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TW 354
M. Jansen, R. Baraniuk, S. Lavu
Multiscale Approximation of Piecewise Smooth Two-Dimensional Functions using Normal Triangulated Meshes
Abstract
Multiresolution triangulation meshes are widely used in computer graphics for representing three-dimensional (3-d) shapes. We propose to use these tools to represent 2-d piecewise smooth functions such as grayscale images, because triangles have potential to more efficiently approximate the discontinuities between the smooth pieces than other standard tools like wavelets. We show that \emph{normal mesh subdivision} is an efficient triangulation, thanks to its local \emph{adaptivity} to the discontinuities. Indeed, for the so-called \emph{horizon class} of functions comprising constant regions separated by smooth discontinuities, we show that the normal mesh representation has an optimal error decay rate as the number of terms in the representation grows. This optimal decay rate is possible because normal meshes automatically generate a polyline (piecewise linear) approximation of each discontinuity, unlike the blocky piecewise constant approximation of tensor product wavelets. In this way, the nonlinear multiscale normal mesh representation is closely related to the recently developed wedgelet and curvelet transforms.
report.pdf / mailto: M. Jansen