A fundamental assumption made in formulating optical-ow algorithms is that motion at any point in an image can be represented as a single pattern component undergoing a simple translation: even complex motion will appear as a uniform displacement when viewed through a su ciently small window.This assumption fails for a number of situations that commonly occur in real world images. For example, transparent surfaces moving past one another yield two motion components at a point. More important, it fails along the boundary between two di erently moving image regions. Even local motion analysis must be performed within a window of nite size. This window c o n tains two motion components when it falls on a motion boundary.We propose an alternative formulation of the local motion assumption in which there may b e t wo distinct patterns undergoing coherent e.g., a ne motion within a given local analysis region. We then present an algorithm for the analysis of two-component motion in which tracking and nulling mechanisms applied to three consecutive image frames separate and estimate the individual components. Precise results are obtained even for components that di er only slightly in velocity, and for a faint component i n the presence of a dominant, masking, component.We demonstrate that the algorithm provides precise motion estimates for a set of elementary two-motion con gurations, and show that it is robust in the presence of noise.
We describe a set of pyramid transforms that decompose an image into a set of basis functions that are (a) spatial-frequency tuned, (b) orientation tuned, (c) spatially localized, and (d) self-similar. For computational reasons the set is also (e) orthogonal and lends itself to (f) rapid computation. The systems are derived from concepts in matrix algebra, but are closely connected to decompositions based on quadrature mirror filters. Our computations take place hierarchically, leading to a pyramid representation in which all of the basis functions have the same basic shape, and appear at many scales. By placing the high-pass and low-pass kernels on staggered grids, we can derive odd-tap QMF kernels that are quite compact. We have developed pyramids using separable, quincunx, and hexagonal kernels. Image data compression with the pyramids gives excellent results, both in terms of MSE and visual appearance. A non-orthogonal variant allows good performance with 3-tap basis kernels and the appropriate inverse sampling kernels.
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