发表机构
KTH Royal Institute of Technology; Ericsson Research（皇家理工学院; 爱立信研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出近场超分辨率数学理论，引入支持均匀二次相位孔径准则，证明全变差最小化可精确恢复稀疏测度，并构造有限谐波Bessel-Vandermonde提升，远场时退化为傅里叶型。
AI 中文摘要
有限孔径近场传感导致了一种与平移不变傅里叶设置根本不同的超分辨率几何。在菲涅耳区域，波前曲率引入了距离相关的二次孔径相位，因此可分辨性由形式为 \\( \sum_{n=0}^{N_r-1} a_n e^{i(\omega_1 n+\omega_2 n^2)} \\) 的不完全二次指数和决定，而非仅由角度间隔决定。我们为在有限网格上具有距离和连续角度的稀疏测度发展了一种确定性恢复理论，并引入了一种支持均匀的二次相位孔径准则，取代经典的最小间隔。在该准则下，全变差最小化能够精确恢复可支持类中的每个稀疏测度，且对所有非零复幅度一致成立。证明过程发展了有限二次和的非渐近支持均匀界，并将其与一个带度量的Hermite对偶证书相结合，以控制插值、局部曲率和支撑外泄漏。我们进一步构造了一个具有显式截断误差的有限谐波Bessel-Vandermonde提升。在远场极限下，二次相位消失，该理论退化为傅里叶型角度超分辨率。
英文摘要
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.
