Estimating the finite-dimensional state of dynamic systems using modern sensors such as cameras, lidar, and radar involves processing increasingly high-dimensional observations. In this paper, we exploit concepts from the theory of infinite-dimensional systems to examine state estimation in the continuum limit of infinite-dimensional observations. Specifically, we investigate state estimation in discrete-time linear systems with finite-dimensional states and infinite-dimensional observations corrupted by additive noise. In contrast to previous derivations of the Kalman filter for infinite-dimensional observations, we are able to derive an explicit solution for the optimal Kalman gain by modeling the infinite-dimensional observation noise as a stationary Gaussian Process. We demonstrate the utility of our Kalman filter in a simulation of a linearized system derived from the pinhole camera model.