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![$ S_m(\Delta )$](img509.png)
![$ \tilde{\Delta }=\{y_i\}_{i=1}^{2m+N}$](img510.png)
![$\displaystyle y_{m+i}=x_i, i=1,\ldots,N, x_1<\cdots<x_N$](img511.png)
![$ m$](img512.png)
![$ y_1\leq \cdots \leq y_{m} \leq
c,\quad d \leq y_{m+N+1}\leq \cdots \leq y_{2m+N}$](img513.png)
![$ \tilde{\Delta }$](img438.png)
![$ S_m(\Delta )$](img509.png)
![$ \{B_{m,j}\}_{j=1}^{m+N}$](img514.png)
![$ B_{m,j}$](img515.png)
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The following theorem paves the way for the construction of dictionaries for
![$ S_m(\Delta )$](img509.png)
![$ \char93 $](img520.png)
Theorem 2
Let
be partitions of
and
. We denote the B-spline basis for
as
.
Accordingly, a dictionary,
, for
can be
constructed as
so as to satisfy
When
,
is reduced to the
B-spline basis of
.
![$ \Delta_j, j=1,\ldots,n$](img521.png)
![$ [c,d]$](img495.png)
![$ \Delta=\cup_{j=1}^n \Delta_j$](img522.png)
![$ S_m(\Delta_j)$](img523.png)
![$ \{B_{m,k}^{(j)}:k=1,\ldots,m+\char93 \Delta_j\}$](img524.png)
![$ {\mathcal
D}_m(\Delta: \cup_{j=1}^n\Delta _j)$](img525.png)
![$ S_m(\Delta )$](img509.png)
![$\displaystyle {\mathcal D}_m(\Delta: \cup_{j=1}^n\Delta _j) :=
\cup_{j=1}^n \{B_{m,k}^{(j)}:k=1,\ldots, m+\char93 \Delta_j \},
$](img526.png)
![$\displaystyle {\rm span}\{{\mathcal D}_m(\Delta: \cup_{j=1}^n\Delta _j)
\} = S_m(\Delta).
$](img527.png)
![$ n=1$](img528.png)
![$ {\mathcal D}_m(\Delta:\Delta _1)$](img529.png)
![$ S_m(\Delta )$](img509.png)
Remark 4
Note that the number of functions in the above defined dictionary is
equal to
, which is larger than
. Hence, excluding the trivial case
, the dictionary constitutes a redundant dictionary for
.
According to Theorem 2, to build a dictionary for
![$ n\cdot m+\sum_{j=1}^n\char93 \Delta_j$](img530.png)
![$ \dim S_m(\Delta)=m+\char93 \Delta$](img531.png)
![$ n=1$](img528.png)
![$ S_m(\Delta )$](img509.png)
![$ S_m(\Delta )$](img509.png)
![$ n$](img532.png)
![$ \Delta_j\in
\Delta$](img533.png)
![$ \cup_{j=1}^n\Delta_j=\Delta$](img534.png)
![$ [0,4]$](img535.png)
![$\displaystyle \Delta := \{0=x_0<x_1<\cdots <x_{6}<x_{7}=4\},
$](img536.png)
![$\displaystyle \Delta _1 := \{0=x_0<x_1<x_3<x_{5}<x_{7}=4\}, \quad
\Delta _2 := \{0=x_0<x_2<x_4<x_{6}<x_{7}=4\}$](img537.png)
![$ \Delta _1$](img538.png)
![$ \Delta _2$](img539.png)
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