Highly Nonlinear Approximations for Sparse Signal Representation


Example 2

Assume that the signal $ f$ to be analyzed here is the superposition of two signals, $ f=f_1+f_2$, each component being produced by a different phenomenon we want to discriminate. Let us assume further that $ f_1 \in {\cal{V}}$ and $ f_2 \in {\cal{W}^\bot}$ with $ {\cal{V}}$ and $ {\cal{W}^\bot}$ disjoint subspaces. Thus, we can obtain, $ f_1$ say, from $ f$, by an oblique projector onto $ {\cal{V}}$ and along $ {\cal{W}^\bot}$. The projector will map to zero the component $ f_2$ to produce

$\displaystyle f_1= \hat{E}_{{\cal{V}}{\cal{W}^\bot}} f.$