Particle image velocimetry (PIV) has become one of the most important tools for experimentally investigating the physics of fluid flows. In PIV, image-processing algorithms estimate flow velocity by measuring the displacements of flow-tracer particles suspended in a fluid. The fundamental operation in PIV is the cross correlation (CC), which measures the displacement between two similar patterns. These measurements can fail under circumstances that arise due to the nature of the underlying flow field (e.g., vortices and boundary layers, where particle patterns not only translate but also rotate, stretch, and shear) or of the images (e.g., X-ray images, with comparatively low signal to noise ratios). Despite these shortcomings, fairly little attention has been paid to fundamentally improving measurements at the level of the CC. The objective of this dissertation is to demonstrate specific modifications to the correlation kernel of PIV that increase its accuracy and in certain cases extend its utility to classes of flows and image types that were previously unresolvable. First, we present a new PIV correlation algorithm called the Fourier-Mellin correlation (FMC) that reduces velocity errors by an order of magnitude in rotating flows (chapter 1). Second, we develop a model of PIV cross correlations that explains the fundamental sources of several major drivers of error in these measurements. We show how the shapes of the tracer particles and the distributions of their individual displacements affect the correlation signal to noise ratio (SNR), whose effects have previously been described only heuristically. We use this insight to create an algorithm that automatically creates a Fourier-based weighting filter, and demonstrate that our algorithm reduces bias and RMS errors in multiple types of PIV experiments (chapter 2). Finally, we apply principles from our insights to measure blood flows in the hearts of grasshoppers using X-ray PIV, and discovered flow kinematics that were unexpected according to the current prevailing understanding of the heart as a peristaltic pump that produces directional flows. Our results suggest that flow production in insect hearts may be more complex than once thought (chapter 3).