High-resolution fluorescence microscopy is an indispensable tool in biological studies. Dueto the diffraction of light, images acquired using conventional fluorescence microscopy techniques exhibit limited resolution. This diffraction limit restricts the level of fine details accessible to the users of a fluorescence microscope, and bypassing this limit would enable the investigation of finer structures, and unlock additional insights for biological studies.

Over the past few decades, an array of super-resolution fluorescence microscopy techniques that bypass the diffraction limit have been proposed, developed, and popularized. These techniques typically exploit the physical properties of the fluorophores used to label the sample, and require the use of specialized illumination setups or photo-switchable fluorophores. These requirements can hinder the adoption of super-resolution microscopy by the larger biological research community,and it is therefore beneficial to attempt to achieve super-resolution while circumventing some or all of these special requirements. It has recently been shown that super-resolution can indeed be achieved by using non-switchable fluorophores and numerical post-processing. Despite being a simpler system, this avenue of achieving super-resolution has received less research effort. While some earlier proof-of-concept experiments have been performed in less practical settings, it is not until much more recently that a proof-of-concept experiment was demonstrated in a biological fluorescence microscopy setting.

Despite recent progress, there are open questions that need to be answered before this computational super-resolution approach can be adopted by a wider audience. In this thesis work, three questions will be addressed: 1) what is the most suitable (i.e., achieves the highest resolution)processing scheme for this super-resolution technique? 2) is the recovered image the only possible solution (i.e., is the solution unique)? and 3) given these promising super-resolution results, what conclusions can be drawn in the analysis of the information transfer ability of a microscope?

For the first question, a careful examination of the overall imaging system is conducted. It is found that the non-negative least squares (NNLS), used so far in the proof-of-concept work, is not optimal in terms of achieving the highest resolution. This is due to the fact that NNLS cannot:1) properly account for the dominant noise model in a typical biological fluorescence microscopy image, and 2) account for the prior information of object sparsity. A properly designed processing scheme based on the correct noise model, and that takes advantage of prior information, is shown to achieve improved super-resolution accuracy. It is first developed using numerical simulation, andthen verified in experiment using a commercially-available calibration sample. A 60nm resolution is shown to be achieved, which is approximately four times beyond the resolution limit. Experimental results, obtained using this imaging technique with up-conversion nanoparticles (UCNPs) as the contrast agents, are also presented and compared with a ground-truth image obtained using transmission electron microscopy (TEM). It is shown that the super-resolved image are a close match to the ground-truth TEM image.

The second question is necessitated by the fact that the inverse problems that need to be solved to recover the super-resolved objects do not seem to have a unique solution, even in noiseless scenarios. This argument can be supported by either: 1) the fact that the microscope rejects some spatial frequencies completely, or 2) the fact that the matrices used in the recovery of super-resolved objects have non-empty null spaces. However, in the existing literature on computational super-resolution, it is generally an accepted conclusion that the solution is unique, at least in the noiseless case. This apparent contradiction hinders the wider adoption of this technique. In this thesis, it is shown that whether the solution is unique or not is not a clear-cut issue. Instead, it is dependent on the sparsity level of the true solution: 1) if the true solution is sparse, the solution is more likely to be unique, 2) if the true solution is somewhat sparse, the solution is non-unique, butthe non-unique solutions do not interfere much with achievable super-resolution, and 3) if the true solution is dense, or non-sparse, the solution is non-unique, and no meaningful super-resolution can be realized. Two methods of determining if the solution is unique are proposed in this thesis, and used to analyze the uniqueness of solution for a range of sparsity conditions.

Finally, the last question stems naturally from the fact that the images being processed are diffraction-limited. Because of this, conventional Fourier optics analysis suggests that no information beyond the diffraction limit can be recovered. This is again in contradiction with a wide array of successful and verifiable computational super-resolution results, presented in existing literature,as well as in this thesis. It is shown in this thesis that while Fourier optics is a good model for image formation in a microscope, it is not a good model for information transfer in a microscope. The main inadequacy of Fourier optics is that it fails to consider the inherent sparsity that is a result of the chemically-specific labeling in fluorescence microscopy. As a result, Fourier optics is not entirely appropriate in analyzing the information transfer in a microscope. It is shown that,if the object exhibits sparsity, information about sample fine details (that are beyond the diffraction limit) is still transmitted via the diffraction-limited images, and that this information can be used by a properly designed processing scheme to reconstruct the super-resolved image. Based on existing research in this topic, a singular value decomposition (SVD) based analysis, which can be shown to be a generalization of Fourier optics analysis, is revisited. By incorporating sparsity in this SVD-based analysis, a more general tool can be obtained that provides a more complete description of the information transfer in a microscope.