Although the edge response of nonlinear filters has been the topic of much research, the behavior of many filters at 2D structures such as sharp corners is not as well understood. We introduce a new technique that defines the response of a 2D filter at corners of any angle. The response measure may be expressed either as fractional preservation or as an attenuation in decibels. Plotting the response measure in polar form illustrates the corner response of a filter both intuitively and quantitatively. The corner response may be computed for a particular size and shape of filter window on a discrete lattice, or may be determined in continuous space for a filter with a specific window shape. The continuous space response measure gives the response of the filter to corners in general, and usually corresponds closely with that determined for the filter on a discrete lattice. The corner responses of several widely used filters (median, averaging, and morphological) are compared.