Spline wavelet

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.

:

:$N\_m(x)=\backslash int\_0^1\; N\_\{m-1\}(x-t)dt$, 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.

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
3. $\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; N\_m(x-k)=1$ for all $x$.
4. The cardinal B-splines of orders m and m-1 are related by the identity: $N\_m(x)=\backslash frac\{x\}\{m-1\}N\_\{m-1\}(x)\; +\; \backslash frac\{m-x\}\{m-1\}N\_\{m-1\}(x-1)$.
5. The function $N\_m(x)$ is symmetrical about $x=\backslash frac\{m\}\{2\}$, that is, $N\_m\backslash left(\backslash frac\{m\}\{2\}-x\backslash right)=N\_m\backslash left(\backslash frac\{m\}\{2\}+x\backslash right)$.
6. The derivative of $N\_m(x)$ is given by $N\_m^\backslash prime(x)=N\_\{m-1\}(x)-N\_\{m-1\}(x-1)$.
7. $\backslash int\_\{-\backslash infty\}^\backslash infty\; N\_m(x)\backslash ,\; dx\; =1$

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

:$N\_m(x)=\backslash sum\_\{k=0\}^m\; 2^\{-m+1\}\{m\; \backslash choose\; k\}N\_m(2x-k)$.

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

:$A\; \backslash left\backslash Vert\; \backslash \{c\_k\; \backslash \}\; \backslash right\backslash Vert^2\; \backslash le\; \backslash left\; \backslash Vert\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; c\_k\; N\_m(x-k)\; \backslash right\backslash Vert^2\; \backslash le\; B\; \backslash left\backslash Vert\backslash \{c\_k\backslash \}\backslash right\backslash Vert^2$

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

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.

The B-spline of order 1, namely $N\_1(x)$, is the constant B-spline. It is defined by

:

The two-scale relation for this B-spline is

:$N\_1(x)=N\_1(2x)+N\_1(2x-1)$

The B-spline of order 2, namely $N\_2(x)$, is the linear B-spline. It is given by

:

The two-scale relation for this wavelet is

:$N\_2(x)=\backslash frac\{1\}\{2\}N\_2(2x)+N\_2(2x-1)+\backslash frac\{1\}\{2\}N\_2(2x-2)$

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

:$N\_3(x)=\backslash frac\{1\}\{4\}N\_3(2x)+\backslash frac\{3\}\{4\}N\_3(2x-1)+\backslash frac\{3\}\{4\}N\_3(2x-2)+\backslash frac\{1\}\{4\}N\_3(2x-3)$

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_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)

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)

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}

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 $\backslash \{\; N\_\{m,kj\}(x):\; j=\backslash cdots,-2,-1,0,1,2,\backslash cdots\backslash \}$.

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

1. The spaces $V\_k$ satisfy the property: $\backslash cdots\; \backslash subset\; V\_\{-2\}\backslash subset\; V\_\{-1\}\backslash subset\; V\_0\; \backslash subset\; V\_1\backslash subset\; V\_2\; \backslash subset\; \backslash cdots$.
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 $\backslash \{N\_\{m,kj\}(x):\; j=\; \backslash cdots,-2,-1,0,1,2,\backslash cdots\backslash \}$ 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.{{cite book|last1=Christopher Heil|title=A Basis Theory Primer|year=2011|url=https://archive.org/details/basistheoryprime00chei|url-access=limited|publisher=Birkhauser|pages=[https://archive.org/details/basistheoryprime00chei/page/n203 177]–188|isbn=9780817646868 }})

Let m be a fixed positive integer and $N\_m(x)$ be the cardinal B-spline of order m. A function $\backslash psi\_m(x)$ 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 $\backslash \{\backslash psi\_m(x-j):j=\backslash cdots,\; -2,-1,0,1,2,\backslash cdots\backslash \}$ (this closure is denoted by $W\_0$) is the orthogonal complement of $V\_0$ in $V\_1$. The subscript m in $\backslash psi\_m(x)$ is used to indicate that $\backslash psi\_m(x)$ is a basic wavelet relative the cardinal B-spline of order m. There is no unique basic wavelet $\backslash psi\_m(x)$ relative to the cardinal B-spline $N\_m(x)$. Some of these are discussed in the following sections.

Let m be a fixed positive integer and let $N\_m(x)$ be the cardinal B-spline of order m. Given a sequence $\backslash \{f\_j:j=\backslash cdots,\; -2,-1,0,1,2,\backslash cdots\; \backslash \}$ of real numbers, the problem of finding a sequence $\backslash \{c\_\{m,k\}:\; k=\backslash cdots,\; -2,-1,0,1,2,\backslash cdots\; \backslash \}$ of real numbers such that

:$\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; c\_\{m,k\}\; N\_m\backslash left(j+\backslash frac\{m\}\{2\}-k\backslash right)\; =\; f\_j$ for all $j$,

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

:,

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

:$L\_m(x)\; =\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; c\_\{m,k\}\; N\_m\backslash left(x+\backslash frac\{m\}\{2\}-k\backslash right)$

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

:$\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; c\_\{m,k\}\; N\_m\backslash left(j+\backslash frac\{m\}\{2\}-k\backslash right)\; =\; \backslash delta\_\{0j\}$

The fundamental cardinal interpolatory spline $L\_m(x)$ can be determined using Z-transforms. Using the following notations

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

:$B\_m(z)=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; N\_m\backslash left(k+\backslash frac\{m\}\{2\}\backslash right)z^k,$

:$C\_m(z)=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; c\_\{m,k\}\; z^k,$

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

:$C\_m(z)=\backslash frac\{1\}\{B\_m(z)\}$.

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

As a concrete example, the case $L\_4(x)$ may be investigated. The definition of $B\_m(z)$ implies that

:$B\_4(x)=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; N\_4(2+k)z^k$

The only nonzero values of $N\_4(k+2)$ are given by $k\; =-1,0,1$ and the corresponding values are

:$N\_4(1)=\; \backslash frac\{1\}\{6\},\; N\_4(2)\; =\; \backslash frac\{4\}\{6\},\; N\_4(3)=\backslash frac\{1\}\{6\}.$

Thus $B\_4(z)$ reduces to

:$B\_4(z)=\backslash frac\{1\}\{6\}z^\{-1\}+\backslash frac\{4\}\{6\}z^0+\backslash frac\{1\}\{6\}z^1=\backslash frac\{1+4z+z^2\}\{6z\}$

This yields the following expression for $C\_4(z)$.

:$C\_4(z)=\backslash frac\{6z\}\{1+4z+z^2\}$

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

:$L\_4(x)=\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\; (-1)^k\; \backslash sqrt\{3\}(2-\backslash sqrt\{3\})^$

For a positive integer m, the function $\backslash psi\_m(x)$ defined by

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

is a basic wavelet relative to the cardinal B-spline of order $N\_m(x)$. The subscript I in $\backslash psi\_\{I,m\}$ is used to indicate that it is based in the interpolatory spline formula. This basic wavelet is not compactly supported.

The wavelet of order 2 using interpolatory spline is given by

:$\backslash psi\_\{I,2\}(x)=\backslash frac\{d^2\}\{dx^2\}L\_4(2x-1)$

The expression for $L\_4(x)$ now yields the following formula:

:$\backslash psi\_\{I,2\}(x)=\backslash frac\{d^2\}\{dx^2\}\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; (-1)^k\; \backslash sqrt\{3\}(2-\backslash sqrt\{3\})^$

Now, using the expression for the derivative of $N\_m(x)$ in terms of $N\_\{m-1\}(x)$ the function $\backslash psi\_2(x)$ can be put in the following form:

:$\backslash psi\_\{I,2\}(x)=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; (-1)^k\; 4\; \backslash sqrt\{3\}(2-\backslash sqrt\{3\})^$

The following piecewise linear function is the approximation to $\backslash psi\_2(x)$ obtained by taking the sum of the terms corresponding to $k=-3,\; \backslash ldots,\; 3$ in the infinite series expression for $\backslash psi\_2(x)$.

:$$

\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

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

-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}

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

:$\backslash psi\_\{I,m\}(x)=\backslash sum\_\{-\backslash infty\}^\backslash infty\; q\_nN\_m(2x-n)$ where $q\_n=\; \backslash sum\_\{j=0\}^m\; (-1)^j\{m\; \backslash choose\; j\}c\_\{m+n-j-1\}.$

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.{{cite book|last1=Charles K Chui|title=An Introduction to Wavelets|date=1992|publisher=Academic Press|page=249|ref=Chui}} 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 $\backslash psi\_\{C,m\}(x)$, has as its support the interval $[0,\; 2m-1]$. These wavelets are essentially unique in a certain sense explained below.

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

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

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

:

which is the well-known Haar wavelet.

1. The support of $\backslash psi\_\{C,m\}(x)$ is the closed interval $[0,\; 2m-1]$.
2. The wavelet $\backslash psi\_\{C,m\}(x)$ is the unique wavelet with minimum support in the following sense: If $\backslash eta(x)\; \backslash in\; W\_0$ generates $W\_0$ and has support not exceeding $2m-1$ in length then $\backslash eta(x)=c\_0\backslash psi\_\{C,m\}(x-n\_0)$ for some nonzero constant $c\_0$ and for some integer $n\_0$.{{cite book|last1=Charles K Chui|title=An Introduction to Wavelets|date=1992|publisher=Academic Press|page=184}}
3. $\backslash psi\_\{C,m\}(x)$ is symmetric for even m and antisymmetric for odd m.

$\backslash psi\_m(x)$ satisfies the two-scale relation:

:$\backslash psi\_\{C,m\}(x)=\backslash sum\_\{n=0\}^\{3m-2\}q\_nN\_m(2x-n)$ where $q\_n=\backslash frac\{(-1)^n\}\{2^\{m-1\}\}\backslash sum\_\{j=0\}^m\; \{m\; \backslash choose\; j\}N\_\{2m\}(n-j+1)$.

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

:$N\_m(2x-l)\; =\; \backslash sum\_\{k=-\backslash infty\}^\{\backslash infty\}\; \backslash left[\; a\_\{m,\; l-2k\}N\_m(x-k)\; +\; b\_\{m,\; l-2k\}\backslash psi\_\{C,m\}(x-k)\backslash right]$

where the coefficients $a\_\{m,j\}$ and $b\_\{m,j\}$ are given by

: $a\_\{m,j\}=\; -\; \backslash frac\{(-1)^j\}\{2\}\backslash sum\_\{l=-\backslash infty\}^\backslash infty\; q\_\{-j+2m-2l+1\}c\_\{2m,l\},$

: $b\_\{m,j\}=\; \backslash frac\{(-1)^j\}\{2\}\backslash sum\_\{l=-\backslash infty\}^\backslash infty\; p\_\{-j+2m-2l+1\}c\_\{2m,l\}.$

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

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

:$\backslash psi\_\{C,1\}(x)=\; N\_1(2x)-N\_1(2x-1)$

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}

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

:$\backslash psi\_\{C,2\}(x)=\; \backslash frac\{1\}\{12\}\backslash left(N\_2(2x)-6\; N\_2(2x-1)+\; 10\; N\_2(2x-2)-6\; N\_2(2x-3)+\; N\_2(2x-4)\backslash right)$

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}

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

:$\backslash psi\_\{C,3\}(x)=\; \backslash frac\{1\}\{480\}\backslash Big[\; (N\_3(2x)-29\; N\_3(2x-1)+\; 147\; N\_3(2x-2)-\; 303\; N\_3(2x-3)+$

:::::$303N\_3(2x-4)\; -\; 147N\_3(2x-5)\; +\; 29\; N\_3(2x-6)\; -\; N\_3(2x-7)\backslash Big]$

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}

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

:$\backslash psi\_\{C,4\}(x)=\; \backslash frac\{1\}\{40320\}\backslash Big[\; N\_4(2x)-\; 124\; N\_4(2x-1)+\; 1677\; N\_4(2x-2)-\; 7904\; N\_4(2x-3)+\; 18482\; N\_4(2x-4)\; -$

:::::$24264\; N\_4(2x-5)\; +\; 18482N\_4(2x-6)\; -\; 7904\; N\_4(2x-7)\; +\; 1677\; N\_4(2x-8)\; -\; 124N\_4(2x-9)\; +\; N\_4(2x-10)\backslash Big]$

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}

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

:$\backslash psi\_\{C,5\}(x)=\; \backslash frac\{1\}\{5806080\}\backslash Big[N\_5(2x)-507\; N\_5(2x-1)+17128\; N\_5(2x-2)-166304\; N\_5(2x-3)+\; 748465N\_5(2x-4)$

:::::$-1900115N\_5(2x-5)+2973560\; N\_5(2x-6)-2973560\; N\_5(2x-7)+1900115N\_5(2x-8)$

:::::$-748465\; N\_5(2x-9)+\; 166304\; N\_5(2x-10)-17128N\_5(2x-11)+507N\_5(2x-12)-N\_5(2x-13)\backslash Big]$

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}{2903040} & 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}

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 $\backslash hat\{F\}(\backslash omega)$.

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 $\backslash hat\{N\}\_m(\backslash omega)$. The scaling function $\backslash phi\_m(t)$ for the m-th order Battle-Lemarie wavelet is that function whose Fourier transform is

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

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

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

{{reflist}}

• {{cite journal|last1=Amir Z Averbuch and Valery A Zheludev|title=Wavelet transforms generated by splines|journal=International Journal of Wavelets, Multiresolution and Information Processing|date=2007|volume=257|issue=5|url=http://www.cs.tau.ac.il/~zhel/PS/splitr3AA.pdf|access-date=21 December 2014}}
• {{cite book|last1=Amir Z. Averbuch, Pekka Neittaanmaki, and Valery A. Zheludev|title=Spline and Spline Wavelet Methods with Applications to Signal and Image Processing Volume I|date=2014|publisher=Springer|isbn=978-94-017-8925-7}}

Category:Wavelets

Category:Continuous wavelets

Category:Splines (mathematics)