Functional Multi-Target Detection via Bispectrum Inversion
基于双谱反演的功能性多目标检测
Anna Little, Daniel Sanz-Alonso, Mikhail Sweeney, Ruiyi Yang
AI总结 针对含未知平移的多目标检测问题,提出基于自相关分析的无初始化恢复算法,通过去偏三阶经验自相关估计双谱,并利用频率推进或Kotlarski反卷积公式恢复信号,证明非渐近恢复保证。
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本文发展了多目标检测的功能性理论,其中从包含信号多个未知平移的单个含噪观测中恢复紧支撑信号。我们的公式允许连续、非网格平移和相关平稳高斯过程噪声,超越了先前工作中常见的离散、网格对齐、白噪声模型。我们分析了两种基于自相关分析的无初始化恢复算法;特别地,两种算法首先通过去偏三阶经验自相关估计信号的双谱。然后利用功能性频率推进方案或Kotlarski型反卷积公式从估计的双谱中恢复信号。对于两种算法,我们在无带限假设下证明了紧支撑信号的非渐近恢复保证。得到的误差界依赖于信号的光滑性和双谱估计的精度,后者由噪声特性和信号出现次数决定。数值实验验证了我们的理论,并展示了在低信噪比条件下的准确恢复。
This paper develops a functional theory for multi-target detection, where a compactly supported signal is recovered from a single noisy observation containing many unknown translations of the signal. Our formulation allows continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond the discrete, grid-aligned, white-noise models common in prior work. We analyze two uninitialized recovery algorithms based on autocorrelation analysis; in particular, both algorithms first estimate the signal's bispectrum via a debiased third-order empirical autocorrelation. The signal is then recovered from the estimated bispectrum using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both algorithms, we prove non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The resulting error bounds depend on the smoothness of the signal and the accuracy of bispectrum estimation, with the latter governed by the noise characteristics and the number of signal occurrences. Numerical experiments validate our theory and demonstrate accurate recovery in low-SNR regimes.