Spline wavelet

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.

Cardinal B-splines

Let n be a fixed non-negative integer. Let Cⁿ 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, x₀, x₁, x₂, . . . 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 Cⁿ 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.

${\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}$

${\displaystyle N_{m}(x)=\int _{0}^{1}N_{m-1}(x-t)dt}$, for ${\displaystyle m>1}$.

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

Elementary properties

1. The support of ${\displaystyle N_{m}(x)}$ is the closed interval ${\displaystyle [0,m]}$.
2. The function ${\displaystyle N_{m}(x)}$ is non-negative, that is, ${\displaystyle N_{m}(x)>0}$ for ${\displaystyle 0<x<m}$.
3. ${\displaystyle \sum _{k=-\infty }^{\infty }N_{m}(x-k)=1}$ for all ${\displaystyle x}$.
4. The cardinal B-splines of orders m and m-1 are related by the identity: ${\displaystyle N_{m}(x)={\frac {x}{m-1}}N_{m-1}(x)+{\frac {m-x}{m-1}}N_{m-1}(x-1)}$.
5. The function ${\displaystyle N_{m}(x)}$ is symmetrical about ${\displaystyle x={\frac {m}{2}}}$, that is, ${\displaystyle N_{m}\left({\frac {m}{2}}-x\right)=N_{m}\left({\frac {m}{2}}+x\right)}$.
6. The derivative of ${\displaystyle N_{m}(x)}$ is given by ${\displaystyle N_{m}^{\prime }(x)=N_{m-1}(x)-N_{m-1}(x-1)}$.
7. ${\displaystyle \int _{-\infty }^{\infty }N_{m}(x)\,dx=1}$

Two-scale relation

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

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

Riesz property

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

${\displaystyle A\left\Vert \{c_{k}\}\right\Vert ^{2}\leq \left\Vert \sum _{k=-\infty }^{\infty }c_{k}N_{m}(x-k)\right\Vert ^{2}\leq B\left\Vert \{c_{k}\}\right\Vert ^{2}}$

where ${\displaystyle \Vert \cdot \Vert }$ is the norm in the ℓ²-space.

Cardinal B-splines of small orders

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

Constant B-spline

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

${\displaystyle N_{1}(x)={\begin{cases}1&0\leq x<1\\0&{\text{otherwise}}\end{cases}}}$

The two-scale relation for this B-spline is

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

Constant B-spline ${\displaystyle N_{1}(x)}$

Linear B-spline

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

${\displaystyle N_{2}(x)={\begin{cases}x&0\leq x<1\\-x+2&1\leq x<2\\0&{\text{otherwise}}\end{cases}}}$

The two-scale relation for this wavelet is

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

Linear B-spline ${\displaystyle N_{2}(x)}$

Quadratic B-spline

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

${\displaystyle N_{3}(x)={\begin{cases}{\frac {1}{2}}x^{2}&0\leq x<1\\-x^{2}+3x-{\frac {3}{2}}&1\leq x<2\\{\frac {1}{2}}x^{2}-3x+{\frac {9}{2}}&2\leq x<3\\0&{\text{otherwise}}\end{cases}}}$

The two-scale relation for this wavelet is

${\displaystyle N_{3}(x)={\frac {1}{4}}N_{3}(2x)+{\frac {3}{4}}N_{3}(2x-1)+{\frac {3}{4}}N_{3}(2x-2)+{\frac {1}{4}}N_{3}(2x-3)}$

Quadratic B-spline ${\displaystyle N_{3}(x)}$

Cubic B-spline

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

${\displaystyle N_{4}(x)={\begin{cases}{\frac {1}{6}}x^{3}&0\leq x<1\\-{\frac {1}{2}}x^{3}+2x^{2}-2x+{\frac {2}{3}}&1\leq x<2\\{\frac {1}{2}}x^{3}-4x^{2}+10x-{\frac {22}{3}}&2\leq x<3\\-{\frac {1}{6}}x^{3}+2x^{2}-8x+{\frac {32}{3}}&3\leq x<4\\0&{\text{otherwise}}\end{cases}}}$

The two-scale relation for the cubic B-spline is

${\displaystyle N_{4}(x)={\frac {1}{8}}N_{4}(2x)+{\frac {1}{2}}N_{4}(2x-1)+{\frac {3}{4}}N_{4}(2x-2)+{\frac {1}{2}}N_{4}(2x-3)+{\frac {1}{8}}N_{4}(2x-4)}$

Cubic B-spline ${\displaystyle N_{4}(x)}$

Remark: Legend for yellow graph should be ${\displaystyle {\frac {1}{4}}N_{4}(2x-4)}$

Bi-quadratic B-spline

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

${\displaystyle N_{5}(x)={\begin{cases}{\frac {1}{24}}x^{4}&0\leq x<1\\-{\frac {1}{6}}x^{4}+{\frac {5}{6}}x^{3}-{\frac {5}{4}}x^{2}+{\frac {5}{6}}x-{\frac {5}{24}}&1\leq x<2\\{\frac {1}{4}}x^{4}-{\frac {5}{2}}x^{3}+{\frac {35}{4}}x^{2}-{\frac {25}{2}}x+{\frac {155}{24}}&2\leq x<3\\-{\frac {1}{6}}x^{4}+{\frac {5}{2}}x^{3}-{\frac {55}{4}}x^{2}+{\frac {65}{2}}x-{\frac {655}{24}}&3\leq x<4\\{\frac {1}{24}}x^{4}-{\frac {5}{6}}x^{3}+{\frac {25}{4}}x^{2}-{\frac {125}{6}}x+{\frac {625}{24}}&4\leq x<5\\0&{\text{otherwise}}\end{cases}}}$

The two-scale relation is

${\displaystyle N_{5}(x)={\frac {1}{16}}N_{5}(2x)+{\frac {5}{16}}N_{5}(2x-1)+{\frac {10}{16}}N_{5}(2x-2)+{\frac {10}{16}}N_{5}(2x-3)+{\frac {5}{16}}N_{5}(2x-4)+{\frac {1}{16}}N_{5}(2x-5)}$

Quintic B-spline

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

${\displaystyle N_{6}(x)={\begin{cases}{\frac {1}{120}}x^{5}&0\leq x<1\\-{\frac {1}{24}}x^{5}+{\frac {1}{4}}x^{4}-{\frac {1}{2}}x^{3}+{\frac {1}{2}}x^{2}-{\frac {1}{4}}x+{\frac {1}{20}}&1\leq x<2\\{\frac {1}{12}}x^{5}-x^{4}+{\frac {9}{2}}x^{3}-{\frac {19}{2}}x^{2}+{\frac {39}{4}}x-{\frac {79}{20}}&2\leq x<3\\-{\frac {1}{12}}x^{5}+{\frac {3}{2}}x^{4}-{\frac {21}{2}}x^{3}+{\frac {71}{2}}x^{2}-{\frac {231}{4}}x+{\frac {731}{20}}&3\leq x<4\\{\frac {1}{24}}x^{5}-x^{4}+{\frac {19}{2}}x^{3}-{\frac {89}{2}}x^{2}+{\frac {409}{4}}x-{\frac {1829}{20}}&4\leq x<5\\-{\frac {1}{120}}x^{5}+{\frac {1}{4}}x^{4}-3x^{3}+18x^{2}-54x+{\frac {324}{5}}&5\leq x<6\\0&{\text{otherwise}}\end{cases}}}$

Multi-resolution analysis generated by cardinal B-splines

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

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

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

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

Let m be a fixed positive integer and ${\displaystyle N_{m}(x)}$ be the cardinal B-spline of order m. A function ${\displaystyle \psi _{m}(x)}$ in ${\displaystyle L^{2}(R)}$ is a basic wavelet relative to the cardinal B-spline function ${\displaystyle N_{m}(x)}$ if the closure in ${\displaystyle L^{2}(R)}$ of the linear span of the set ${\displaystyle \{\psi _{m}(x-j):j=\cdots ,-2,-1,0,1,2,\cdots \}}$ (this closure is denoted by ${\displaystyle W_{0}}$) is the orthogonal complement of ${\displaystyle V_{0}}$ in ${\displaystyle V_{1}}$. The subscript m in ${\displaystyle \psi _{m}(x)}$ is used to indicate that ${\displaystyle \psi _{m}(x)}$ is a basic wavelet relative the cardinal B-spline of order m. There is no unique basic wavelet ${\displaystyle \psi _{m}(x)}$ relative to the cardinal B-spline ${\displaystyle N_{m}(x)}$. 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 ${\displaystyle N_{m}(x)}$ be the cardinal B-spline of order m. Given a sequence ${\displaystyle \{f_{j}:j=\cdots ,-2,-1,0,1,2,\cdots \}}$ of real numbers, the problem of finding a sequence ${\displaystyle \{c_{m,k}:k=\cdots ,-2,-1,0,1,2,\cdots \}}$ of real numbers such that

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

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

${\displaystyle \delta _{ij}={\begin{cases}1,&{\text{ if }}i=j\\0,&{\text{ if }}i\neq j\end{cases}}}$,

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

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

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

${\displaystyle \sum _{k=-\infty }^{\infty }c_{m,k}N_{m}\left(j+{\frac {m}{2}}-k\right)=\delta _{0j}}$

Procedure to find the fundamental cardinal interpolatory spline

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

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

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

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

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

${\displaystyle B_{m}(z)C_{m}(z)=A(z)}$

from which we get

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

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

Example

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

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

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

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

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

${\displaystyle B_{4}(z)={\frac {1}{6}}z^{-1}+{\frac {4}{6}}z^{0}+{\frac {1}{6}}z^{1}={\frac {1+4z+z^{2}}{6z}}}$

This yields the following expression for ${\displaystyle C_{4}(z)}$.

${\displaystyle C_{4}(z)={\frac {6z}{1+4z+z^{2}}}}$

Splitting this expression into partial fractions and expanding each term in powers of z in an annular region the values of ${\displaystyle c_{4,k}}$ can be computed. These values are then substituted in the expression for ${\displaystyle L_{4}(x)}$ to yield

${\displaystyle L_{4}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(x+2-k)}$

Wavelet using fundamental interpolatory spline

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

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

is a basic wavelet relative to the cardinal B-spline of order ${\displaystyle N_{m}(x)}$. The subscript I in ${\displaystyle \psi _{I,m}}$ 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

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

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

${\displaystyle \psi _{I,2}(x)={\frac {d^{2}}{dx^{2}}}\sum _{k=-\infty }^{\infty }(-1)^{k}{\sqrt {3}}(2-{\sqrt {3}})^{|k|}N_{4}(2x+1-k)}$

Now, using the expression for the derivative of ${\displaystyle N_{m}(x)}$ in terms of ${\displaystyle N_{m-1}(x)}$ the function ${\displaystyle \psi _{2}(x)}$ can be put in the following form:

${\displaystyle \psi _{I,2}(x)=\sum _{k=-\infty }^{\infty }(-1)^{k}4{\sqrt {3}}(2-{\sqrt {3}})^{|k|}{\Big (}(N_{2}(2x+k-1)-2N_{2}(2x+k-2)+N_{2}(2x+k-3){\Big )}}$

The following piecewise linear function is the approximation to ${\displaystyle \psi _{2}(x)}$ obtained by taking the sum of the terms corresponding to ${\displaystyle k=-3,\ldots ,3}$ in the infinite series expression for ${\displaystyle \psi _{2}(x)}$.

${\displaystyle \psi _{I,2}(x)\approx {\begin{cases}0.07142668x+0.17856670&-2.5<x\leq -2\\-0.48084803x-0.92598272&-2<x\leq -1.5\\2.0088293x+2.8085333&-1.5<x\leq -1\\-7.5684795x-6.7687755&-1<x\leq -0.5\\28.245949x+11.138439&-0.5<x\leq 0\\-57.415316x+11.138439&0<x\leq 0.5\\57.415316x-46.276878&0.5<x\leq 1\\-28.245949x+39.384388&1<x\leq 1.5\\7.5684795x-14.337255&1.5<x\leq 2\\-2.0088293x+4.8173625&2<x\leq 2.5\\0.48084803x-1.4068308&2.5<x\leq 3\\-0.07142668x+0.24999338&3<x\leq 3.5\\0&{otherwise}\end{cases}}}$

Two-scale relation

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

${\displaystyle \psi _{I,m}(x)=\sum _{-\infty }^{\infty }q_{n}N_{m}(2x-n)}$ where ${\displaystyle q_{n}=\sum _{j=0}^{m}(-1)^{j}{m \choose j}c_{m+n-j-1}.}$

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

Definition

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

${\displaystyle \psi _{C,m}(x)={\frac {1}{2^{2m-1}}}\sum _{j=0}^{2m-2}(-1)^{j}N_{2m}(j+1){\frac {d^{m}}{dx^{m}}}N_{2m}(2x-j)}$

This is an m-th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is

${\displaystyle \psi _{C,1}(x)={\frac {1}{2}}N_{2}(1){\frac {d}{dx}}N_{2}(2x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}$

which is the well-known Haar wavelet.

Properties

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

Two-scale relation

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

${\displaystyle \psi _{C,m}(x)=\sum _{n=0}^{3m-2}q_{n}N_{m}(2x-n)}$ where ${\displaystyle q_{n}={\frac {(-1)^{n}}{2^{m-1}}}\sum _{j=0}^{m}{m \choose j}N_{2m}(n-j+1)}$.

Decomposition relation

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

${\displaystyle N_{m}(2x-l)=\sum _{k=-\infty }^{\infty }\left[a_{m,l-2k}N_{m}(x-k)+b_{m,l-2k}\psi _{C,m}(x-k)\right]}$

where the coefficients ${\displaystyle a_{m,j}}$ and ${\displaystyle b_{m,j}}$ are given by

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

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

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

Compactly supported B-spline wavelets of small orders

Compactly supported B-spline wavelet of order 1

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

${\displaystyle \psi _{C,1}(x)=N_{1}(2x)-N_{1}(2x-1)}$

The closed form expression for compactly supported B-spline wavelet of order 1 is

${\displaystyle \psi _{C,1}(x)={\begin{cases}1&0\leq x<{\frac {1}{2}}\\-1&{\frac {1}{2}}\leq x<1\\0&{\text{otherwise}}\end{cases}}}$

Compactly supported B-spline wavelet of order 2

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

${\displaystyle \psi _{C,2}(x)={\frac {1}{12}}\left(N_{2}(2x)-6N_{2}(2x-1)+10N_{2}(2x-2)-6N_{2}(2x-3)+N_{2}(2x-4)\right)}$

The closed form expression for compactly supported B-spline wavelet of order 2 is

${\displaystyle \psi _{C,2}(x)={\begin{cases}{\frac {1}{6}}x&0\leq x<{\frac {1}{2}}\\-{\frac {7}{6}}x+{\frac {2}{3}}&{\frac {1}{2}}\leq x<1\\{\frac {8}{3}}x-{\frac {19}{6}}&1\leq x<{\frac {3}{2}}\\-{\frac {8}{3}}x+{\frac {29}{6}}&{\frac {3}{2}}\leq x<2\\{\frac {7}{6}}x-{\frac {17}{6}}&2\leq x<{\frac {5}{2}}\\-{\frac {1}{6}}x+{\frac {1}{2}}&{\frac {5}{2}}\leq x<3\\0&{\text{otherwise}}\end{cases}}}$

Compactly supported B-spline wavelet of order 3

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

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

${\displaystyle 303N_{3}(2x-4)-147N_{3}(2x-5)+29N_{3}(2x-6)-N_{3}(2x-7){\Big ]}}$

The closed form expression for compactly supported B-spline wavelet of order 3 is

${\displaystyle \psi _{C,3}(x)={\begin{cases}{\frac {1}{240}}x^{2}&0\leq x<{\frac {1}{2}}\\-{\frac {31}{240}}x^{2}+{\frac {2}{15}}x-{\frac {1}{30}}&{\frac {1}{2}}\leq x<1\\{\frac {103}{120}}x^{2}-{\frac {221}{120}}x+{\frac {229}{240}}&1\leq x<{\frac {3}{2}}\\-{\frac {313}{120}}x^{2}+{\frac {1027}{120}}x-{\frac {1643}{240}}&{\frac {3}{2}}\leq x<2\\{\frac {22}{5}}x^{2}-{\frac {779}{40}}x+{\frac {339}{16}}&2\leq x<{\frac {5}{2}}\\-{\frac {22}{5}}x^{2}+{\frac {981}{40}}x-{\frac {541}{16}}&{\frac {5}{2}}\leq x<3\\{\frac {313}{120}}x^{2}-{\frac {701}{40}}x+{\frac {2341}{80}}&3\leq x<{\frac {7}{2}}\\-{\frac {103}{120}}x^{2}+{\frac {809}{120}}x-{\frac {3169}{240}}&{\frac {7}{2}}\leq x<4\\{\frac {31}{240}}x^{2}-{\frac {139}{120}}x+{\frac {623}{240}}&4\leq x<{\frac {9}{2}}\\-{\frac {1}{240}}x^{2}+{\frac {1}{24}}x-{\frac {5}{48}}&{\frac {9}{2}}\leq x<5\\0&{\text{otherwise}}\end{cases}}}$

Compactly supported B-spline wavelet of order 4

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

${\displaystyle \psi _{C,4}(x)={\frac {1}{40320}}{\Big [}N_{4}(2x)-124N_{4}(2x-1)+1677N_{4}(2x-2)-7904N_{4}(2x-3)+18482N_{4}(2x-4)-}$

${\displaystyle 24264N_{4}(2x-5)+18482N_{4}(2x-6)-7904N_{4}(2x-7)+1677N_{4}(2x-8)-124N_{4}(2x-9)+N_{4}(2x-10){\Big ]}}$

The closed form expression for compactly supported B-spline wavelet of order 4 is

${\displaystyle \psi _{C,4}(x)={\begin{cases}{\frac {1}{30240}}x^{3}&0\leq x<{\frac {1}{2}}\\-{\frac {127}{30240}}x^{3}+{\frac {2}{315}}x^{2}-{\frac {1}{315}}x+{\frac {1}{1890}}&{\frac {1}{2}}\leq x<1\\{\frac {19}{280}}x^{3}-{\frac {47}{224}}x^{2}+{\frac {2147}{10080}}x-{\frac {103}{1440}}&1\leq x<{\frac {3}{2}}\\-{\frac {1109}{2520}}x^{3}+{\frac {465}{224}}x^{2}-{\frac {32413}{10080}}x+{\frac {16559}{10080}}&{\frac {3}{2}}\leq x<2\\{\frac {5261}{3360}}x^{3}-{\frac {33463}{3360}}x^{2}+{\frac {42043}{2016}}x-{\frac {145193}{10080}}&2\leq x<{\frac {5}{2}}\\-{\frac {35033}{10080}}x^{3}+{\frac {93577}{3360}}x^{2}-{\frac {148517}{2016}}x+{\frac {216269}{3360}}&{\frac {5}{2}}\leq x<3\\{\frac {4832}{945}}x^{3}-{\frac {27691}{560}}x^{2}+{\frac {113923}{720}}x-{\frac {28145}{168}}&3\leq x<{\frac {7}{2}}\\-{\frac {4832}{945}}x^{3}+{\frac {58393}{1008}}x^{2}-{\frac {52223}{240}}x+{\frac {2048227}{7560}}&{\frac {7}{2}}\leq x<4\\{\frac {35033}{10080}}x^{3}-{\frac {75827}{1680}}x^{2}+{\frac {981101}{5040}}x-{\frac {234149}{840}}&4\leq x<{\frac {9}{2}}\\-{\frac {5261}{3360}}x^{3}+{\frac {38509}{1680}}x^{2}-{\frac {112487}{1008}}x+{\frac {30347}{168}}&{\frac {9}{2}}\leq x<5\\{\frac {1109}{2520}}x^{3}-{\frac {24077}{3360}}x^{2}+{\frac {78311}{2016}}x-{\frac {141311}{2016}}&5\leq x<{\frac {11}{2}}\\-{\frac {19}{280}}x^{3}+{\frac {1361}{1120}}x^{2}-{\frac {14617}{2016}}x+{\frac {4151}{288}}&{\frac {11}{2}}\leq x<6\\{\frac {127}{30240}}x^{3}-{\frac {55}{672}}x^{2}+{\frac {5359}{10080}}x-{\frac {11603}{10080}}&6\leq x<{\frac {13}{2}}\\-{\frac {1}{30240}}x^{3}+{\frac {1}{1440}}x^{2}-{\frac {7}{1440}}x+{\frac {49}{4320}}&{\frac {13}{2}}\leq x<7\\0&{\text{otherwise}}\end{cases}}}$

Compactly supported B-spline wavelet of order 5

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

${\displaystyle \psi _{C,5}(x)={\frac {1}{5806080}}{\Big [}N_{5}(2x)-507N_{5}(2x-1)+17128N_{5}(2x-2)-166304N_{5}(2x-3)+748465N_{5}(2x-4)}$

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

${\displaystyle -748465N_{5}(2x-9)+166304N_{5}(2x-10)-17128N_{5}(2x-11)+507N_{5}(2x-12)-N_{5}(2x-13){\Big ]}}$

The closed form expression for compactly supported B-spline wavelet of order 5 is

${\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}{290304}}&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

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

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

Definition

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

${\displaystyle {\hat {\phi }}_{m}(\omega )={\frac {{\hat {N}}_{m}(\omega )}{\left(\sum _{k=-\infty }^{\infty }\vert {\hat {N}}_{m}(\omega +2\pi k)\vert ^{2}\right)^{1/2}}}.}$

The m-th order Battle-Lemarie wavelet is the function ${\displaystyle \psi _{BL,m}(t)}$ whose Fourier transform is

${\displaystyle {\hat {\psi }}_{BL,m}(\omega )=-{\frac {e^{-i\omega /2}\,\,{\overline {{\hat {\phi }}_{m}(\omega +2\pi )}}\,\,{\hat {\phi }}_{m}\left({\frac {\omega }{2}}\right)}{\overline {{\hat {\phi }}_{m}\left({\frac {\omega }{2}}+\pi \right)}}}}$

References

1. Michael Unser (1997). "Ten good reasons for using spline wavelets" (PDF). Proc. SPIE Vol. 3169, Wavelets Applications in Signal and Image Processing V: 422–431. Retrieved 21 December 2014.
2. Chui, Charles K, and Jian-zhong Wang (1991). "A cardinal spline approach to wavelets" (PDF). Proceedings of the American Mathematical Society. 113 (3): 785–793. doi:10.2307/2048616. JSTOR 2048616. Retrieved 22 January 2015.CS1 maint: multiple names: authors list (link)
3. Charles K. Chui and Jian-Zhong Wang (April 1992). "On Compactly Supported Spline Wavelets and a Duality Principle" (PDF). Transactions of the American Mathematical Society. 330 (2): 903–915. doi:10.1090/s0002-9947-1992-1076613-3. Retrieved 21 December 2014.
4. Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 177.
5. Ingrid Daubechies (1992). Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics. pp. 146–153.
6. Christopher Heil (2011). A Basis Theory Primer. Birkhauser. pp. 177–188.
7. Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 249.
8. Charles K Chui (1992). An Introduction to Wavelets. Academic Press. p. 184.

Further reading

• 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.CS1 maint: multiple names: authors list (link)