# Highly Nonlinear Approximations for Sparse Signal Representation

## Building B-spline dictionaries

Let us start by recalling that an extended partition with single inner knots associated with is a set such that

and the first and last points can be arbitrarily chosen. With each fixed extended partition there is associated a unique B-spline basis for , that we denote as . The B-spline can be defined by the recursive formulae [23]:

The following theorem paves the way for the construction of dictionaries for . We use the symbol to indicate the cardinality of a set.

Theorem 2   Let be partitions of and . We denote the B-spline basis for as . Accordingly, a dictionary, , for can be constructed as

so as to satisfy

When , is reduced to the B-spline basis of .

Remark 4   Note that the number of functions in the above defined dictionary is equal to , which is larger than . Hence, excluding the trivial case , the dictionary constitutes a redundant dictionary for .

According to Theorem 2, to build a dictionary for we need to choose -subpartitions such that . This gives a great deal of freedom for the actual construction of a non-uniform B-spline dictionary. Fig.7 shows some examples which are produced by generating a random partition of with 6 interior knots. From an arbitrary partition

we generate two subpartitions as

and join together the B-spline basis for (light lines in the right graphs of Fig.7) and (dark lines in the same graphs)