Highly Nonlinear Approximations for Sparse Signal Representation
Building Bspline dictionaries
Let us start by recalling that an extended partition with single inner knots associated with 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 Bspline basis for
as
.
Accordingly, a dictionary,
, for
can be
constructed as
so as to satisfy
When ,
is reduced to the
Bspline 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 nonuniform Bspline
dictionary. Fig.7 shows some examples which are produced by
generating a random partition of with 6 interior knots.
From an arbitrary partition
