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.

$N_1(x)=\begin{cases}1 & 0\le x <1 \\ 0 & \text{otherwise}\end{cases}$

$N_m(x)=\int_0^1 N_{m-1}(x-t)dt$, for $m>1$.

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

Two-scale relation

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,

$A \left\Vert \{c_k \} \right\Vert^2 \le \left \Vert \sum_{k=-\infty}^\infty c_k N_m(x-k) \right\Vert^2 \le B \left\Vert\{c_k\}\right\Vert^2$

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

$N_1(x)=\begin{cases}1 & 0\le x < 1 \\ 0 &\text{otherwise}\end{cases}$

The two-scale relation for this B-spline is

$N_1(x)=N_1(2x)+N_1(2x-1)$

Constant B-spline $N_1(x)$

Linear B-spline

The B-spline of order 2, namely $N_2(x)$, is the linear B-spline. It is given by

$N_2(x)=\begin{cases}x & 0\le x < 1 \\ -x+2 & 1\le x<2 \\ 0 &\text{otherwise}\end{cases}$

The two-scale relation for this wavelet is

$N_2(x)=\frac{1}{2}N_2(2x)+N_2(2x-1)+\frac{1}{2}N_2(2x-2)$

Linear B-spline $N_2(x)$

Quadratic B-spline

The B-spline of order 3, namely $N_3(x)$, is the quadratic B-spline. It is given by

$N_3(x)= \begin{cases} \frac{1}{2}x^2 & 0\le x < 1 \\ -x^2 +3x-\frac{3}{2} & 1\le x<2 \\ \frac{1}{2}x^2 -3x + \frac{9}{2} & 2\le x<3 \\ 0 &\text{otherwise}\end{cases}$

The two-scale relation for this wavelet is

$N_3(x)=\frac{1}{4}N_3(2x)+\frac{3}{4}N_3(2x-1)+\frac{3}{4}N_3(2x-2)+\frac{1}{4}N_3(2x-3)$

Quadratic B-spline $N_3(x)$

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:

$N_4(x)= \begin{cases} \frac{1}{6}x^3 & 0\le x < 1 \\ -\frac{1}{2}x^3+2x^2-2x+\frac{2}{3} & 1\le x < 2 \\ \frac{1}{2}x^3-4x^2+10x-\frac{22}{3} & 2\le x< 3 \\ - \frac{1}{6}x^3 +2x^2 -8x +\frac{32}{3} & 3\le x < 4 \\ 0 & \text{otherwise} \end{cases}$

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

$N_4(x)=\frac{1}{8}N(2x)+\frac{1}{2}N(2x-1)+\frac{3}{4}N(2x-2)+\frac{1}{2}N_4(2x-3)+\frac{1}{8}N(2x-4)$

Cubic B-spline $N_4(x)$

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

$N_5(x)= \begin{cases} \frac{1}{24}x^4 & 0 \le x < 1 \\ -\frac{1}{6}x^4+\frac{5}{6}x^3-\frac{5}{4}x^2 +\frac{5}{6}x-\frac{5}{24} & 1\le x < 2 \\ \frac{1}{4}x^4 -\frac{5}{2} x^3 +\frac{35}{4}x^2 -\frac{25}{2}x +\frac{155}{24} & 2\le x < 3 \\ -\frac{1}{6}x^4 +\frac{5}{2}x^3 -\frac{55}{4}x^2 +\frac{65}{2}x -\frac{655}{24} & 3 \le x < 4 \\ \frac{1}{24}x^4 - \frac{5}{6}x^3 + \frac{25}{4}x^2 - \frac{125}{6} x + \frac{625}{24} & 4 \le x < 5 \\ 0 & \text{otherwise} \end{cases}$

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

$N_6(x) = \begin{cases} \frac{1}{120}x^5 & 0\le x < 1 \\ -\frac{1}{24}x^5+\frac{1}{4}x^4 -\frac{1}{2}x^3 +\frac{1}{2}x^2 - \frac{1}{4}x +\frac{1}{20} & 1 \le x < 2 \\ \frac{1}{12}x^5 - x^4 +\frac{9}{2} x^3 -\frac{19}{2}x^2 +\frac{39}{4}x -\frac{79}{20} & 2 \le x < 3 \\ -\frac{1}{12}x^5 +\frac{3}{2}x^4 - \frac{21}{2}x^3 +\frac{71}{2}x^2 -\frac{231}{4}x+\frac{731}{20} & 3 \le x < 4 \\ \frac{1}{24}x^5 -x^4 +\frac{19}{2}x^3 - \frac{89}{2}x^2 +\frac{409}{4}x -\frac{1829}{20} & 4 \le x < 5 \\ -\frac{1}{120}x^5 +\frac{1}{4}x^4 -3x^3 +18x^2 -54 x +\frac{324}{5} & 5 \le x < 6 \\ 0 & \text{otherwise} \end{cases}$

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 $N_{m,kj}(x)=N_m(2^kx-j)$.
• 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:

1. The spaces $V_k$ satisfy the property: $\cdots \subset V_{-2}\subset V_{-1}\subset V_0 \subset V_1\subset V_2 \subset \cdots$.
2. The closure in $L^2(R)$ of the union of all the subspaces $V_k$ is the whole space $L^2(R)$.
3. The intersection of all the subspaces $V_k$ is the singleton set containing only the zero function.
4. 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

$\delta_{ij}=\begin{cases}1,&\text{ if } i=j \\ 0, & \text{ if } i\ne j \end{cases}$,

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 $\psi_2(x)$.

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

Two-scale relation

The two-scale relation for the wavelet function $\psi_m(x)$ is given by

$\psi_{I,m}(x)=\sum_{-\infty}^\infty q_nN_m(2x-n)$ 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 Wong 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

$\psi_{C,m}(x)=\frac{1}{2^{2m-1}}\sum_{j=0}^{2m-2} (-1)^j N_{2m}(j+1)\frac{d^m}{dx^m}N_{2m}(2x-j)$

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

$\psi_{C,1}(x)=\frac{1}{2}N_2(1)\frac{d}{dx}N_2(2x) = \begin{cases}1 & 0\le x < \frac{1}{2} \\ -1 & \frac{1}{2} \le x < 1 \\ 0 & \text{otherwise}\end{cases}$

which is the well-known Haar wavelet.

Properties

1. The support of $\psi_{C,m}(x)$ is the closed interval $[0, 2m-1]$.
2. 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]
3. $\psi_{C,m}(x)$ is symmetric for even m and antisymmetric for odd m.

Two-scale relation

$\psi_m(x)$ satisfies the two-scale relation:

$\psi_{C,m}(x)=\sum_{n=0}^{3m-2}q_nN_m(2x-n)$ 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

$\psi_{C,1}(x)= \begin{cases} 1 & 0\le x < \frac{1}{2} \\ -1 & \frac{1}{2} \le x < 1\\ 0 & \text{otherwise} \end{cases}$

Compactly supported B-spline wavelet of order 2

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

$\psi_{C,2}(x)= \frac{1}{12}\left(N_2(2x)-6 N_2(2x-1)+ 10 N_2(2x-2)-6 N_2(2x-3)+ N_2(2x-4)\right)$

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

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

Compactly supported B-spline wavelet of order 3

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

$\psi_{C,3}(x)= \frac{1}{480}\Big[ (N_3(2x)-29 N_3(2x-1)+ 147 N_3(2x-2)- 303 N_3(2x-3)+$

$303N_3(2x-4) - 147N_3(2x-5) + 29 N_3(2x-6) - N_3(2x-7)\Big]$

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

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

Compactly supported B-spline wavelet of order 4

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

$\psi_{C,4}(x)= \frac{1}{40320}\Big[ N_4(2x)- 124 N_4(2x-1)+ 1677 N_4(2x-2)- 7904 N_4(2x-3)+ 18482 N_4(2x-4) -$

$24264 N_4(2x-5) + 18482N_4(2x-6) - 7904 N_4(2x-7) + 1677 N_4(2x-8) - 124N_4(2x-9) + N_4(2x-10)\Big]$

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

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

Compactly supported B-spline wavelet of order 5

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

$\psi_{C,5}(x)= \frac{1}{5806080}\Big[N_5(2x)-507 N_5(2x-1)+17128 N_5(2x-2)-166304 N_5(2x-3)+ 748465N_5(2x-4)$

$-1900115N_5(2x-5)+2973560 N_5(2x-6)-2973560 N_5(2x-7)+1900115N_5(2x-8)$

$-748465 N_5(2x-9)+ 166304 N_5(2x-10)-17128N_5(2x-11)+507N_5(2x-12)-N_5(2x-13)\Big]$

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\le 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} \le 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\le 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} \le 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\le 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}\le 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\le 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}\le 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\le 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}\le 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\le 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}\le 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\le 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} \le 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\le 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} \le 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\le 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} \le 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, $F(t)$, 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

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 American Mathematical Society 113: 785–793. doi:10.2307/2048616. 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.
6. Christopher Heil. 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.

Additional reading

• Amir Z Averbuch and Valery A Zheludev (2007). "Wavelet transforms generated by splines" (PDF). Int. J. Wavelets Multiresolut Inf. Process 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.