Highly Nonlinear Approximations for Sparse Signal Representation
Oblique projectors in the context of signal reconstruction were introduced in  and further analyzed in . The application to signal splitting, also termed structured noise filtering, amongst a number of other applications, is discussed in . For advanced theoretical studies of oblique projector operators in infinite dimensional spaces see [5,6]. We restrict our consideration to numerical constructions in finite dimension, with the aim of addressing the problem of signal splitting when the problem is ill posed.
Given and disjoint, i.e., such that in order to provide a prescription for constructing we proceed as follows. Firstly we define as the direct sum of and , which we express as
Considering that is a spanning set for a spanning set for is obtained as
Denoting as the standard orthonormal basis in , i.e., the Euclidean inner product , we define the operators and as
Clearly for the operator to map to zero every vector in the dual vectors must span . This entails that for each there exists a set of coefficients such that
which guarantees that every is orthogonal to all vectors in and therefore is included in the null space of . Moreover, since every signal, say, in can be written as with and , the fact that implies . Hence, , which implies that the null space of restricted to is .
In order for to be a projector it is necessary that . As will be shown in the next proposition, if the coefficients are the matrix elements of a generalised inverse of the matrix this condition is fulfilled.
and using the operators and , as given above, the left hand side in (5) can be expressed as
and the right hand side as
Assuming that is a generalised inverse of indicated as it satisfies (c.f. Section )
and therefore, from (7), the right hand side of (5) follows. Since and , we have . Hence, if the elements determining in (6) are the matrix elements of a generalised inverse of the matrix representation of , the corresponding vectors obtained by (4) yield an operator of the form (3), which is an oblique projector.