Alpert Multiwavelets from Gabriel Peyre

from:

Introduction to the Alpert Multiwavelets

The Alpert transform is a multiwavelets transform based on orthogonal polynomials. It was originaly designed for the resolution of partial differential and integral equations, since it avoid boundary artifact and can be used with an arbitrary sampling.

The reference for the numerical algorithm:

• Bradley K. Alpert, Wavelets and Other Bases for Fast Numerical Linear Algebra, in Wavelets: A Tutorial in Theory and Applications, C. K. Chui, editor, Academic Press, New York, 1992.

And more theoretical (continuous setting):

• B.K. Alpert, A class of bases in L^2 for the sparse representatiion of integral operators, in SIAM J. Math. Anal., 24 (1993), 246-262.

The strengh of this transform is that you can transform data sampled irregularly. Of course this algorithm runs in linear time, i.e. O(n). The use of multiwavelets remove any boundary artifact (which are common with wavelet of support > 1, e.g. Daubechies wavelets), but the price to pay is that the wavelets functions are not continue, they look like the Haar basis function. So don't use this transform to compress data that will be seen by human eyes (although the reconstruction error can be very low, the reconstructed function can have some ugly steps-like artifacts).