The standard architecture of a medical ultrasound transducer is a linear phased array of piezoelectric elements in a compact, hand-held form. Acoustic energy not directly reflected back towards the transducer elements during a transmit-receive cycle amounts to lost information for image reconstruction. To mitigate this loss, a large, flexible transducer array which conforms to contours of the subject's body would result in a greater effective aperture and an increase in received image data. However, in this reconfigurable array design, element distributions are irregular and an organized arrangement can no longer be assumed. Phased array architecture also has limited scalability potential for large 2D arrays.

This research work investigates a multistatic, stepped-FMCW modality as an alternative to array phasing in order to accommodate the flexible and reconfigurable nature of an array. A space-time reconstruction algorithm was developed for the imaging system. We include ultrasound imaging experiments and describe a simulation method for quickly predicting imaging performance for any given target and array configuration. Lastly, we demonstrate two reconstruction techniques for improving image resolution. The first takes advantage of the statistical significance of pixel contributions prior to the final summation, and the second corrects data errors originating from the stepped-FMCW quadrature receiver.

Optical microscopy offers the unique possibility to study living samples under conditions akin to their native state. However, the technique is not void of inherent problems such as optical blur due to light diffraction, contamination with out-of-focus light from adjacent focal planes, and spherical aberrations. Furthermore, with a dearth of techniques that are capable of imaging multiple focal sections in quick succession, the multi-dimensional capture of dynamically changing samples remains a challenge of its own. Computational techniques that use auxiliary knowledge about the imaging system and the sample to mitigate these problems are hence of great interest in optical microscopy.

The first part of this thesis deals with the design of a discrete model to characterize light propagation. Following the scalar diffraction theory in optics, we propose a discrete algorithm, based on generalized sampling theory, to reverse the coherent diffraction process via back propagation. The algorithm consists of a wavelet-based model for the spherical waves emanating from the object of interest and an optimized multi-rate filtering protocol for reconstruction from the diffraction data recorded by non-ideal detectors.

The second part of this thesis describes a spatial registration tool designed for multi-view microscopy. Here, the imaged sample is rotated about a lateral axis for the acquisition of multiple 3D datasets from different views in order to subsequently alleviate the severe axial blur found in each such dataset. Automatic algorithms that only rely on maximizing pixel-based similarity provide poor results in such applications owing to the anisotropic point-spread-function (PSF) of optical microscopes. We propose a pyramid-based spatial registration algorithm that re-blurs the multi-view datasets with transformed forms of the PSF in order to make them comparable, before maximizing their pixel-based similarity for registration.

The third part of this thesis describes a fast converging iterative multi-view deconvolution technique that can be applied to the spatially registered forms of the 3D datasets acquired using multi-view microscopy. Our sparsity based algorithm solves a non-linear objective function to jointly deconvolve and fuse the multi-view datasets to finally produce a single deblurred 3D result that has nearly isotropic spatial resolution.

The fourth part of this thesis addresses problems due to spherical aberrations encountered during the imaging of thick samples in optical microscopy. The depth-varying nature of the optical blur found in such cases renders fast and efficient shift-invariant deconvolution techniques to be inapplicable. Here, we propose a fast iterative-shrinkage-thresholding shift-variant 3D deconvolution method that uses depth-dependent PSFs to reconstruct a 3D deblurred form of the imaged thick specimen.

The final part of this thesis describes a non-rigid temporal registration tool that aids in the multi-dimensional imaging of quasi-periodic processes such as cardiac cycles. We propose a variant of dynamic time warping that is capable of both temporally warping and wrapping an input sequence by allowing for jump discontinuities in the non-linear temporal alignment function akin to those found in wrapped phase functions.

This work provides a new set of tools for spatio-temporal reconstruction in optical microscopy and we anticipate them to be useful for a wide range of problems in practice.

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.