-field image resolution.
- •For radar system designers: Develop algorithms to recover target positions precisely in near-field scenarios where classical far-field assumptions fail.
A theory result. No direct application yet.
Authors
Sajad Daei, Gabor Fodor, Mikael Skoglund
Abstract
Finite-aperture near-field sensing leads to a super-resolution geometry fundamentally different from the translation-invariant Fourier setting. In the Fresnel regime, wavefront curvature introduces a range-dependent quadratic aperture phase, so distinguishability is governed by incomplete quadratic exponential sums of the form \( \sum_{n=0}^{N_r-1} a_n e^{i(ω_1 n+ω_2 n^2)} \), rather than by angular separation alone. We develop a deterministic recovery theory for sparse measures with ranges on a finite grid and continuous angles, and introduce a support-uniform quadratic-phase aperture criterion replacing classical minimum separation. Under this criterion, total-variation minimization exactly recovers every sparse measure in the admissible support class, uniformly over all nonzero complex amplitudes. The proof develops nonasymptotic support-uniform bounds for finite quadratic sums and combines them with a gauged Hermite dual certificate controlling interpolation, local curvature, and off-support leakage. We further construct a finite-harmonic Bessel-Vandermonde lift with explicit truncation error. In the far-field limit, the quadratic phase disappears and the theory reduces to Fourier-type angular super-resolution.