The increasing density and smaller length scales in integrated circuits (ICs) create resolution challenges for optical failure analysis techniques. Due to flip-chip bonding and dense metal layers on the front side, optical analysis of ICs is restricted to backside imaging through the silicon substrate, which limits the spatial resolution due to the minimum wavelength of transmission and refraction at the planar interface. The state-of-the-art backside analysis approach is to use aplanatic solid immersion lenses in order to achieve the highest possible numerical aperture of the imaging system. Signal processing algorithms are essential to complement the optical microscopy efforts to increase resolution through hardware modifications in order to meet the resolution requirements of new IC technologies. The focus of this thesis is the development of sparsity-based image reconstruction techniques to improve resolution of static IC images and dynamic optical measurements of device activity. A physics-based observation model is exploited in order to take advantage of polarization diversity in high numerical aperture systems. Multiple-polarization observation data are combined to produce a single enhanced image with higher resolution. In the static IC image case, two sparsity paradigms are considered. The first approach, referred to as analysis-based sparsity, creates enhanced resolution imagery by solving a linear inverse problem while enforcing sparsity through non-quadratic regularization functionals appropriate to IC features. The second approach, termed synthesis-based sparsity, is based on sparse representations with respect to overcomplete dictionaries. The domain of IC imaging is particularly suitable for the application of overcomplete dictionaries because the images are highly structured; they contain predictable building blocks derivable from the corresponding computer-aided design layouts. This structure provides a strong and natural a-priori dictionary for image reconstruction. In the dynamic case, an extension of the synthesis-based sparsity paradigm is formulated. Spatial regions of active areas with the same behavior over time or over frequency are coupled by an overcomplete dictionary consisting of space-time or space-frequency blocks. This extended dictionary enables resolution improvement through sparse representation of dynamic measurements. Additionally, extensions to darkfield subsurface microscopy of ICs and focus determination based on image stacks are provided. The resolution improvement ability of the proposed methods has been validated on both simulated and experimental data.