Prasad Sudhakar, Luo Xuan
Center for Advanced Studies and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA.
Opt Express. 2009 Dec 7;17(25):23213-33. doi: 10.1364/OE.17.023213.
We analyze the problem of optical superresolution (OSR) of a one-dimensional (1D) incoherent spatial signal from undersampled data when the support of the signal is known in advance. The present paper corrects and extends our previous work on the calculation of Fisher information (FI) and the associated Cramer-Rao lower bound (CRB) on the minimum error for estimating the signal intensity distribution and its Fourier components at spatial frequencies lying beyond the optical band edge. The faint-signal and bright-signal limits emerge from a unified noise analysis in which we include both additive noise of detection and shot noise of photon counting via an approximate Gaussian statistical distribution. For a large space-bandwidth product, we derive analytical approximations to the exact expressions for FI and CRB in the faint-signal limit and use them to argue why achieving any significant amount o unbiased bandwidth extension in the presence of noise is a uniquely challenging proposition. Unlike previous theoretical work on the subject of support-assisted bandwidth extension, our approach is not restricted to specific forms of the system transfer functions, and provides a unified analysis of both digital and optical superresolution of undersampled data.
当一维（1D）非相干空间信号的支撑预先已知时，我们分析从欠采样数据中对其进行光学超分辨率（OSR）的问题。本文修正并扩展了我们之前关于费希尔信息（FI）计算以及相关克拉美 - 罗下界（CRB）的工作，该下界针对估计位于光学频带边缘之外空间频率处的信号强度分布及其傅里叶分量时的最小误差。微弱信号和强信号极限源自统一的噪声分析，其中我们通过近似高斯统计分布纳入了检测的加性噪声和光子计数的散粒噪声。对于大的空间带宽积，我们在微弱信号极限下推导了FI和CRB精确表达式的解析近似，并利用它们论证了为何在存在噪声的情况下实现任何显著的无偏带宽扩展是一个极具挑战性的独特命题。与先前关于支撑辅助带宽扩展主题的理论工作不同，我们的方法不限于系统传递函数的特定形式，并提供了对欠采样数据的数字和光学超分辨率的统一分析。
