Application to sparse signal representation

Given a signal,

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

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

are chosen as the initial knots of $\Delta$ . In order to obtain more knots we apply subdivision between consecutive knots in

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

-subpartitions $\Delta_j\in \Delta$ such that $\cup_{j=1}^n\Delta_j=\Delta$ . As an example we suggest a simple method for producing

-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

we set

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

e.g. if

and

, 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.

Subsections

Highly Nonlinear Approximations for Sparse Signal Representation

Application to sparse signal representation