Highly Nonlinear Approximations for Sparse Signal Representation
is a set such that
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 asso as to satisfyWhen , 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