  Fast methods for convolution and correlation underlie a variety of
applications in computer vision and graphics, including efficient filtering,
analysis, and simulation. However, standard convolution and correlation are
inherently limited to fixed filters: spatial adaptation is impossible without
sacrificing efficient computation. In early work, Freeman and Adelson have
shown how steerable filters can address this limitation, providing a way for
rotating the filter as it is passed over the signal. In this work, we provide a
general, representation-theoretic, framework that allows for spatially varying
linear transformations to be applied to the filter. This framework allows for
efficient implementation of extended convolution and correlation for
transformation groups such as rotation (in 2D and 3D) and scale, and provides a
new interpretation for previous methods including steerable filters and the
generalized Hough transform. We present applications to pattern matching, image
feature description, vector field visualization, and adaptive image filtering.
