Highly Nonlinear Approximations for Sparse Signal Representation


Oblique and Orthogonal Projector

When $ {\cal{N}}(\hat{E})$ happens to be equal to $ {\cal{R}}(\hat{E})^\bot$, which indicates the orthogonal complement of $ {\cal{R}}(\hat{E})$, the projector is called orthogonal projector onto $ {\cal{R}}(\hat{E})$. This is the case if and only if the projector is self adjoint.

A projector which is not orthogonal is called an oblique projector and we need two subscripts to represent it. One subscript to indicate the range of the projector and another to represent the subspace along which the projection is performed. Hence the projector onto $ {\cal{V}}$ along $ {\cal{W}^\bot}$ is indicated as $ \hat{E}_{{\cal{V}}{\cal{W}^\bot}}$.

The particular case $ \hat{E}_{{\cal{V}}{\cal{V}}^\bot}$ corresponds to an orthogonal projector and we use the special notation $ \hat{P}_{{\cal{V}}}$ to indicate such a projector. When a projector onto $ {\cal{V}}$ is used for signal processing, $ {\cal{W}^\bot}$ can be chosen according to the processing task. The examples below illustrate two different situations.