Convolution Pyramids

We present a novel approach for rapid numerical approximation of convolutions with filters of large support. Our approach consists of a multiscale scheme, fashioned after the wavelet transform, which computes the approximation in linear time. Given a specific large target filter to approximate, we first use numerical optimization to design a set of small kernels, which are then used to perform the analysis and synthesis steps of our multiscale transform. Once the optimization has been done, the resulting transform can be applied to any signal in linear time. We demonstrate that our method is well suited for tasks such as gradient field integration, seamless image cloning, and scattered data interpolation, outperforming existing state-of-the-art methods.