SPARCOM: Sparsity Based Super-resolution Correlation Microscopy

Abstract

In traditional optical imaging systems, the spatial resolution is limited by the physics of diffraction, which acts as a low-pass filter. The information on sub-wavelength features is carried by evanescent waves, never reaching the camera, thereby posing a hard limit on resolution: the so-called diffraction limit. Modern microscopic methods enable super-resolution, by employing florescence techniques. State-of-the-art localization based fluorescence subwavelength imaging techniques such as PALM and STORM achieve sub-diffraction spatial resolution of several tens of nano-meters. However, they require tens of thousands of exposures, which limits their temporal resolution. We have recently proposed SPARCOM (sparsity based super-resolution correlation microscopy), which exploits the sparse nature of the fluorophores distribution, alongside a statistical prior of uncorrelated emissions, and showed that SPARCOM achieves spatial resolution comparable to PALM/STORM, while capturing the data hundreds of times faster. Here, we provide a detailed mathematical formulation of SPARCOM, which in turn leads to an efficient numerical implementation, suitable for large-scale problems. We further extend our method to a general framework for sparsity based super-resolution imaging, in which sparsity can be assumed in other domains such as wavelet or discrete-cosine, leading to improved reconstructions in a variety of physical settings.

Learned SPARCOM: Unfolded Deep Super-Resolution Microscopy

“…The relation between the low-resolution M × M frame f (t) acquired at time t and the location of the emitters on the high resolution grid X can be formulated considering the blinking and diffraction phenomena as follows [6,7]:…”

“…In SPARCOM [6,7], Solomon et al assume that emissions by different emitters are uncorrelated over time and space, providing further prior information to exploit for solving the compressed sensing MMV problem given in (3).…”

“…To circumvent the long acquisition periods required for SMLM methods, a variety of techniques have emerged, which enable the use of a smaller number of frames for reconstructing the 2-D super-resolved image [3][4][5][6][7][8][9]. These techniques take advantage of prior information regarding either the optical setup, the geometry of the sample, or the statistics of the emitters.…”

“…In practice, however, the use of statistical orders higher than two is limited due to signal-to-noise ratio (SNR), dynamic range expansion, and temporal resolution considerations, leaving the spatial resolution practically offered by SOFI significantly lower than PALM and STORM. Solomon et al have recently suggested combining the ideas of sparse recovery and SOFI, leading to a sparsity-based approach for superresolution microscopy from correlation information of high emitter-density frames, dubbed SPARCOM [6,7]. SPARCOM utilizes sparsity in the correlation domain, while assuming that the blinking emitters are uncorrelated over time and space.…”

“…In this paper, we utilize for the first time the deep algorithm unfolding concept to transform the correlation-domain sparsity-based approach suggested by Solomon et al [6,7] into a simple deep learning, parameter-free framework, dubbed LSPARCOM (learned SPARCOM), which we train on a single field of view (FOV). Our method is robust, generalizes well, is interpretable, and requires only a small number of layers, without relying on explicit knowledge of the optical setup or requiring fine-tuning of optimization parameters.…”