There has been significant recent interest in fast imaging with sparse sampling. Conventional imaging methods are based on Shannon-Nyquist sampling theory. As such, the number of required samples often increases exponentially with the dimensionality of the image, which limits achievable resolution in high-dimensional scenarios. The partially-separable function (PSF) model has previously been proposed to enable sparse data sampling in this context. Existing methods to leverage PSF structure utilize tailored data samplingstrategies, which enable a specialized two-step reconstruction procedure. This work formulates the PSF reconstruction problem using the matrix-recovery framework. The explicit matrix formulation provides new opportunities for data acquisition and image reconstruction with rank constraints. Theoretical results from the emerging field of low-rank matrix recovery (which generalizes theory from sparse-vector recovery) and our empirical results illustrate the potential of this new approach.