Optical microscopy is an essential tool for biological research, as it allows for non-invasive imaging of small animals. However, optical microscopy has its limits. Due to the low light level, fluorescence microscopy prohibits high speed imaging, making it difficult to study fast dynamic biological processes. In addition, optical blur due to the diffraction of light results in limited spatial resolution, particularly when using objective lenses with low numerical apertures. In this thesis, we propose computational imaging methods to overcome these limitations using a combination of novel image acquisition procedures and reconstruction algorithms.

The first part of this thesis deals with improving temporal resolution in fluorescence microscopy to image rapid, repeating processes. We take advantage of multiple acquisitions, each taken with different time delays or temporally modulated illumination patterns, to recover high frequency information that is lost with traditional imaging. We demonstrate our method to image the beating heart in live embryonic zebrafish with reduced motion blur and high resolution in time.

The second part of this thesis deals with reducing spatial blur in optical projection tomography, a form of optical microscopy that uses multiple 2D projections to reconstruct a 3D image of an object. We propose a method to reduce the optical distortion (as characterized by the system's optical point spread function) that can be implemented with a scanning acquisition approach combined with a modified filtered backprojection algorithm for reconstruction. We demonstrate our method to image blood vessels in larval zebrafish with high spatial resolution and reduced out-of-focus blur.

The final part of this thesis deals with the dimensional limitation of 2D sensors for measuring 3D motion in microscopy. We propose a method to combine two-dimensional motion estimates from multiple views to recover out-of-plane velocity and reconstruct a divergence-free, three-dimensional velocity field. We demonstrate our method to measure, for the first time, dynamic blood flow in 3D inside the beating heart of a live zebrafish using optical microscopy.

This thesis provides new tools that integrate custom image acquisition procedures and image reconstruction algorithms to overcome the resolution limitations -- temporal, spatial, and out-of-plane velocity resolution -- in optical microscopy. The methods presented in this thesis, in particular the single camera, active illumination method for temporal superresolution in fluorescence microscopy, will be directly applicable to a broad range of biological studies and will open up new perspectives for imaging small organisms in 3D (and time) with high spatio-temporal resolution.

This work presents first-principles theoretical studies on two topics of condensed matter physics. The first topic is the adsorption of metal adatoms on graphene. Graphene, a two-dimensional material made of carbon atoms arranged in a honeycomb lattice, has many outstanding properties that can be enhanced or tailored by adsorbing adatoms on its surface. The second topic involves the coupling of spins or electrons to phonons in a solid. The interaction between different degrees of freedom of a material complicates the study of its properties but also leads to fascinating phenomena, such as superconductivity, and potential device applications. This dissertation is organized into six chapters:

· In Chapter One, we give an overview of this work and review the first-principles theory and methods used in our studies.

· Chapter Two focuses on structural, energetic, and electronic properties for a variety of adatom species adsorbed on the graphene surface. We classify different species as having mostly ionic or covalent character of bonding to graphene. For ionically bonded adatoms, charge transfer between the adatom and graphene is signficant. We find general trends relating the surface dipole moment, work function, and atomic ionization potential of the adatom species.

· In Chapter Three, we study the electronic structure of adatoms on graphene when a gate voltage is applied to control the number of electrons in the system. Lithium on graphene, a prototype system, and cobalt on graphene, an experimentally relevant case, are studied. We find that localized states on the adatom can be charged or discharged by the application of gate voltage, and we study the changes in potential and charge density of the system as electrons are added or removed.

· In Chapter Four, we extend the work in Chapter Three to consider the possibility of transforming the electronic structure of one species of adatom on graphene into that of another by applying a gate voltage. We find that within our model, such transformations are possible for certain adatom species.

· In Chapter Five, the zone-center phonons for the frustrated antiferromagnetic compound ZnCr2O4 are calculated. We find that the transition from nonmagnetic to antiferromagnetic ordering causes a splitting of certain degnerate phonon frequencies, in agreement with experimental results.

· In Chapter Six, the pressure dependence of electron-phonon coupling and the superconducting transition temperature (Tc ) in elemental arsenic is studied. We find that an experimentally observed peak in Tc as a function of pressure is related to a structural transition and can be explained mainly by changes in electronic structure and phonon frequencies with pressure.

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