Spline wavelet

Animation showing the compactly supported cardinal B-spline wavelets of orders 1, 2, 3, 4 and 5.

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]

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 ${\displaystyle N_{4}(x)}$

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

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

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

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

which is the well-known Haar wavelet.

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}{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

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). Aldroubi, Akram; Laine, Andrew F.; Unser, Michael A. (eds.). "Ten good reasons for using spline wavelets" (PDF). Proc. SPIE Vol. 3169, Wavelets Applications in Signal and Image Processing V. Wavelet Applications in Signal and Image Processing V. 3169: 422–431. Bibcode:1997SPIE.3169..422U. doi:10.1117/12.292801. S2CID   12705597 . 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.
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. ISBN   9780898712742.
6. Christopher Heil (2011). A Basis Theory Primer . Birkhauser. pp.  177–188. ISBN   9780817646868.
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.