Highly Nonlinear Approximations for Sparse Signal Representation


Application to sparse signal representation

Given a signal, $ f$ say, we address now the issue of determining a partition $ \Delta $, and sub-partitions $ \Delta_j,j=1,\ldots,n$, such that: a) $ \cup_{j=1}^n\Delta_j=\Delta$ and b) the partitions are suitable for generating a sparse representation of the signal in hand. As a first step we propose to tailor the partition to the signal $ f$ by setting $ \Delta $ taking into account the critical points of the curvature function of the signal, i.e.,

$\displaystyle T  :=  \{t: \left(\frac{f''}{(1-f'^2)^{3/2}}\right)'(t)=0\}.

Usually the entries in $ T$ are chosen as the initial knots of $ \Delta $. In order to obtain more knots we apply subdivision between consecutive knots in $ T$ thereby obtaining a partition $ \Delta $ with the decided number of knots. An algorithm for implementing such procedure can be found in [21]. According to Theorem 2, in order to build a dictionary for $ S_m(\Delta )$ we need to choose $ n$-subpartitions $ \Delta_j\in
\Delta$ such that $ \cup_{j=1}^n\Delta_j=\Delta$. As an example we suggest a simple method for producing $ n$-subpartitions $ \Delta_j\in
\Delta$, which is used in the numerical simulations of the next section. Considering the partition $ \Delta =\{x_0,x_1,\ldots,x_{N+1}\}$ such that $ c=x_0<x_1<\cdots
<x_{N+1}=d$, for each integer $ j$ in $ [1,n]$ we set

$\displaystyle \Delta _j  :=   \{c,d\}\cup \{x_{k} : k\in [1,N]$    and $\displaystyle k\!\!\mod n=j-1\},

e.g. if $ N=10$ and $ n=3$, we have $ \Delta _1  =   \{c, x_3,
x_6, x_9, d\},   \Delta _2  =   \{c, x_1, x_4, x_7, x_{10},
d\}$ $    $ $ \Delta _3  =   \{c, x_2, x_5, x_8, d\}. $ The codes for creating a partition adapted to a given signal are ProducePartition.m and FinalProducePartition.m and the one code for creating the dictionaries for the space is CutDic.m.