Ondícula spline

En la teoría matemática de las ondículas , una ondícula spline es una ondícula construida utilizando una función spline . ^{[ 1 ]} Existen diferentes tipos de ondículas spline. Las ondículas spline interpoladoras introducidas por CK Chui y JZ Wang se basan en una fórmula de interpolación spline específica. ^[²^] Aunque estas ondículas son ortogonales , no tienen soportes compactos . Existe una clase particular de ondículas, únicas en cierto sentido, construidas utilizando B-splines y con soportes compactos. Si bien estas ondículas no son ortogonales, poseen propiedades especiales que las han hecho muy populares. ^[³^] El término ondícula spline se utiliza a veces para referirse a las ondículas de esta clase. Estas ondículas especiales también se denominan ondículas B-spline y ondículas B-spline cardinales . ^[⁴^] Las ondículas de Battle-Lemarie también son ondículas construidas utilizando funciones spline. ^[⁵^]

Splines B cardinales

Sea n un entero fijo no negativo . Sea C ⁿ el conjunto de todas las funciones de valor real definidas sobre el conjunto de números reales tales que cada función en el conjunto, así como sus primeras n derivadas, son continuas en todas partes. Una secuencia bi-infinita . . . x ₋₂ , x ₋₁ , x ₀ , x ₁ , x ₂ , . . . tal que x _r < x _{r +1} para todo r y tal que x _r se aproxima a ±∞ cuando r se aproxima a ±∞ se dice que define un conjunto de nudos. Una spline de orden n con un conjunto de nudos { x _r } es una función S ( x ) en C ⁿ tal que, para cada r , la restricción de S ( x ) al intervalo [ x _r , x _{r +1} ) coincide con un polinomio con coeficientes reales de grado como máximo n en x .

Si la separación x _{r +1} - x _r , donde r es cualquier entero, entre los nodos sucesivos del conjunto de nodos es constante, la spline se denomina spline cardinal . El conjunto de enteros Z = {..., -2, -1, 0, 1, 2, ...} es una elección estándar para el conjunto de nodos de una spline cardinal. Salvo que se especifique lo contrario, generalmente se asume que el conjunto de nodos es el conjunto de enteros.

Una B-spline cardinal es un tipo especial de spline cardinal. Para cualquier entero positivo m, la B-spline cardinal de orden m , denotada por N _m ( x ), se define recursivamente de la siguiente manera.

N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{en otro caso}}\end{cases}}

N_{m}(x)=\int _{0}^{1}N_{m-1}(xt)dt

, para

m>1

.

Más adelante en este artículo se presentan expresiones concretas para las B-splines cardinales de todos los órdenes hasta 5 y sus gráficas.

Propiedades de las B-splines cardinales

Propiedades elementales

El apoyo de $N_{m}(x)$ es el intervalo cerrado $[0,m]$ .
La función $N_{m}(x)$ es no negativo, es decir, $N_{m}(x)>0$ para $0<x<m$ .
$\sum _{k=-\infty }^{\infty }N_{m}(xk)=1$ a pesar de $x$ .
Las B-splines cardinales de órdenes m y m-1 están relacionadas por la identidad: $N_{m}(x)={\frac {x}{m}}N_{m-1}(x)+{\frac {m+1-x}{m}}N_{m-1}(x-1)$ .
La función $N_{m}(x)$ es simétrico respecto a $x={\frac {m}{2}}$ , eso es, $N_{m}\left({\frac {m}{2}}-x\right)=N_{m}\left({\frac {m}{2}}+x\right)$ .
El derivado de $N_{m}(x)$ es dado por $N_{m}^{\prime }(x)=N_{m-1}(x)-N_{m-1}(x-1)$ .
$\int _{-\infty }^{\infty }N_{m}(x)\,dx=1$

Relación de dos escalas

The cardinal B-spline of order m satisfies the following two-scale relation:

N_{m}(x)=\sum _{k=0}^{m}2^{-m+1}{m \choose k}N_{m}(2x-k)

.

Riesz property

The cardinal B-spline of order m satisfies the following property, known as the Riesz property: There exists two positive real numbers $A$ and $B$ such that for any square summable two-sided sequence $\{c_{k}\}_{k=-\infty }^{\infty }$ and for any x,

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 $\Vert \cdot \Vert$ is the norm in the ℓ²-space.

Cardinal B-splines of small orders

The cardinal B-splines are defined recursively starting from the B-spline of order 1, namely $N_{1}(x)$ , 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.

Constant B-spline

The B-spline of order 1, namely $N_{1}(x)$ , is the constant B-spline. It is defined by

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}(2x)+N_{1}(2x-1)

Linear B-spline

The B-spline of order 2, namely $N_{2}(x)$ , is the linear B-spline. It is given by

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)={\frac {1}{2}}N_{2}(2x)+N_{2}(2x-1)+{\frac {1}{2}}N_{2}(2x-2)

Quadratic B-spline

The B-spline of order 3, namely $N_{3}(x)$ , is the quadratic B-spline. It is given by

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)={\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)

Cubic B-spline

The cubic B-spline is the cardinal B-spline of order 4, denoted by $N_{4}(x)$ . It is given by the following expressions:

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)={\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)

Bi-quadratic B-spline

The bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by $N_{5}(x)$ . It is given by

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)={\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)

Quintic B-spline

The quintic B-spline is the cardinal B-spline of order 6 denoted by $N_{6}(x)$ . It is given by

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

The cardinal B-spline $N_{m}(x)$ of order m generates a multi-resolution analysis. In fact, from the elementary properties of these functions enunciated above, it follows that the function $N_{m}(x)$ is square integrable and is an element of the space $L^{2}(R)$ of square integrable functions. To set up the multi-resolution analysis the following notations used.

For any integers $k,j$ , define the function $N_{m,kj}(x)=N_{m}(2^{k}x-j)$ .
For each integer $k$ , define the subspace $V_{k}$ of $L^{2}(R)$ as the closure of the linear span of the set $\{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}$ .

That these define a multi-resolution analysis follows from the following:

The spaces $V_{k}$ satisfy the property: $\cdots \subset V_{-2}\subset V_{-1}\subset V_{0}\subset V_{1}\subset V_{2}\subset \cdots$ .
The closure in $L^{2}(R)$ of the union of all the subspaces $V_{k}$ is the whole space $L^{2}(R)$ .
The intersection of all the subspaces $V_{k}$ is the singleton set containing only the zero function.
For each integer $k$ the set $\{N_{m,kj}(x):j=\cdots ,-2,-1,0,1,2,\cdots \}$ is an unconditional basis for $V_{k}$ . (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

Let m be a fixed positive integer and $N_{m}(x)$ be the cardinal B-spline of order m. A function $\psi _{m}(x)$ in $L^{2}(R)$ is a basic wavelet relative to the cardinal B-spline function $N_{m}(x)$ if the closure in $L^{2}(R)$ of the linear span of the set $\{\psi _{m}(x-j):j=\cdots ,-2,-1,0,1,2,\cdots \}$ (this closure is denoted by $W_{0}$ ) is the orthogonal complement of $V_{0}$ in $V_{1}$ . The subscript m in $\psi _{m}(x)$ is used to indicate that $\psi _{m}(x)$ is a basic wavelet relative the cardinal B-spline of order m. There is no unique basic wavelet $\psi _{m}(x)$ relative to the cardinal B-spline $N_{m}(x)$ . Some of these are discussed in the following sections.

Wavelets relative to cardinal B-splines using fundamental interpolatory splines

Fundamental interpolatory spline

Definitions

Let m be a fixed positive integer and let $N_{m}(x)$ be the cardinal B-spline of order m. Given a sequence $\{f_{j}:j=\cdots ,-2,-1,0,1,2,\cdots \}$ of real numbers, the problem of finding a sequence $\{c_{m,k}:k=\cdots ,-2,-1,0,1,2,\cdots \}$ of real numbers such that

\sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=f_{j}

for all

j

,

is known as the cardinal spline interpolation problem. The special case of this problem where the sequence $\{f_{j}\}$ is the sequence $\delta _{0j}$ , where $\delta _{ij}$ is the Kronecker delta function $\delta _{ij}$ defined by

\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 $L_{m}(x)$ and is given by

L_{m}(x)=\sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(x+{\frac {m}{2}}-k\right)

where the sequence $\{c_{m,k}\}$ is now the solution of the following system of equations:

\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

The fundamental cardinal interpolatory spline $L_{m}(x)$ can be determined using Z-transforms. Using the following notations

A(z)=\sum _{k=-\infty }^{\infty }\delta _{k0}z^{k}=1,

B_{m}(z)=\sum _{k=-\infty }^{\infty }N_{m}\left(k+{\frac {m}{2}}\right)z^{k},

C_{m}(z)=\sum _{k=-\infty }^{\infty }c_{m,k}z^{k},

it can be seen from the equations defining the sequence $c_{m,k}$ that

B_{m}(z)C_{m}(z)=A(z)

from which we get

C_{m}(z)={\frac {1}{B_{m}(z)}}

.

This can be used to obtain concrete expressions for $c_{m,k}$ .

Example

As a concrete example, the case $L_{4}(x)$ may be investigated. The definition of $B_{m}(z)$ implies that

B_{4}(x)=\sum _{k=-\infty }^{\infty }N_{4}(2+k)z^{k}

The only nonzero values of $N_{4}(k+2)$ are given by $k=-1,0,1$ and the corresponding values are

N_{4}(1)={\frac {1}{6}},N_{4}(2)={\frac {4}{6}},N_{4}(3)={\frac {1}{6}}.

Thus $B_{4}(z)$ reduces to

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 $C_{4}(z)$ .

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 $c_{4,k}$ can be computed. These values are then substituted in the expression for $L_{4}(x)$ to yield

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

For a positive integer m, the function $\psi _{m}(x)$ defined by

\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 $N_{m}(x)$ . The subscript I in $\psi _{I,m}$ is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported.

Example

The wavelet of order 2 using interpolatory spline is given by

\psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}L_{4}(2x-1)

The expression for $L_{4}(x)$ now yields the following formula:

\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 $N_{m}(x)$ in terms of $N_{m-1}(x)$ the function $\psi _{2}(x)$ can be put in the following form:

\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 $\psi _{2}(x)$ obtained by taking the sum of the terms corresponding to $k=-3,\ldots ,3$ in the infinite series expression for $\psi _{2}(x)$ .

\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}}

Two-scale relation

The two-scale relation for the wavelet function $\psi _{m}(x)$ is given by

\psi _{I,m}(x)=\sum _{-\infty }^{\infty }q_{n}N_{m}(2x-n)

where

q_{n}=\sum _{j=0}^{m}(-1)^{j}{m \choose j}c_{m+n-j-1}.

Compactly supported B-spline wavelets

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 $N_{m}(x)$ of order m discovered by Chui and Wang and denoted by $\psi _{C,m}(x)$ , has as its support the interval $[0,2m-1]$ . These wavelets are essentially unique in a certain sense explained below.

Definition

The compactly supported B-spline wavelet of order m is given by

\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

\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.

Properties

The support of $\psi _{C,m}(x)$ is the closed interval $[0,2m-1]$ .
The wavelet $\psi _{C,m}(x)$ is the unique wavelet with minimum support in the following sense: If $\eta (x)\in W_{0}$ generates $W_{0}$ and has support not exceeding $2m-1$ in length then $\eta (x)=c_{0}\psi _{C,m}(x-n_{0})$ for some nonzero constant $c_{0}$ and for some integer $n_{0}$ .^[8]
$\psi _{C,m}(x)$ is symmetric for even m and antisymmetric for odd m.

Two-scale relation

$\psi _{m}(x)$ satisfies the two-scale relation:

\psi _{C,m}(x)=\sum _{n=0}^{3m-2}q_{n}N_{m}(2x-n)

where

q_{n}={\frac {(-1)^{n}}{2^{m-1}}}\sum _{j=0}^{m}{m \choose j}N_{2m}(n-j+1)

.

Decomposition relation

The decomposition relation for the compactly supported B-spline wavelet has the following form:

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 $a_{m,j}$ and $b_{m,j}$ are given by

a_{m,j}=-{\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }q_{-j+2m-2l+1}c_{2m,l},

b_{m,j}={\frac {(-1)^{j}}{2}}\sum _{l=-\infty }^{\infty }p_{-j+2m-2l+1}c_{2m,l}.

Here the sequence $c_{2m,l}$ is the sequence of coefficients in the fundamental interpolatoty cardinal spline wavelet of order m.

Compactly supported B-spline wavelets of small orders

Compactly supported B-spline wavelet of order 1

The two-scale relation for the compactly supported B-spline wavelet of order 1 is

\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

\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

The two-scale relation for the compactly supported B-spline wavelet of order 2 is

\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

\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

The two-scale relation for the compactly supported B-spline wavelet of order 3 is

\psi _{C,3}(x)={\frac {1}{480}}{\Big [}(N_{3}(2x)-29N_{3}(2x-1)+147N_{3}(2x-2)-303N_{3}(2x-3)+

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

\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

The two-scale relation for the compactly supported B-spline wavelet of order 4 is

\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)-

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

\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

The two-scale relation for the compactly supported B-spline wavelet of order 5 is

\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)

-1900115N_{5}(2x-5)+2973560N_{5}(2x-6)-2973560N_{5}(2x-7)+1900115N_{5}(2x-8)

-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

\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

Battle-Lemarie wavelets

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, $F(t)$ , is denoted by ${\hat {F}}(\omega )$ .

Definition

Let m be a positive integer and let $N_{m}(x)$ be the cardinal B-spline of order m. The Fourier transform of $N_{m}(x)$ is ${\hat {N}}_{m}(\omega )$ . The scaling function $\phi _{m}(t)$ for the m-th order Battle-Lemarie wavelet is that function whose Fourier transform is

{\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 $\psi _{BL,m}(t)$ whose Fourier transform is

{\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)}}}

References

↑ Michael Unser (1997). "Diez buenas razones para usar ondículas spline" (PDF) . En Aldroubi, Akram; Laine, Andrew F.; Unser, Michael A. (eds.). Aplicaciones de ondículas en el procesamiento de señales e imágenes V. Vol. 3169. pp. 422–431 . Bibcode : 1997SPIE.3169..422U . doi : 10.1117/12.292801 . S2CID 12705597. Recuperado el 21 de diciembre de 2014 .
↑ Chui, Charles K y Jian-zhong Wang (1991). "Un enfoque de spline cardinal para ondículas" (PDF) . Actas de la Sociedad Matemática Americana . 113 (3): 785–793 . doi : 10.2307/2048616 . JSTOR 2048616. Consultado el 22 de enero de 2015 . {{cite journal}}: CS1 maint: varios nombres: lista de autores ( enlace )
1 2 Charles K. Chui y Jian-Zhong Wang (abril de 1992). "Sobre ondículas spline con soporte compacto y un principio de dualidad" (PDF) . Transactions of the American Mathematical Society . 330 (2): 903–915 . doi : 10.1090/s0002-9947-1992-1076613-3 . Recuperado el 21 de diciembre de 2014 .
↑ Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 177.
↑ Ingrid Daubechies (1992). Diez conferencias sobre ondículas . Filadelfia: Society for Industrial and Applied Mathematics. págs. 146-153 . ISBN 9780898712742.
↑ Christopher Heil (2011). Introducción a la teoría básica . Birkhauser. págs. 177 –188. ISBN 9780817646868.
↑ Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 249.
↑ Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 184.

Lecturas adicionales

Amir Z Averbuch y Valery A Zheludev (2007). "Transformadas wavelet generadas por splines" (PDF) . Revista Internacional de Wavelets, Multiresolución y Procesamiento de la Información . 257 (5) . Recuperado el 21 de diciembre de 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)

[ten-1] Michael Unser (1997). "Diez buenas razones para usar ondículas spline" (PDF) . En Aldroubi, Akram; Laine, Andrew F.; Unser, Michael A. (eds.). Aplicaciones de ondículas en el procesamiento de señales e imágenes V. Vol. 3169. pp. 422–431 . Bibcode : 1997SPIE.3169..422U . doi : 10.1117/12.292801 . S2CID 12705597. Recuperado el 21 de diciembre de 2014 .

[2] Chui, Charles K y Jian-zhong Wang (1991). "Un enfoque de spline cardinal para ondículas" (PDF) . Actas de la Sociedad Matemática Americana . 113 (3): 785–793 . doi : 10.2307/2048616 . JSTOR 2048616. Consultado el 22 de enero de 2015 . {{cite journal}}: CS1 maint: varios nombres: lista de autores ( enlace )

[ChuiCompact-3] 1 2 Charles K. Chui y Jian-Zhong Wang (abril de 1992). "Sobre ondículas spline con soporte compacto y un principio de dualidad" (PDF) . Transactions of the American Mathematical Society . 330 (2): 903–915 . doi : 10.1090/s0002-9947-1992-1076613-3 . Recuperado el 21 de diciembre de 2014 .

[4] Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 177.

[Daubechies-5] Ingrid Daubechies (1992). Diez conferencias sobre ondículas . Filadelfia: Society for Industrial and Applied Mathematics. págs. 146-153 . ISBN 9780898712742.

[6] Christopher Heil (2011). Introducción a la teoría básica . Birkhauser. págs. 177 –188. ISBN 9780817646868.

[7] Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 249.

[8] Charles K Chui (1992). Una introducción a las ondículas . Academic Press. pág. 184.

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