Compression of Higher Dimensional Functions Containing Smooth Discontinuities

摘要

Discontinuities in data often represent the keyrninformation of interest. Efficient representations for suchrndiscontinuities are important for many signal processingrnapplications, including compression, but standard Fourierrnand wavelet representations fail to efficiently capture thernstructure of the discontinuities. These issues have beenrnmost notable in image processing, where progress hasrnbeen made on modeling and representing one-dimensionalrnedge discontinuities along C~2 curves. Little work, however,rnhas been done on efficient representations for higherrndimensional functions or on handling higher orders ofrnsmoothness in discontinuities. In this paper, we considerrnthe class of N-dimensional Horizon functions containing arnC~K smooth singularity in N-1 dimensions, which servesrnas a manifold boundary between two constant regions; wernfirst derive the optimal rate-distortion function for thisrnclass. We then introduce the surflet representation for approximationrnand compression of Horizon-class functions.rnSurflets enable a multiscale, piecewise polynomial approximationrnof the discontinuity. We propose a compressionrnalgorithm using surflets that achieves the optimal asymptoticrnrate-distortion performance for this function class.rnEqually important, the algorithm can be implemented usingrnknowledge of only the N-dimensional function, withoutrnexplicitly estimating the (N-1)-dimensional discontinuity.