Nonlinear scale-space filtering and multiresolution system

We derive and demonstrate a nonlinear scale-space filter and its application in generating a nonlinear multiresolution system. For each datum in a signal, a neighborhood of weighted data is used for clustering. The cluster center becomes the filter output. The filter is governed by a single scale parameter that dictates the spatial extent of nearby data used for clustering. This, together with the local characteristic of the signal, determines the scale parameter in the output space, which dictates the influences of these data on the output. This filter is thus adaptive and data driven. It provides a mechanism for (a) removing impulsive noise, (b) improved smoothing of nonimpulsive noise, and (c) preserving edges. Comparisons with Gaussian scale-space filtering and median filters are made using real images. Using the architecture of the Laplacian pyramid and this nonlinear filter for interpolation, we construct a nonlinear multiresolution system that has two features: (1) edges are well preserved at low resolutions, and (2) difference signals are small and spatially localized. This filter implicitly presents a new mechanism for detecting discontinuities differing from techniques based on local gradients and line processes. This work shows that scale-space filtering, nonlinear filtering, and scale-space clustering are closely related and provides a framework within which further image processing, image coding, and computer vision problems can be investigated