KDE Paring and a Faster Mean Shift Algorithm

The Kernel Density Estimate (KDE) is a non-parametric density estimate which has broad application in computer vision and pattern recognition. In particular, the Mean Shift procedure uses the KDE structure in order to cluster or segment data, including images and video. The usefulness of these twin techniques – KDEs and Mean Shift – on large datasets is hampered by the large space or description complexity of the KDE, which in turn leads to a large time complexity of the Mean Shift procedure, that is superlinear in the number of points. In this paper, we propose a sampling technique for KDE paring, i.e. the construction of a compactly represented KDE with much smaller description complexity. We prove that this technique has good properties, in that the pared-down KDE so constructed is close to the original KDE in a precise mathematical sense. We then show how to use this pared-down KDE to devise a considerably faster Mean Shift algorithm, whose time complexity we analyze formally. Experiments show that image and video segmentation results of the proposed fast Mean Shift method are similar to those based on the standard Mean shift procedure, with the typical speed-up several orders of magnitude for large data sets. Finally, we present an application of the Fast Mean Shift method to the efficient construction of multiscale graph structures for images, which can be used as a pre-processing step for more sophisticated segmentation algorithms.