Within the family of super-resolution (SR) fluorescence microscopy, single-molecule localization microscopies (PALM, STORM and their derivatives) afford among the highest spatial resolution (approximately 5 to 10 nm), but often with moderate temporal resolution. The high spatial resolution relies on the adequate accumulation of precise localizations of bright fluorophores, which requires the bright fluorophores to possess a relatively low spatial density. Several methods have demonstrated localization at higher densities in both two dimensions (2D)[3, 4] and three dimensions (3D)[5-7]. Additionally, with further advancements, such as functional super-resolution[8, 9] and point spread function (PSF) engineering with[8-11] or without multi-channel observations, extra information (spectra, dipole orientation) can be encoded and recovered at the single molecule level. However, such advancements are not fully extended for high-density localizations in 3D. In this work, we adopt sparse recovery using simple matrix/vector operations, and propose a systematic progressive refinement method (dubbed as PRIS) for 3D high-density reconstruction. Our method allows for localization reconstruction using experimental PSFs that include the spatial aberrations and fingerprint patterns of the PSFs. We generalized the method for PSF engineering, multi-channel and multi-species observations using different forms of matrix concatenations. Reconstructions with both double-helix and astigmatic PSFs, for both single and biplane settings are demonstrated, together with the recovery capability for a mixture of two different color species.
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