Multiscale signal-to-noise thresholding

Abstract

The basic idea behind selective m ultiscale reconstruction of functions from error{a ected data is outlined on the sphere. The selective reconstruction mechanism is based on the premise that multiscale approximation can be well{represented in terms of only a relatively small number of expansion coe cients at various resolution levels. An attempt is made within a tree algorithm (pyramid scheme) to remove the noise component from each scale coe cient using a priori statistical information (provided by an error covariance kernel of a Gaussian, stationary stochastic model).

Wavelet modelling of the spherical inverse source problem with application to geomagnetism

“…Applying the curl to equation ( 10) yields ∇ ∧ b = j + ∇(∇ • a). But the last term vanishes in R 3 because of ∇ • a = 0 which follows directly from (11) by partial integration and by j being of zero divergence. Thus, we get that j is the source field of b, provided that b is given by ( 8).…”

“…This linear technique is circumvented in this paper by an equivalent bilinear two-step method using vector kernel functions and two different types of convolutions. Similar approaches have already been proposed in [4,11,18].…”