Compression of Higher Dimensional Functions Containing Smooth Discontinuities
Baraniuk, Richard G.
Discontinuities in data often represent the key information of interest. Efficient representations for such discontinuities are important for many signal processing applications, including compression, but standard Fourier and wavelet representations fail to efficiently capture the structure of the discontinuities. These issues have been most notable in image processing, where progress has been made on modeling and representing one-dimensional edge discontinuities along <i>C²</i> curves. Little work, however, has been done on efficient representations for higher dimensional functions or on handling higher orders of smoothness in discontinuities. In this paper, we consider the class of <i>N</i>-dimensional Horizon functions containing a <i>C<sup>K</sup></i> smooth singularity in N-1 dimensions, which serves as a manifold boundary between two constant regions; we first derive the optimal rate-distortion function for this class. We then introduce the <i>surflet</i> representation for approximation and compression of Horizon-class functions. Surflets enable a multiscale, piecewise polynomial approximation of the discontinuity. We propose a compression algorithm using surflets that achieves the optimal asymptotic rate-distortion performance for this function class. Equally important, the algorithm can be implemented using knowledge of only the <i>N</i>-dimensional function, without explicitly estimating the (<i>N</i>-1)-dimensional discontinuity.