This work proposes a novel approach to approximate optimal linear filters for discrete-time linear Gaussian systems with infinite-dimensional measurements and finite- dimensional states. Assuming scalar-valued states for simplicity, we formulate the problem in terms of optimally selecting $N$ points at which to sample the infinite-dimensional measurement, in order to minimize the mean-squared filtering error. We show that for large N, this problem can be expressed using the notion of an asymptotic point density function from the field of high-resolution quantization theory. To the best of the authors' knowledge, this method has not been considered in infinite- dimensional filtering previously. This leads to a characterization in terms of an Urysohn integral equation, which can be solved numerically to yield an asymptotically optimal $N$-point filter. The mean-squared approximation error is proportional to N^-4, which is faster than the typical N^-2 decay of high-resolution quantization and suggests that this approximation method will be useful even for moderate or small N. These properties are verified by simulations based on a linearized pinhole camera measurement model.