Non-uniform multiresolution analysis for surfaces and applications

Haar wavelet can exactly represent any piecewise constant function. Beam and Warming proved later that the supercompact wavelets can exactly represent any piecewise polynomial function in one variable, attaining higher level of accuracy by increasing the polynomial order of the supercompact wavelets. The approach of Beam and Warming, which is based on multiwavelets (family of wavelets) constructed in a one dimensional context, uses orthogonal basis defined over sequences of uniform partitions of [0,1]. The work of Beam and Warming has been recently extended by Fortes and Moncayo to the case of surfaces by using orthogonal basis defined over sequences of uniform triangulations of [0,1]2. In that work the authors propose applications to data compression and to discontinuities detection, but both applications have the constraint that it is necessary to know information (at least) at the vertex of the triangulation, and so the data must be uniformly distributed. In the present work we overcome this constraint by considering a multiresolution scheme based on non-uniform triangulations. We develop the multiresolution algorithms and present two examples of the application of the algorithms to compress data and to detect discontinuities of data sets which need not to be uniformly distributed.