Closed-Form Expression of the Fourier Ring-Correlation for Single-Molecule Localization Microscopy

Single-molecule localization microscopy (SMLM) is a popular microscopic technique that achieves super resolution imaging by localizing individual blinking molecules in thousands of frames. Therefore, the reconstructed high-resolution image is a combination of millions of point sources. This particular computational reconstruction leads to the question of the estimation of the image resolution. Fourier-ring correlation (FRC) is the standard tool for assessing the resolution. It has been proposed for SMLM by computing a discrete correlation in the Fourier domain. In this work, we derive a closed-form expression to compute the continuous FRC. Our implementation provides an exact FRC and an alternative to compute a parameter-free FRC. In addition, it gives insights on the discrepancy of the discrete FRC and yields a rule to select its parameters such as the spatial sampling step or the width of the kernel used as density estimator.