Imaging through turbid environments is experimentally challenging due to multiple light scattering. Structured laser illumination has proven to be effective to minimize errors arising from this phenomenon, allowing the interior of optically dense media to be observed. However, in order to preserve the image spatial resolution while suppressing the intensity contribution from multiple light scattering, the method relies on multiple acquisitions and thus sequential illumination. These requirements significantly limit the usefulness of structured illumination when imaging highly transient events. Here we present a method for achieving snapshot visualizations using structured illumination, where the spatial frequency domain is increased by a factor of two compared to past structured illumination snapshots. Our approach uses two crossed intensity-modulated patterns, allowing us to expand the spatial frequency response of the extracted data. The snapshot capability of this imaging approach allows tracking single particles and opens up for the extraction of velocity vectors by combining it with standard particle tracking/image velocimetry (PTV or PIV) equipment. In this paper we demonstrate the capabilities of this new method and, for the first time, use structured illumination to extract velocity vectors in 2D in a transient turbid medium, in this case an optically dense atomizing spray.