# Highly Nonlinear Approximations for Sparse Signal Representation

## Construction of Oblique Projectors for signal splitting

Oblique projectors in the context of signal reconstruction were introduced in [2] and further analyzed in [3]. The application to signal splitting, also termed structured noise filtering, amongst a number of other applications, is discussed in [4]. 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

Let be the orthogonal complement of in . Thus we have

The operations and are termed direct and orthogonal sum, respectively.

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

Thus the adjoint operators and are

It follows that and hence, defined as:

is self-adjoint and its matrix representation, , has elements

From now on we restrict our signal space to be , since we would like to build the oblique projector onto and along having the form

 (3)

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

 (4)

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.

Proposition 1   If the coefficients in (4) are the matrix elements of a generalised inverse of the matrix , which has elements , the operator in (3) is a projector.

Proof. For the measurement vectors in (4) to yield a projector of the form (3), the corresponding operator should be idempotent, i.e.,

 (5)

Defining

 (6)

and using the operators and , as given above, the left hand side in (5) can be expressed as

 (7)

and the right hand side as

 (8)

Assuming that is a generalised inverse of indicated as it satisfies (c.f. Section )

 (9)

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.

Property 1   Let be the oblique projector onto and along and the orthogonal projector onto . Then the following relation holds

Proof. given in (3) can be recast, in terms of operator and , as:

Applying both sides of the equation we obtain:

which is a well known form for the orthogonal projector onto .

Remark 2   Notice that the operative steps for constructing an oblique projector are equivalent to those for constructing an orthogonal one. The difference being that in general the spaces are different. For the special case , , both sets of vectors span and we have an orthogonal projector onto along .

Subsections