In the mathematical theory of wavelets , a spline wavelet is a wavelet constructed using a spline function .[ 1] There are different types of spline wavelets. The interpolatory spline wavelets introduced by C.K. Chui and J.Z. Wang are based on a certain spline interpolation formula.[ 2] Though these wavelets are orthogonal , they do not have compact supports . There is a certain class of wavelets, unique in some sense, constructed using B-splines and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular.[ 3] The terminology spline wavelet is sometimes used to refer to the wavelets in this class of spline wavelets. These special wavelets are also called B-spline wavelets and cardinal B-spline wavelets .[ 4] The Battle-Lemarie wavelets are also wavelets constructed using spline functions.[ 5]
Animation showing the compactly supported cardinal B-spline wavelets of orders 1, 2, 3, 4 and 5.
Let n be a fixed non-negative integer . Let C n denote the set of all real-valued functions defined over the set of real numbers such that each function in the set as well its first n derivatives are continuous everywhere. A bi-infinite sequence . . . x −2 , x −1 , x 0 , x 1 , x 2 , . . . such that x r < x r +1 for all r and such that x r approaches ±∞ as r approaches ±∞ is said to define a set of knots. A spline of order n with a set of knots {x r } is a function S (x ) in C n such that, for each r , the restriction of S (x ) to the interval [x r , x r +1 ) coincides with a polynomial with real coefficients of degree at most n in x .
If the separation x r +1 - x r , where r is any integer, between the successive knots in the set of knots is a constant, the spline is called a cardinal spline . The set of integers Z = {. . ., -2, -1, 0, 1, 2, . . .} is a standard choice for the set of knots of a cardinal spline. Unless otherwise specified, it is generally assumed that the set of knots is the set of integers.
A cardinal B-spline is a special type of cardinal spline. For any positive integer m the cardinal B-spline of order m , denoted by N m (x ), is defined recursively as follows.
N
1
(
x
)
=
{
1
0
≤
x
<
1
0
otherwise
{\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}
N
m
(
x
)
=
∫
0
1
N
m
−
1
(
x
−
t
)
d
t
{\displaystyle N_{m}(x)=\int _{0}^{1}N_{m-1}(x-t)dt}
, for
m
>
1
{\displaystyle m>1}
.
Concrete expressions for the cardinal B-splines of all orders up to 5 and their graphs are given later in this article.
Properties of the cardinal B-splines
edit
Elementary properties
edit
The support of
N
m
(
x
)
{\displaystyle N_{m}(x)}
is the closed interval
[
0
,
m
]
{\displaystyle [0,m]}
.
The function
N
m
(
x
)
{\displaystyle N_{m}(x)}
is non-negative, that is,
N
m
(
x
)
>
0
{\displaystyle N_{m}(x)>0}
for
0
<
x
<
m
{\displaystyle 0<x<m}
.
∑
k
=
−
∞
∞
N
m
(
x
−
k
)
=
1
{\displaystyle \sum _{k=-\infty }^{\infty }N_{m}(x-k)=1}
for all
x
{\displaystyle x}
.
The cardinal B-splines of orders m and m-1 are related by the identity:
N
m
(
x
)
=
x
m
−
1
N
m
−
1
(
x
)
+
m
−
x
m
−
1
N
m
−
1
(
x
−
1
)
{\displaystyle N_{m}(x)={\frac {x}{m-1}}N_{m-1}(x)+{\frac {m-x}{m-1}}N_{m-1}(x-1)}
.
The function
N
m
(
x
)
{\displaystyle N_{m}(x)}
is symmetrical about
x
=
m
2
{\displaystyle x={\frac {m}{2}}}
, that is,
N
m
(
m
2
−
x
)
=
N
m
(
m
2
+
x
)
{\displaystyle N_{m}\left({\frac {m}{2}}-x\right)=N_{m}\left({\frac {m}{2}}+x\right)}
.
The derivative of
N
m
(
x
)
{\displaystyle N_{m}(x)}
is given by
N
m
′
(
x
)
=
N
m
−
1
(
x
)
−
N
m
−
1
(
x
−
1
)
{\displaystyle N_{m}^{\prime }(x)=N_{m-1}(x)-N_{m-1}(x-1)}
.
∫
−
∞
∞
N
m
(
x
)
d
x
=
1
{\displaystyle \int _{-\infty }^{\infty }N_{m}(x)\,dx=1}
The cardinal B-spline of order m satisfies the following two-scale relation:
N
m
(
x
)
=
∑
k
=
0
m
2
−
m
+
1
(
m
k
)
N
m
(
2
x
−
k
)
{\displaystyle N_{m}(x)=\sum _{k=0}^{m}2^{-m+1}{m \choose k}N_{m}(2x-k)}
.
The cardinal B-spline of order m satisfies the following property, known as the Riesz property: There exists two positive real numbers
A
{\displaystyle A}
and
B
{\displaystyle B}
such that for any square summable two-sided sequence
{
c
k
}
k
=
−
∞
∞
{\displaystyle \{c_{k}\}_{k=-\infty }^{\infty }}
and for any x ,
A
‖
{
c
k
}
‖
2
≤
‖
∑
k
=
−
∞
∞
c
k
N
m
(
x
−
k
)
‖
2
≤
B
‖
{
c
k
}
‖
2
{\displaystyle A\left\Vert \{c_{k}\}\right\Vert ^{2}\leq \left\Vert \sum _{k=-\infty }^{\infty }c_{k}N_{m}(x-k)\right\Vert ^{2}\leq B\left\Vert \{c_{k}\}\right\Vert ^{2}}
where
‖
⋅
‖
{\displaystyle \Vert \cdot \Vert }
is the norm in the ℓ2 -space.
Cardinal B-splines of small orders
edit
The cardinal B-splines are defined recursively starting from the B-spline of order 1, namely
N
1
(
x
)
{\displaystyle N_{1}(x)}
, which takes the value 1 in the interval [0, 1) and 0 elsewhere. Computer algebra systems may have to be employed to obtain concrete expressions for higher order cardinal B-splines. The concrete expressions for cardinal B-splines of all orders up to 6 are given below. The graphs of cardinal B-splines of orders up to 4 are also exhibited. In the images, the graphs of the terms contributing to the corresponding two-scale relations are also shown. The two dots in each image indicate the extremities of the interval supporting the B-spline.
The B-spline of order 1, namely
N
1
(
x
)
{\displaystyle N_{1}(x)}
, is the constant B-spline. It is defined by
N
1
(
x
)
=
{
1
0
≤
x
<
1
0
otherwise
{\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}
The two-scale relation for this B-spline is
N
1
(
x
)
=
N
1
(
2
x
)
+
N
1
(
2
x
−
1
)
{\displaystyle N_{1}(x)=N_{1}(2x)+N_{1}(2x-1)}
Constant B-spline
N
1
(
x
)
{\displaystyle N_{1}(x)}
The B-spline of order 2, namely
N
2
(
x
)
{\displaystyle N_{2}(x)}
, is the linear B-spline. It is given by
N
2
(
x
)
=
{
x
0
≤
x
<
1
−
x
+
2
1
≤
x
<
2
0
otherwise
{\displaystyle N_{2}(x)={\begin{cases}x&0\leq x<1\\-x+2&1\leq x<2\\0&{\text{otherwise}}\end{cases}}}
The two-scale relation for this wavelet is
N
2
(
x
)
=
1
2
N
2
(
2
x
)
+
N
2
(
2
x
−
1
)
+
1
2
N
2
(
2
x
−
2
)
{\displaystyle N_{2}(x)={\frac {1}{2}}N_{2}(2x)+N_{2}(2x-1)+{\frac {1}{2}}N_{2}(2x-2)}
Linear B-spline
N
2
(
x
)
{\displaystyle N_{2}(x)}
The B-spline of order 3, namely
N
3
(
x
)
{\displaystyle N_{3}(x)}
, is the quadratic B-spline. It is given by
N
3
(
x
)
=
{
1
2
x
2
0
≤
x
<
1
−
x
2
+
3
x
−
3
2
1
≤
x
<
2
1
2
x
2
−
3
x
+
9
2
2
≤
x
<
3
0
otherwise
{\displaystyle N_{3}(x)={\begin{cases}{\frac {1}{2}}x^{2}&0\leq x<1\\-x^{2}+3x-{\frac {3}{2}}&1\leq x<2\\{\frac {1}{2}}x^{2}-3x+{\frac {9}{2}}&2\leq x<3\\0&{\text{otherwise}}\end{cases}}}
The two-scale relation for this wavelet is
N
3
(
x
)
=
1
4
N
3
(
2
x
)
+
3
4
N
3
(
2
x
−
1
)
+
3
4
N
3
(
2
x
−
2
)
+
1
4
N
3
(
2
x
−
3
)
{\displaystyle N_{3}(x)={\frac {1}{4}}N_{3}(2x)+{\frac {3}{4}}N_{3}(2x-1)+{\frac {3}{4}}N_{3}(2x-2)+{\frac {1}{4}}N_{3}(2x-3)}
Quadratic B-spline
N
3
(
x
)
{\displaystyle N_{3}(x)}
The cubic B-spline is the cardinal B-spline of order 4, denoted by
N
4
(
x
)
{\displaystyle N_{4}(x)}
. It is given by the following expressions:
N
4
(
x
)
=
{
1
6
x
3
0
≤
x
<
1
−
1
2
x
3
+
2
x
2
−
2
x
+
2
3
1
≤
x
<
2
1
2
x
3
−
4
x
2
+
10
x
−
22
3
2
≤
x
<
3
−
1
6
x
3
+
2
x
2
−
8
x
+
32
3
3
≤
x
<
4
0
otherwise
{\displaystyle N_{4}(x)={\begin{cases}{\frac {1}{6}}x^{3}&0\leq x<1\\-{\frac {1}{2}}x^{3}+2x^{2}-2x+{\frac {2}{3}}&1\leq x<2\\{\frac {1}{2}}x^{3}-4x^{2}+10x-{\frac {22}{3}}&2\leq x<3\\-{\frac {1}{6}}x^{3}+2x^{2}-8x+{\frac {32}{3}}&3\leq x<4\\0&{\text{otherwise}}\end{cases}}}
The two-scale relation for the cubic B-spline is
N
4
(
x
)
=
1
8
N
4
(
2
x
)
+
1
2
N
4
(
2
x
−
1
)
+
3
4
N
4
(
2
x
−
2
)
+
1
2
N
4
(
2
x
−
3
)
+
1
8
N
4
(
2
x
−
4
)
{\displaystyle N_{4}(x)={\frac {1}{8}}N_{4}(2x)+{\frac {1}{2}}N_{4}(2x-1)+{\frac {3}{4}}N_{4}(2x-2)+{\frac {1}{2}}N_{4}(2x-3)+{\frac {1}{8}}N_{4}(2x-4)}
Cubic B-spline
N
4
(
x
)
{\displaystyle N_{4}(x)}
Bi-quadratic B-spline
edit
The bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by
N
5
(
x
)
{\displaystyle N_{5}(x)}
. It is given by
N
5
(
x
)
=
{
1
24
x
4
0
≤
x
<
1
−
1
6
x
4
+
5
6
x
3
−
5
4
x
2
+
5
6
x
−
5
24
1
≤
x
<
2
1
4
x
4
−
5
2
x
3
+
35
4
x
2
−
25
2
x
+
155
24
2
≤
x
<
3
−
1
6
x
4
+
5
2
x
3
−
55
4
x
2
+
65
2
x
−
655
24
3
≤
x
<
4
1
24
x
4
−
5
6
x
3
+
25
4
x
2
−
125
6
x
+
625
24
4
≤
x
<
5
0
otherwise
{\displaystyle N_{5}(x)={\begin{cases}{\frac {1}{24}}x^{4}&0\leq x<1\\-{\frac {1}{6}}x^{4}+{\frac {5}{6}}x^{3}-{\frac {5}{4}}x^{2}+{\frac {5}{6}}x-{\frac {5}{24}}&1\leq x<2\\{\frac {1}{4}}x^{4}-{\frac {5}{2}}x^{3}+{\frac {35}{4}}x^{2}-{\frac {25}{2}}x+{\frac {155}{24}}&2\leq x<3\\-{\frac {1}{6}}x^{4}+{\frac {5}{2}}x^{3}-{\frac {55}{4}}x^{2}+{\frac {65}{2}}x-{\frac {655}{24}}&3\leq x<4\\{\frac {1}{24}}x^{4}-{\frac {5}{6}}x^{3}+{\frac {25}{4}}x^{2}-{\frac {125}{6}}x+{\frac {625}{24}}&4\leq x<5\\0&{\text{otherwise}}\end{cases}}}
The two-scale relation is
N
5
(
x
)
=
1
16
N
5
(
2
x
)
+
5
16
N
5
(
2
x
−
1
)
+
10
16
N
5
(
2
x
−
2
)
+
10
16
N
5
(
2
x
−
3
)
+
5
16
N
5
(
2
x
−
4
)
+
1
16
N
5
(
2
x
−
5
)
{\displaystyle N_{5}(x)={\frac {1}{16}}N_{5}(2x)+{\frac {5}{16}}N_{5}(2x-1)+{\frac {10}{16}}N_{5}(2x-2)+{\frac {10}{16}}N_{5}(2x-3)+{\frac {5}{16}}N_{5}(2x-4)+{\frac {1}{16}}N_{5}(2x-5)}
The quintic B-spline is the cardinal B-spline of order 6 denoted by
N
6
(
x
)
{\displaystyle N_{6}(x)}
. It is given by
N
6
(
x
)
=
{
1
120
x
5
0
≤
x
<
1
−
1
24
x
5
+
1
4
x
4
−
1
2
x
3
+
1
2
x
2
−
1
4
x
+
1
20
1
≤
x
<
2
1
12
x
5
−
x
4
+
9
2
x
3
−
19
2
x
2
+
39
4
x
−
79
20
2
≤
x
<
3
−
1
12
x
5
+
3
2
x
4
−
21
2
x
3
+
71
2
x
2
−
231
4
x
+
731
20
3
≤
x
<
4
1
24
x
5
−
x
4
+
19
2
x
3
−
89
2
x
2
+
409
4
x
−
1829
20
4
≤
x
<
5
−
1
120
x
5
+
1
4
x
4
−
3
x
3
+
18
x
2
−
54
x
+
324
5
5
≤
x
<
6
0
otherwise
{\displaystyle N_{6}(x)={\begin{cases}{\frac {1}{120}}x^{5}&0\leq x<1\\-{\frac {1}{24}}x^{5}+{\frac {1}{4}}x^{4}-{\frac {1}{2}}x^{3}+{\frac {1}{2}}x^{2}-{\frac {1}{4}}x+{\frac {1}{20}}&1\leq x<2\\{\frac {1}{12}}x^{5}-x^{4}+{\frac {9}{2}}x^{3}-{\frac {19}{2}}x^{2}+{\frac {39}{4}}x-{\frac {79}{20}}&2\leq x<3\\-{\frac {1}{12}}x^{5}+{\frac {3}{2}}x^{4}-{\frac {21}{2}}x^{3}+{\frac {71}{2}}x^{2}-{\frac {231}{4}}x+{\frac {731}{20}}&3\leq x<4\\{\frac {1}{24}}x^{5}-x^{4}+{\frac {19}{2}}x^{3}-{\frac {89}{2}}x^{2}+{\frac {409}{4}}x-{\frac {1829}{20}}&4\leq x<5\\-{\frac {1}{120}}x^{5}+{\frac {1}{4}}x^{4}-3x^{3}+18x^{2}-54x+{\frac {324}{5}}&5\leq x<6\\0&{\text{otherwise}}\end{cases}}}
Multi-resolution analysis generated by cardinal B-splines
edit
The cardinal B-spline
N
m
(
x
)
{\displaystyle N_{m}(x)}
of order m generates a multi-resolution analysis . In fact, from the elementary properties of these functions enunciated above, it follows that the function
N
m
(
x
)
{\displaystyle N_{m}(x)}
is square integrable and is an element of the space
L
2
(
R
)
{\displaystyle L^{2}(R)}
of square integrable functions. To set up the multi-resolution analysis the following notations used.
For any integers
k
,
j
{\displaystyle k,j}
, define the function
N
m
,
k
j
(
x
)
=
N
m
(
2
k
x
−
j
)
{\displaystyle N_{m,kj}(x)=N_{m}(2^{k}x-j)}
.
For each integer
k
{\displaystyle k}
, define the subspace
V
k
{\displaystyle V_{k}}
of
L
2
(
R
)
{\displaystyle L^{2}(R)}
as the closure of the linear span of the set
{
N
m
,
k
j
(
x
)
:
j
=
⋯
,
−
2
,
−
1
,
0
,
1
,
2
,
⋯
}
{\displaystyle \{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}}
.
That these define a multi-resolution analysis follows from the following:
The spaces
V
k
{\displaystyle V_{k}}
satisfy the property:
⋯
⊂
V
−
2
⊂
V
−
1
⊂
V
0
⊂
V
1
⊂
V
2
⊂
⋯
{\displaystyle \cdots \subset V_{-2}\subset V_{-1}\subset V_{0}\subset V_{1}\subset V_{2}\subset \cdots }
.
The closure in
L
2
(
R
)
{\displaystyle L^{2}(R)}
of the union of all the subspaces
V
k
{\displaystyle V_{k}}
is the whole space
L
2
(
R
)
{\displaystyle L^{2}(R)}
.
The intersection of all the subspaces
V
k
{\displaystyle V_{k}}
is the singleton set containing only the zero function.
For each integer
k
{\displaystyle k}
the set
{
N
m
,
k
j
(
x
)
:
j
=
⋯
,
−
2
,
−
1
,
0
,
1
,
2
,
⋯
}
{\displaystyle \{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}}
is an unconditional basis for
V
k
{\displaystyle V_{k}}
. (A sequence {x n } in a Banach space X is an unconditional basis for the space X if every permutation of the sequence {x n } is also a basis for the same space X .[ 6] )
Wavelets from cardinal B-splines
edit
Let m be a fixed positive integer and
N
m
(
x
)
{\displaystyle N_{m}(x)}
be the cardinal B-spline of order m . A function
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
in
L
2
(
R
)
{\displaystyle L^{2}(R)}
is a basic wavelet relative to the cardinal B-spline function
N
m
(
x
)
{\displaystyle N_{m}(x)}
if the closure in
L
2
(
R
)
{\displaystyle L^{2}(R)}
of the linear span of the set
{
ψ
m
(
x
−
j
)
:
j
=
⋯
,
−
2
,
−
1
,
0
,
1
,
2
,
⋯
}
{\displaystyle \{\psi _{m}(x-j):j=\cdots ,-2,-1,0,1,2,\cdots \}}
(this closure is denoted by
W
0
{\displaystyle W_{0}}
) is the orthogonal complement of
V
0
{\displaystyle V_{0}}
in
V
1
{\displaystyle V_{1}}
. The subscript m in
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
is used to indicate that
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
is a basic wavelet relative the cardinal B-spline of order m . There is no unique basic wavelet
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
relative to the cardinal B-spline
N
m
(
x
)
{\displaystyle N_{m}(x)}
. Some of these are discussed in the following sections.
Wavelets relative to cardinal B-splines using fundamental interpolatory splines
edit
Fundamental interpolatory spline
edit
Let m be a fixed positive integer and let
N
m
(
x
)
{\displaystyle N_{m}(x)}
be the cardinal B-spline of order m . Given a sequence
{
f
j
:
j
=
⋯
,
−
2
,
−
1
,
0
,
1
,
2
,
⋯
}
{\displaystyle \{f_{j}:j=\cdots ,-2,-1,0,1,2,\cdots \}}
of real numbers, the problem of finding a sequence
{
c
m
,
k
:
k
=
⋯
,
−
2
,
−
1
,
0
,
1
,
2
,
⋯
}
{\displaystyle \{c_{m,k}:k=\cdots ,-2,-1,0,1,2,\cdots \}}
of real numbers such that
∑
k
=
−
∞
∞
c
m
,
k
N
m
(
j
+
m
2
−
k
)
=
f
j
{\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=f_{j}}
for all
j
{\displaystyle j}
,
is known as the cardinal spline interpolation problem . The special case of this problem where the sequence
{
f
j
}
{\displaystyle \{f_{j}\}}
is the sequence
δ
0
j
{\displaystyle \delta _{0j}}
, where
δ
i
j
{\displaystyle \delta _{ij}}
is the Kronecker delta function
δ
i
j
{\displaystyle \delta _{ij}}
defined by
δ
i
j
=
{
1
,
if
i
=
j
0
,
if
i
≠
j
{\displaystyle \delta _{ij}={\begin{cases}1,&{\text{ if }}i=j\\0,&{\text{ if }}i\neq j\end{cases}}}
,
is the fundamental cardinal spline interpolation problem . The solution of the problem yields the fundamental cardinal interpolatory spline of order m . This spline is denoted by
L
m
(
x
)
{\displaystyle L_{m}(x)}
and is given by
L
m
(
x
)
=
∑
k
=
−
∞
∞
c
m
,
k
N
m
(
x
+
m
2
−
k
)
{\displaystyle L_{m}(x)=\sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(x+{\frac {m}{2}}-k\right)}
where the sequence
{
c
m
,
k
}
{\displaystyle \{c_{m,k}\}}
is now the solution of the following system of equations:
∑
k
=
−
∞
∞
c
m
,
k
N
m
(
j
+
m
2
−
k
)
=
δ
0
j
{\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=\delta _{0j}}
Procedure to find the fundamental cardinal interpolatory spline
edit
The fundamental cardinal interpolatory spline
L
m
(
x
)
{\displaystyle L_{m}(x)}
can be determined using Z-transforms . Using the following notations
A
(
z
)
=
∑
k
=
−
∞
∞
δ
k
0
z
k
=
1
,
{\displaystyle A(z)=\sum _{k=-\infty }^{\infty }\delta _{k0}z^{k}=1,}
B
m
(
z
)
=
∑
k
=
−
∞
∞
N
m
(
k
+
m
2
)
z
k
,
{\displaystyle B_{m}(z)=\sum _{k=-\infty }^{\infty }N_{m}\left(k+{\frac {m}{2}}\right)z^{k},}
C
m
(
z
)
=
∑
k
=
−
∞
∞
c
m
,
k
z
k
,
{\displaystyle C_{m}(z)=\sum _{k=-\infty }^{\infty }c_{m,k}z^{k},}
it can be seen from the equations defining the sequence
c
m
,
k
{\displaystyle c_{m,k}}
that
B
m
(
z
)
C
m
(
z
)
=
A
(
z
)
{\displaystyle B_{m}(z)C_{m}(z)=A(z)}
from which we get
C
m
(
z
)
=
1
B
m
(
z
)
{\displaystyle C_{m}(z)={\frac {1}{B_{m}(z)}}}
.
This can be used to obtain concrete expressions for
c
m
,
k
{\displaystyle c_{m,k}}
.
As a concrete example, the case
L
4
(
x
)
{\displaystyle L_{4}(x)}
may be investigated. The definition of
B
m
(
z
)
{\displaystyle B_{m}(z)}
implies that
B
4
(
x
)
=
∑
k
=
−
∞
∞
N
4
(
2
+
k
)
z
k
{\displaystyle B_{4}(x)=\sum _{k=-\infty }^{\infty }N_{4}(2+k)z^{k}}
The only nonzero values of
N
4
(
k
+
2
)
{\displaystyle N_{4}(k+2)}
are given by
k
=
−
1
,
0
,
1
{\displaystyle k=-1,0,1}
and the corresponding values are
N
4
(
1
)
=
1
6
,
N
4
(
2
)
=
4
6
,
N
4
(
3
)
=
1
6
.
{\displaystyle N_{4}(1)={\frac {1}{6}},N_{4}(2)={\frac {4}{6}},N_{4}(3)={\frac {1}{6}}.}
Thus
B
4
(
z
)
{\displaystyle B_{4}(z)}
reduces to
B
4
(
z
)
=
1
6
z
−
1
+
4
6
z
0
+
1
6
z
1
=
1
+
4
z
+
z
2
6
z
{\displaystyle B_{4}(z)={\frac {1}{6}}z^{-1}+{\frac {4}{6}}z^{0}+{\frac {1}{6}}z^{1}={\frac {1+4z+z^{2}}{6z}}}
This yields the following expression for
C
4
(
z
)
{\displaystyle C_{4}(z)}
.
C
4
(
z
)
=
6
z
1
+
4
z
+
z
2
{\displaystyle C_{4}(z)={\frac {6z}{1+4z+z^{2}}}}
Splitting this expression into partial fractions and expanding each term in powers of z in an annular region the values of
c
4
,
k
{\displaystyle c_{4,k}}
can be computed. These values are then substituted in the expression for
L
4
(
x
)
{\displaystyle L_{4}(x)}
to yield
L
4
(
x
)
=
∑
k
=
−
∞
∞
(
−
1
)
k
3
(
2
−
3
)
|
k
|
N
4
(
x
+
2
−
k
)
{\displaystyle L_{4}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(x+2-k)}
Wavelet using fundamental interpolatory spline
edit
For a positive integer m , the function
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
defined by
ψ
I
,
m
(
x
)
=
d
m
d
x
m
L
2
m
(
2
x
−
1
)
{\displaystyle \psi _{I,m}(x)={\frac {d^{m}}{dx^{m}}}L_{2m}(2x-1)}
is a basic wavelet relative to the cardinal B-spline of order
N
m
(
x
)
{\displaystyle N_{m}(x)}
. The subscript I in
ψ
I
,
m
{\displaystyle \psi _{I,m}}
is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported.
The wavelet of order 2 using interpolatory spline is given by
ψ
I
,
2
(
x
)
=
d
2
d
x
2
L
4
(
2
x
−
1
)
{\displaystyle \psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}L_{4}(2x-1)}
The expression for
L
4
(
x
)
{\displaystyle L_{4}(x)}
now yields the following formula:
ψ
I
,
2
(
x
)
=
d
2
d
x
2
∑
k
=
−
∞
∞
(
−
1
)
k
3
(
2
−
3
)
|
k
|
N
4
(
2
x
+
1
−
k
)
{\displaystyle \psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(2x+1-k)}
Now, using the expression for the derivative of
N
m
(
x
)
{\displaystyle N_{m}(x)}
in terms of
N
m
−
1
(
x
)
{\displaystyle N_{m-1}(x)}
the function
ψ
2
(
x
)
{\displaystyle \psi _{2}(x)}
can be put in the following form:
ψ
I
,
2
(
x
)
=
∑
k
=
−
∞
∞
(
−
1
)
k
4
3
(
2
−
3
)
|
k
|
(
(
N
2
(
2
x
+
k
−
1
)
−
2
N
2
(
2
x
+
k
−
2
)
+
N
2
(
2
x
+
k
−
3
)
)
{\displaystyle \psi _{I,2}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}4{\sqrt {3}}(2-{\sqrt {3}})^{|k|}{\Big (}(N_{2}(2x+k-1)-2N_{2}(2x+k-2)+N_{2}(2x+k-3){\Big )}}
The following piecewise linear function is the approximation to
ψ
2
(
x
)
{\displaystyle \psi _{2}(x)}
obtained by taking the sum of the terms corresponding to
k
=
−
3
,
…
,
3
{\displaystyle k=-3,\ldots ,3}
in the infinite series expression for
ψ
2
(
x
)
{\displaystyle \psi _{2}(x)}
.
ψ
I
,
2
(
x
)
≈
{
0.07142668
x
+
0.17856670
−
2.5
<
x
≤
−
2
−
0.48084803
x
−
0.92598272
−
2
<
x
≤
−
1.5
2.0088293
x
+
2.8085333
−
1.5
<
x
≤
−
1
−
7.5684795
x
−
6.7687755
−
1
<
x
≤
−
0.5
28.245949
x
+
11.138439
−
0.5
<
x
≤
0
−
57.415316
x
+
11.138439
0
<
x
≤
0.5
57.415316
x
−
46.276878
0.5
<
x
≤
1
−
28.245949
x
+
39.384388
1
<
x
≤
1.5
7.5684795
x
−
14.337255
1.5
<
x
≤
2
−
2.0088293
x
+
4.8173625
2
<
x
≤
2.5
0.48084803
x
−
1.4068308
2.5
<
x
≤
3
−
0.07142668
x
+
0.24999338
3
<
x
≤
3.5
0
o
t
h
e
r
w
i
s
e
{\displaystyle \psi _{I,2}(x)\approx {\begin{cases}0.07142668x+0.17856670&-2.5<x\leq -2\\-0.48084803x-0.92598272&-2<x\leq -1.5\\2.0088293x+2.8085333&-1.5<x\leq -1\\-7.5684795x-6.7687755&-1<x\leq -0.5\\28.245949x+11.138439&-0.5<x\leq 0\\-57.415316x+11.138439&0<x\leq 0.5\\57.415316x-46.276878&0.5<x\leq 1\\-28.245949x+39.384388&1<x\leq 1.5\\7.5684795x-14.337255&1.5<x\leq 2\\-2.0088293x+4.8173625&2<x\leq 2.5\\0.48084803x-1.4068308&2.5<x\leq 3\\-0.07142668x+0.24999338&3<x\leq 3.5\\0&{otherwise}\end{cases}}}
The two-scale relation for the wavelet function
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
is given by
ψ
I
,
m
(
x
)
=
∑
−
∞
∞
q
n
N
m
(
2
x
−
n
)
{\displaystyle \psi _{I,m}(x)=\sum _{-\infty }^{\infty }q_{n}N_{m}(2x-n)}
where
q
n
=
∑
j
=
0
m
(
−
1
)
j
(
m
j
)
c
m
+
n
−
j
−
1
.
{\displaystyle q_{n}=\sum _{j=0}^{m}(-1)^{j}{m \choose j}c_{m+n-j-1}.}
Compactly supported B-spline wavelets
edit
The spline wavelets generated using the interpolatory wavelets are not compactly supported. Compactly supported B-spline wavelets were discovered by Charles K. Chui and Jian-zhong Wang and published in 1991.[ 3] [ 7] The compactly supported B-spline wavelet relative to the cardinal B-spline
N
m
(
x
)
{\displaystyle N_{m}(x)}
of order m discovered by Chui and Wong and denoted by
ψ
C
,
m
(
x
)
{\displaystyle \psi _{C,m}(x)}
, has as its support the interval
[
0
,
2
m
−
1
]
{\displaystyle [0,2m-1]}
. These wavelets are essentially unique in a certain sense explained below.
The compactly supported B-spline wavelet of order m is given by
ψ
C
,
m
(
x
)
=
1
2
m
−
1
∑
j
=
0
2
m
−
2
(
−
1
)
j
N
2
m
(
j
+
1
)
d
m
d
x
m
N
2
m
(
2
x
−
j
)
{\displaystyle \psi _{C,m}(x)={\frac {1}{2^{m-1}}}\sum _{j=0}^{2m-2}(-1)^{j}N_{2m}(j+1){\frac {d^{m}}{dx^{m}}}N_{2m}(2x-j)}
This is an m -th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is
ψ
C
,
1
(
x
)
=
N
2
(
1
)
d
d
x
N
2
(
2
x
)
=
{
1
0
≤
x
<
1
2
−
1
1
2
≤
x
<
1
0
otherwise
{\displaystyle \psi _{C,1}(x)=N_{2}(1){\frac {d}{dx}}N_{2}(2x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}
which is the well-known Haar wavelet .
The support of
ψ
C
,
m
(
x
)
{\displaystyle \psi _{C,m}(x)}
is the closed interval
[
0
,
2
m
−
1
]
{\displaystyle [0,2m-1]}
.
The wavelet
ψ
C
,
m
(
x
)
{\displaystyle \psi _{C,m}(x)}
is the unique wavelet with minimum support in the following sense: If
η
(
x
)
∈
W
0
{\displaystyle \eta (x)\in W_{0}}
generates
W
0
{\displaystyle W_{0}}
and has support not exceeding
2
m
−
1
{\displaystyle 2m-1}
in length then
η
(
x
)
=
c
0
ψ
C
,
m
(
x
−
n
0
)
{\displaystyle \eta (x)=c_{0}\psi _{C,m}(x-n_{0})}
for some nonzero constant
c
0
{\displaystyle c_{0}}
and for some integer
n
0
{\displaystyle n_{0}}
.[ 8]
ψ
C
,
m
(
x
)
{\displaystyle \psi _{C,m}(x)}
is symmetric for even m and antisymmetric for odd m .
ψ
m
(
x
)
{\displaystyle \psi _{m}(x)}
satisfies the two-scale relation:
ψ
C
,
m
(
x
)
=
∑
n
=
0
3
m
−
2
q
n
N
m
(
2
x
−
n
)
{\displaystyle \psi _{C,m}(x)=\sum _{n=0}^{3m-2}q_{n}N_{m}(2x-n)}
where
q
n
=
(
−
1
)
n
2
m
−
1
∑
j
=
0
m
(
m
j
)
N
2
m
(
n
−
j
+
1
)
{\displaystyle q_{n}={\frac {(-1)^{n}}{2^{m-1}}}\sum _{j=0}^{m}{m \choose j}N_{2m}(n-j+1)}
.
Decomposition relation
edit
The decomposition relation for the compactly supported B-spline wavelet has the following form:
N
m
(
2
x
−
l
)
=
∑
k
=
−
∞
∞
[
a
m
,
l
−
2
k
N
m
(
x
−
k
)
+
b
m
,
l
−
2
k
ψ
C
,
m
(
x
−
k
)
]
{\displaystyle N_{m}(2x-l)=\sum _{k=-\infty }^{\infty }\left[a_{m,l-2k}N_{m}(x-k)+b_{m,l-2k}\psi _{C,m}(x-k)\right]}
where the coefficients
a
m
,
j
{\displaystyle a_{m,j}}
and
b
m
,
j
{\displaystyle b_{m,j}}
are given by
a
m
,
j
=
−
(
−
1
)
j
2
∑
l
=
−
∞
∞
q
−
j
+
2
m
−
2
l
+
1
c
2
m
,
l
,
{\displaystyle a_{m,j}=-{\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }q_{-j+2m-2l+1}c_{2m,l},}
b
m
,
j
=
(
−
1
)
j
2
∑
l
=
−
∞
∞
p
−
j
+
2
m
−
2
l
+
1
c
2
m
,
l
.
{\displaystyle b_{m,j}={\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }p_{-j+2m-2l+1}c_{2m,l}.}
Here the sequence
c
2
m
,
l
{\displaystyle c_{2m,l}}
is the sequence of coefficients in the fundamental interpolatoty cardinal spline wavelet of order m .
Compactly supported B-spline wavelets of small orders
edit
Compactly supported B-spline wavelet of order 1
edit
The two-scale relation for the compactly supported B-spline wavelet of order 1 is
ψ
C
,
1
(
x
)
=
N
1
(
2
x
)
−
N
1
(
2
x
−
1
)
{\displaystyle \psi _{C,1}(x)=N_{1}(2x)-N_{1}(2x-1)}
The closed form expression for compactly supported B-spline wavelet of order 1 is
ψ
C
,
1
(
x
)
=
{
1
0
≤
x
<
1
2
−
1
1
2
≤
x
<
1
0
otherwise
{\displaystyle \psi _{C,1}(x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}
Compactly supported B-spline wavelet of order 2
edit
The two-scale relation for the compactly supported B-spline wavelet of order 2 is
ψ
C
,
2
(
x
)
=
1
12
(
N
2
(
2
x
)
−
6
N
2
(
2
x
−
1
)
+
10
N
2
(
2
x
−
2
)
−
6
N
2
(
2
x
−
3
)
+
N
2
(
2
x
−
4
)
)
{\displaystyle \psi _{C,2}(x)={\frac {1}{12}}\left(N_{2}(2x)-6N_{2}(2x-1)+10N_{2}(2x-2)-6N_{2}(2x-3)+N_{2}(2x-4)\right)}
The closed form expression for compactly supported B-spline wavelet of order 2 is
ψ
C
,
2
(
x
)
=
{
1
6
x
0
≤
x
<
1
2
−
7
6
x
+
2
3
1
2
≤
x
<
1
8
3
x
−
19
6
1
≤
x
<
3
2
−
8
3
x
+
29
6
3
2
≤
x
<
2
7
6
x
−
17
6
2
≤
x
<
5
2
−
1
6
x
+
1
2
5
2
≤
x
<
3
0
otherwise
{\displaystyle \psi _{C,2}(x)={\begin{cases}{\frac {1}{6}}x&0\leq x<{\frac {1}{2}}\\-{\frac {7}{6}}x+{\frac {2}{3}}&{\frac {1}{2}}\leq x<1\\{\frac {8}{3}}x-{\frac {19}{6}}&1\leq x<{\frac {3}{2}}\\-{\frac {8}{3}}x+{\frac {29}{6}}&{\frac {3}{2}}\leq x<2\\{\frac {7}{6}}x-{\frac {17}{6}}&2\leq x<{\frac {5}{2}}\\-{\frac {1}{6}}x+{\frac {1}{2}}&{\frac {5}{2}}\leq x<3\\0&{\text{otherwise}}\end{cases}}}
Compactly supported B-spline wavelet of order 3
edit
The two-scale relation for the compactly supported B-spline wavelet of order 3 is
ψ
C
,
3
(
x
)
=
1
480
[
(
N
3
(
2
x
)
−
29
N
3
(
2
x
−
1
)
+
147
N
3
(
2
x
−
2
)
−
303
N
3
(
2
x
−
3
)
+
{\displaystyle \psi _{C,3}(x)={\frac {1}{480}}{\Big [}(N_{3}(2x)-29N_{3}(2x-1)+147N_{3}(2x-2)-303N_{3}(2x-3)+}
303
N
3
(
2
x
−
4
)
−
147
N
3
(
2
x
−
5
)
+
29
N
3
(
2
x
−
6
)
−
N
3
(
2
x
−
7
)
]
{\displaystyle 303N_{3}(2x-4)-147N_{3}(2x-5)+29N_{3}(2x-6)-N_{3}(2x-7){\Big ]}}
The closed form expression for compactly supported B-spline wavelet of order 3 is
ψ
C
,
3
(
x
)
=
{
1
240
x
2
0
≤
x
<
1
2
−
31
240
x
2
+
2
15
x
−
1
30
1
2
≤
x
<
1
103
120
x
2
−
221
120
x
+
229
240
1
≤
x
<
3
2
−
313
120
x
2
+
1027
120
x
−
1643
240
3
2
≤
x
<
2
22
5
x
2
−
779
40
x
+
339
16
2
≤
x
<
5
2
−
22
5
x
2
+
981
40
x
−
541
16
5
2
≤
x
<
3
313
120
x
2
−
701
40
x
+
2341
80
3
≤
x
<
7
2
−
103
120
x
2
+
809
120
x
−
3169
240
7
2
≤
x
<
4
31
240
x
2
−
139
120
x
+
623
240
4
≤
x
<
9
2
−
1
240
x
2
+
1
24
x
−
5
48
9
2
≤
x
<
5
0
otherwise
{\displaystyle \psi _{C,3}(x)={\begin{cases}{\frac {1}{240}}x^{2}&0\leq x<{\frac {1}{2}}\\-{\frac {31}{240}}x^{2}+{\frac {2}{15}}x-{\frac {1}{30}}&{\frac {1}{2}}\leq x<1\\{\frac {103}{120}}x^{2}-{\frac {221}{120}}x+{\frac {229}{240}}&1\leq x<{\frac {3}{2}}\\-{\frac {313}{120}}x^{2}+{\frac {1027}{120}}x-{\frac {1643}{240}}&{\frac {3}{2}}\leq x<2\\{\frac {22}{5}}x^{2}-{\frac {779}{40}}x+{\frac {339}{16}}&2\leq x<{\frac {5}{2}}\\-{\frac {22}{5}}x^{2}+{\frac {981}{40}}x-{\frac {541}{16}}&{\frac {5}{2}}\leq x<3\\{\frac {313}{120}}x^{2}-{\frac {701}{40}}x+{\frac {2341}{80}}&3\leq x<{\frac {7}{2}}\\-{\frac {103}{120}}x^{2}+{\frac {809}{120}}x-{\frac {3169}{240}}&{\frac {7}{2}}\leq x<4\\{\frac {31}{240}}x^{2}-{\frac {139}{120}}x+{\frac {623}{240}}&4\leq x<{\frac {9}{2}}\\-{\frac {1}{240}}x^{2}+{\frac {1}{24}}x-{\frac {5}{48}}&{\frac {9}{2}}\leq x<5\\0&{\text{otherwise}}\end{cases}}}
Compactly supported B-spline wavelet of order 4
edit
The two-scale relation for the compactly supported B-spline wavelet of order 4 is
ψ
C
,
4
(
x
)
=
1
40320
[
N
4
(
2
x
)
−
124
N
4
(
2
x
−
1
)
+
1677
N
4
(
2
x
−
2
)
−
7904
N
4
(
2
x
−
3
)
+
18482
N
4
(
2
x
−
4
)
−
{\displaystyle \psi _{C,4}(x)={\frac {1}{40320}}{\Big [}N_{4}(2x)-124N_{4}(2x-1)+1677N_{4}(2x-2)-7904N_{4}(2x-3)+18482N_{4}(2x-4)-}
24264
N
4
(
2
x
−
5
)
+
18482
N
4
(
2
x
−
6
)
−
7904
N
4
(
2
x
−
7
)
+
1677
N
4
(
2
x
−
8
)
−
124
N
4
(
2
x
−
9
)
+
N
4
(
2
x
−
10
)
]
{\displaystyle 24264N_{4}(2x-5)+18482N_{4}(2x-6)-7904N_{4}(2x-7)+1677N_{4}(2x-8)-124N_{4}(2x-9)+N_{4}(2x-10){\Big ]}}
The closed form expression for compactly supported B-spline wavelet of order 4 is
ψ
C
,
4
(
x
)
=
{
1
30240
x
3
0
≤
x
<
1
2
−
127
30240
x
3
+
2
315
x
2
−
1
315
x
+
1
1890
1
2
≤
x
<
1
19
280
x
3
−
47
224
x
2
+
2147
10080
x
−
103
1440
1
≤
x
<
3
2
−
1109
2520
x
3
+
465
224
x
2
−
32413
10080
x
+
16559
10080
3
2
≤
x
<
2
5261
3360
x
3
−
33463
3360
x
2
+
42043
2016
x
−
145193
10080
2
≤
x
<
5
2
−
35033
10080
x
3
+
93577
3360
x
2
−
148517
2016
x
+
216269
3360
5
2
≤
x
<
3
4832
945
x
3
−
27691
560
x
2
+
113923
720
x
−
28145
168
3
≤
x
<
7
2
−
4832
945
x
3
+
58393
1008
x
2
−
52223
240
x
+
2048227
7560
7
2
≤
x
<
4
35033
10080
x
3
−
75827
1680
x
2
+
981101
5040
x
−
234149
840
4
≤
x
<
9
2
−
5261
3360
x
3
+
38509
1680
x
2
−
112487
1008
x
+
30347
168
9
2
≤
x
<
5
1109
2520
x
3
−
24077
3360
x
2
+
78311
2016
x
−
141311
2016
5
≤
x
<
11
2
−
19
280
x
3
+
1361
1120
x
2
−
14617
2016
x
+
4151
288
11
2
≤
x
<
6
127
30240
x
3
−
55
672
x
2
+
5359
10080
x
−
11603
10080
6
≤
x
<
13
2
−
1
30240
x
3
+
1
1440
x
2
−
7
1440
x
+
49
4320
13
2
≤
x
<
7
0
otherwise
{\displaystyle \psi _{C,4}(x)={\begin{cases}{\frac {1}{30240}}x^{3}&0\leq x<{\frac {1}{2}}\\-{\frac {127}{30240}}x^{3}+{\frac {2}{315}}x^{2}-{\frac {1}{315}}x+{\frac {1}{1890}}&{\frac {1}{2}}\leq x<1\\{\frac {19}{280}}x^{3}-{\frac {47}{224}}x^{2}+{\frac {2147}{10080}}x-{\frac {103}{1440}}&1\leq x<{\frac {3}{2}}\\-{\frac {1109}{2520}}x^{3}+{\frac {465}{224}}x^{2}-{\frac {32413}{10080}}x+{\frac {16559}{10080}}&{\frac {3}{2}}\leq x<2\\{\frac {5261}{3360}}x^{3}-{\frac {33463}{3360}}x^{2}+{\frac {42043}{2016}}x-{\frac {145193}{10080}}&2\leq x<{\frac {5}{2}}\\-{\frac {35033}{10080}}x^{3}+{\frac {93577}{3360}}x^{2}-{\frac {148517}{2016}}x+{\frac {216269}{3360}}&{\frac {5}{2}}\leq x<3\\{\frac {4832}{945}}x^{3}-{\frac {27691}{560}}x^{2}+{\frac {113923}{720}}x-{\frac {28145}{168}}&3\leq x<{\frac {7}{2}}\\-{\frac {4832}{945}}x^{3}+{\frac {58393}{1008}}x^{2}-{\frac {52223}{240}}x+{\frac {2048227}{7560}}&{\frac {7}{2}}\leq x<4\\{\frac {35033}{10080}}x^{3}-{\frac {75827}{1680}}x^{2}+{\frac {981101}{5040}}x-{\frac {234149}{840}}&4\leq x<{\frac {9}{2}}\\-{\frac {5261}{3360}}x^{3}+{\frac {38509}{1680}}x^{2}-{\frac {112487}{1008}}x+{\frac {30347}{168}}&{\frac {9}{2}}\leq x<5\\{\frac {1109}{2520}}x^{3}-{\frac {24077}{3360}}x^{2}+{\frac {78311}{2016}}x-{\frac {141311}{2016}}&5\leq x<{\frac {11}{2}}\\-{\frac {19}{280}}x^{3}+{\frac {1361}{1120}}x^{2}-{\frac {14617}{2016}}x+{\frac {4151}{288}}&{\frac {11}{2}}\leq x<6\\{\frac {127}{30240}}x^{3}-{\frac {55}{672}}x^{2}+{\frac {5359}{10080}}x-{\frac {11603}{10080}}&6\leq x<{\frac {13}{2}}\\-{\frac {1}{30240}}x^{3}+{\frac {1}{1440}}x^{2}-{\frac {7}{1440}}x+{\frac {49}{4320}}&{\frac {13}{2}}\leq x<7\\0&{\text{otherwise}}\end{cases}}}
Compactly supported B-spline wavelet of order 5
edit
The two-scale relation for the compactly supported B-spline wavelet of order 5 is
ψ
C
,
5
(
x
)
=
1
5806080
[
N
5
(
2
x
)
−
507
N
5
(
2
x
−
1
)
+
17128
N
5
(
2
x
−
2
)
−
166304
N
5
(
2
x
−
3
)
+
748465
N
5
(
2
x
−
4
)
{\displaystyle \psi _{C,5}(x)={\frac {1}{5806080}}{\Big [}N_{5}(2x)-507N_{5}(2x-1)+17128N_{5}(2x-2)-166304N_{5}(2x-3)+748465N_{5}(2x-4)}
−
1900115
N
5
(
2
x
−
5
)
+
2973560
N
5
(
2
x
−
6
)
−
2973560
N
5
(
2
x
−
7
)
+
1900115
N
5
(
2
x
−
8
)
{\displaystyle -1900115N_{5}(2x-5)+2973560N_{5}(2x-6)-2973560N_{5}(2x-7)+1900115N_{5}(2x-8)}
−
748465
N
5
(
2
x
−
9
)
+
166304
N
5
(
2
x
−
10
)
−
17128
N
5
(
2
x
−
11
)
+
507
N
5
(
2
x
−
12
)
−
N
5
(
2
x
−
13
)
]
{\displaystyle -748465N_{5}(2x-9)+166304N_{5}(2x-10)-17128N_{5}(2x-11)+507N_{5}(2x-12)-N_{5}(2x-13){\Big ]}}
The closed form expression for compactly supported B-spline wavelet of order 5 is
ψ
C
,
5
(
x
)
=
{
1
8709120
x
4
0
≤
x
<
1
2
−
73
1244160
x
4
+
1
8505
x
3
−
1
11340
x
2
+
1
34020
x
−
1
272160
1
2
≤
x
<
1
9581
4354560
x
4
−
19417
2177280
x
3
+
1303
96768
x
2
−
19609
2177280
x
+
6547
2903040
1
≤
x
<
3
2
−
118931
4354560
x
4
+
366119
2177280
x
3
−
186253
483840
x
2
+
121121
311040
x
−
427181
2903040
3
2
≤
x
<
2
759239
4354560
x
4
−
3146561
2177280
x
3
+
6466601
1451520
x
2
−
13202873
2177280
x
+
26819897
8709120
2
≤
x
<
5
2
−
2980409
4354560
x
4
+
5183893
725760
x
3
−
13426333
483840
x
2
+
426589
8960
x
−
12635243
414720
5
2
≤
x
<
3
7873577
4354560
x
4
−
16524079
725760
x
3
+
7385369
69120
x
2
−
17868671
80640
x
+
497668543
2903040
3
≤
x
<
7
2
−
14714327
4354560
x
4
+
108543091
2177280
x
3
−
56901557
207360
x
2
+
1454458651
2177280
x
−
5286189059
8709120
7
2
≤
x
<
4
15619
3402
x
4
−
33822017
435456
x
3
+
15828929
32256
x
2
−
597598433
435456
x
+
277413649
193536
4
≤
x
<
9
2
−
15619
3402
x
4
+
38150335
435456
x
3
−
20157247
32256
x
2
+
859841695
435456
x
−
64472345
27648
9
2
≤
x
<
5
14714327
4354560
x
4
−
4466137
62208
x
3
+
165651247
290304
x
2
−
875490655
435456
x
+
4614904015
1741824
5
≤
x
<
11
2
−
7873577
4354560
x
4
+
30717383
725760
x
3
−
179437319
483840
x
2
+
16606729
11520
x
−
869722273
414720
11
2
≤
x
<
6
2980409
4354560
x
4
−
12698561
725760
x
3
+
16211669
96768
x
2
−
19138891
26880
x
+
3289787993
2903040
6
≤
x
<
13
2
−
759239
4354560
x
4
+
10519741
2177280
x
3
−
10403603
207360
x
2
+
71964499
311040
x
−
3481646837
8709120
13
2
≤
x
<
7
118931
4354560
x
4
−
1774639
2177280
x
3
+
630259
69120
x
2
−
14096161
311040
x
+
245108501
2903040
7
≤
x
<
15
2
−
9581
4354560
x
4
+
21863
311040
x
3
−
407387
483840
x
2
+
9758873
2177280
x
−
25971499
2903040
15
2
≤
x
<
8
73
1244160
x
4
−
4343
2177280
x
3
+
5273
207360
x
2
−
313703
2177280
x
+
380873
1244160
8
≤
x
<
17
2
−
1
8709120
x
4
+
1
241920
x
3
−
1
17920
x
2
+
3
8960
x
−
27
35840
17
2
≤
x
<
9
0
otherwise
{\displaystyle \psi _{C,5}(x)={\begin{cases}{\frac {1}{8709120}}x^{4}&0\leq x<{\frac {1}{2}}\\-{\frac {73}{1244160}}x^{4}+{\frac {1}{8505}}x^{3}-{\frac {1}{11340}}x^{2}+{\frac {1}{34020}}x-{\frac {1}{272160}}&{\frac {1}{2}}\leq x<1\\{\frac {9581}{4354560}}x^{4}-{\frac {19417}{2177280}}x^{3}+{\frac {1303}{96768}}x^{2}-{\frac {19609}{2177280}}x+{\frac {6547}{2903040}}&1\leq x<{\frac {3}{2}}\\-{\frac {118931}{4354560}}x^{4}+{\frac {366119}{2177280}}x^{3}-{\frac {186253}{483840}}x^{2}+{\frac {121121}{311040}}x-{\frac {427181}{2903040}}&{\frac {3}{2}}\leq x<2\\{\frac {759239}{4354560}}x^{4}-{\frac {3146561}{2177280}}x^{3}+{\frac {6466601}{1451520}}x^{2}-{\frac {13202873}{2177280}}x+{\frac {26819897}{8709120}}&2\leq x<{\frac {5}{2}}\\-{\frac {2980409}{4354560}}x^{4}+{\frac {5183893}{725760}}x^{3}-{\frac {13426333}{483840}}x^{2}+{\frac {426589}{8960}}x-{\frac {12635243}{414720}}&{\frac {5}{2}}\leq x<3\\{\frac {7873577}{4354560}}x^{4}-{\frac {16524079}{725760}}x^{3}+{\frac {7385369}{69120}}x^{2}-{\frac {17868671}{80640}}x+{\frac {497668543}{2903040}}&3\leq x<{\frac {7}{2}}\\-{\frac {14714327}{4354560}}x^{4}+{\frac {108543091}{2177280}}x^{3}-{\frac {56901557}{207360}}x^{2}+{\frac {1454458651}{2177280}}x-{\frac {5286189059}{8709120}}&{\frac {7}{2}}\leq x<4\\{\frac {15619}{3402}}x^{4}-{\frac {33822017}{435456}}x^{3}+{\frac {15828929}{32256}}x^{2}-{\frac {597598433}{435456}}x+{\frac {277413649}{193536}}&4\leq x<{\frac {9}{2}}\\-{\frac {15619}{3402}}x^{4}+{\frac {38150335}{435456}}x^{3}-{\frac {20157247}{32256}}x^{2}+{\frac {859841695}{435456}}x-{\frac {64472345}{27648}}&{\frac {9}{2}}\leq x<5\\{\frac {14714327}{4354560}}x^{4}-{\frac {4466137}{62208}}x^{3}+{\frac {165651247}{290304}}x^{2}-{\frac {875490655}{435456}}x+{\frac {4614904015}{1741824}}&5\leq x<{\frac {11}{2}}\\-{\frac {7873577}{4354560}}x^{4}+{\frac {30717383}{725760}}x^{3}-{\frac {179437319}{483840}}x^{2}+{\frac {16606729}{11520}}x-{\frac {869722273}{414720}}&{\frac {11}{2}}\leq x<6\\{\frac {2980409}{4354560}}x^{4}-{\frac {12698561}{725760}}x^{3}+{\frac {16211669}{96768}}x^{2}-{\frac {19138891}{26880}}x+{\frac {3289787993}{2903040}}&6\leq x<{\frac {13}{2}}\\-{\frac {759239}{4354560}}x^{4}+{\frac {10519741}{2177280}}x^{3}-{\frac {10403603}{207360}}x^{2}+{\frac {71964499}{311040}}x-{\frac {3481646837}{8709120}}&{\frac {13}{2}}\leq x<7\\{\frac {118931}{4354560}}x^{4}-{\frac {1774639}{2177280}}x^{3}+{\frac {630259}{69120}}x^{2}-{\frac {14096161}{311040}}x+{\frac {245108501}{2903040}}&7\leq x<{\frac {15}{2}}\\-{\frac {9581}{4354560}}x^{4}+{\frac {21863}{311040}}x^{3}-{\frac {407387}{483840}}x^{2}+{\frac {9758873}{2177280}}x-{\frac {25971499}{2903040}}&{\frac {15}{2}}\leq x<8\\{\frac {73}{1244160}}x^{4}-{\frac {4343}{2177280}}x^{3}+{\frac {5273}{207360}}x^{2}-{\frac {313703}{2177280}}x+{\frac {380873}{1244160}}&8\leq x<{\frac {17}{2}}\\-{\frac {1}{8709120}}x^{4}+{\frac {1}{241920}}x^{3}-{\frac {1}{17920}}x^{2}+{\frac {3}{8960}}x-{\frac {27}{35840}}&{\frac {17}{2}}\leq x<9\\0&{\text{otherwise}}\end{cases}}}
Images of compactly supported B-spline wavelets
edit
B-spline wavelet of order 1
B-spline wavelet of order 2
B-spline wavelet of order 3
B-spline wavelet of order 4
B-spline wavelet of order 5
Battle-Lemarie wavelets
edit
The Battle-Lemarie wavelets form a class of orthonormal wavelets constructed using the class of cardinal B-splines. The expressions for these wavelets are given in the frequency domain; that is, they are defined by specifying their Fourier transforms. The Fourier transform of a function of t , say,
F
(
t
)
{\displaystyle F(t)}
, is denoted by
F
^
(
ω
)
{\displaystyle {\hat {F}}(\omega )}
.
Let m be a positive integer and let
N
m
(
x
)
{\displaystyle N_{m}(x)}
be the cardinal B-spline of order m . The Fourier transform of
N
m
(
x
)
{\displaystyle N_{m}(x)}
is
N
^
m
(
ω
)
{\displaystyle {\hat {N}}_{m}(\omega )}
. The scaling function
ϕ
m
(
t
)
{\displaystyle \phi _{m}(t)}
for the m -th order Battle-Lemarie wavelet is that function whose Fourier transform is
ϕ
^
m
(
ω
)
=
N
^
m
(
ω
)
(
∑
k
=
−
∞
∞
|
N
^
m
(
ω
+
2
π
k
)
|
2
)
1
/
2
.
{\displaystyle {\hat {\phi }}_{m}(\omega )={\frac {{\hat {N}}_{m}(\omega )}{\left(\sum _{k=-\infty }^{\infty }\vert {\hat {N}}_{m}(\omega +2\pi k)\vert ^{2}\right)^{1/2}}}.}
The m -th order Battle-Lemarie wavelet is the function
ψ
B
L
,
m
(
t
)
{\displaystyle \psi _{BL,m}(t)}
whose Fourier transform is
ψ
^
B
L
,
m
(
ω
)
=
−
e
−
i
ω
/
2
ϕ
^
m
(
ω
+
2
π
)
¯
ϕ
^
m
(
ω
2
)
ϕ
^
m
(
ω
2
+
π
)
¯
{\displaystyle {\hat {\psi }}_{BL,m}(\omega )=-{\frac {e^{-i\omega /2}\,\,{\overline {{\hat {\phi }}_{m}(\omega +2\pi )}}\,\,{\hat {\phi }}_{m}\left({\frac {\omega }{2}}\right)}{\overline {{\hat {\phi }}_{m}\left({\frac {\omega }{2}}+\pi \right)}}}}
Amir Z Averbuch and Valery A Zheludev (2007). "Wavelet transforms generated by splines" (PDF) . International Journal of Wavelets, Multiresolution and Information Processing . 257 (5). Retrieved 21 December 2014 .
Amir Z. Averbuch, Pekka Neittaanmaki, and Valery A. Zheludev (2014). Spline and Spline Wavelet Methods with Applications to Signal and Image Processing Volume I . Springer. ISBN 978-94-017-8925-7 . {{cite book }}
: CS1 maint: multiple names: authors list (link )