Analyzing the Effect of the Intra-Pixel Position of Small PSFs for Optimizing the PL of Optical Subpixel Localization

Abstract

Subpixel localization techniques for estimating the positions of point-like images captured by pixelated image sensors have been widely used in diverse optical measurement fields. With unavoidable imaging noise, there is a precision limit (PL) when estimating the target positions on image sensors, which depends on the detected photon count, noise, point spread function (PSF) radius, and PSF’s intra-pixel position. Previous studies have clearly reported the effects of the first three parameters on the PL but have neglected the intra-pixel position information. Here, we develop a localization PL analysis framework for revealing the effect of the intra-pixel position of small PSFs. To accurately estimate the PL in practical applications, we provide effective PSF (ePSF) modeling approaches and apply the Cramér–Rao lower bound. Based on the characteristics of small PSFs, we first derive simplified equations for finding the best PL and the best intra-pixel region for an arbitrary small PSF; we then verify these equations on real PSFs. Next, we use the typical Gaussian PSF to perform a further analysis and find that the final optimum of the PL is achieved at the pixel boundaries when the Gaussian radius is as small as possible, indicating that the optimum is ultimately limited by light diffraction. Finally, we apply the maximum likelihood method. Its combination with ePSF modeling allows us to successfully reach the PL in experiments, making the above theoretical analysis effective. This work provides a new perspective on combining image sensor position control with PSF engineering to make full use of information theory, thereby paving the way for thoroughly understanding and achieving the final optimum of the PL in optical localization.