Highly Nonlinear Approximations for Sparse Signal Representation
An operator is a projector if it is idempotent, i.e.,
such thatThus, if ,
such thatIt is clear then that to reconstruct a signal by means of (1) the involved measurement vectors , that we shall also called henceforth duals, should give rise to an operator of the form (2), which must be a projector onto . Notice that the required operator is not unique, because there exist many projectors onto having different . Thus, for reconstructing signals in the range of the projector its null space can be chosen arbitrarily. However, the null space becomes extremely important when the projector acts on signals outside its range. A popular projector is the orthogonal one.