Multiresolution analysis and supercompact multiwavelets for surfaces

It is a well-known fact that Haar wavelet can exactly represent any piecewise constant function. Beam and Warming proved later, in 2000, that the supercompact wavelets can exactly represent any piecewise polynomial function in one variable. Higher level of accuracy is attained by higher order polynomials of supercompact wavelets. The initial approach of Beam and Warming, which is based on multiwavelets (family of wavelets) constructed in a one dimensional context, was later extended to the case of multidimensional multiwavelets (3D). The orthogonal basis used by these authors was defined as separable functions given by the product of three Legendre polynomials. In this paper we propose an extension of these previous works to the case of surfaces by using non separable orthogonal functions. Our construction keeps the same advantages attained by the just referenced articles in relation with orthogonality, short support, approximation of surfaces with no border effects, detection of discontinuities, higher degree of accuracy and compressibility, as it is shown in the presented graphical and numerical examples. In this sense, our work may be regarded as a new contribution to supercompact multiwavelets’ theory with great applicability to the approximation of surfaces.