Interpolating wavelets and their application to incremental signal reconstruction from scattered data. Learn how to build a wavelet tree such that approximation error locally only depends on a local sample density.
Wavelet expansion is an efficient way to represent functions and operators. In such an expansion, a function is decomposed into fluctuations at various resolutions plus an averaged information at the coarsest resolution. Their main advantage over traditional Fourier methods is their hability to handle situations where the target signal contains discontinuities and sharp spikes.
In this project we will take a closer look to interpolating wavelets and their application to incremental signal reconstruction from scattered data. We will build a wavelet tree such that approximation error locally only depends on a local sample density.
For this application we will use dyadic interpolating Deslauriers-Dubuc wavelets up to a certain order p. We can generate ϕkp(x) at resolution 2−k by recursively applying k moving Lagrange interpolation of order p to initial ϕ0p(xi)=δi0 where xi=i∀i∈[[−2p+1,2p+1]].
For each subinterval [xk,xk+1] we generate a new point xk+1/2 by fitting a polynomial Πk(x) of degree p to surrounding points (not centered on interval at boundaries) and by evaluating Πk(xk+1/2).
The resulting scaling functions ϕp(x) are symmetric with support [−2p+1/2,2p−1/2] and have 2p vanishing moments.
By defining the mother wavelet Ψp(x)=ϕp(2x−1) and the wavelet family Ψpjk=Ψp(2jx−k) we can define our wavelet basis at order p by Bp0={Ψpjk|j=0 or (j,k)∈N∗×(2Z+1)}.
As we want to apply approximation to the finite interval Ω=[0,1], we would normally need to construct explicit boundary wavelets as linear combinations of wavelets truncated at the boundary. Here we will follow a simpler approach which is to take boundary filters of lower order.
