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In the
mathematical theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
of
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the n ...
s, a spline wavelet is a wavelet constructed using a spline function. 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 In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
formula. Though these wavelets are
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
, they do not have
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
support Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...
s. There is a certain class of wavelets, unique in some sense, constructed using
B-spline In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...
s and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular. 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. The Battle-Lemarie wavelets are also wavelets constructed using spline functions.


Cardinal B-splines

Let ''n'' be a fixed non-negative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
. Let ''C''''n'' denote the set of all
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
s defined over the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s such that each function in the set as well its first ''n''
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
everywhere. A
bi-infinite sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. . . ''x''−2, ''x''−1, ''x''0, ''x''1, ''x''2, . . . 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 is a function ''S''(''x'') in ''C''''n'' 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'' = 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)=\begin1 & 0\le x <1 \\ 0 & \text\end :N_m(x)=\int_0^1 N_(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

# The support (mathematics)">support Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...
of N_m(x) is the closed interval [0,m]. # The function N_m(x) is non-negative, that is, N_m(x)>0 for 0. # \sum_^\infty N_m(x-k)=1 for all x. # The cardinal B-splines of orders ''m'' and ''m-1'' are related by the identity: N_m(x)=\fracN_(x) + \fracN_(x-1). # The function N_m(x) is symmetrical about x=\frac, that is, N_m\left(\frac-x\right)=N_m\left(\frac+x\right). # The derivative of N_m(x) is given by N_m^\prime(x)=N_(x)-N_(x-1). # \int_^\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_^m 2^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 \_^\infty and for any ''x'', :A \left\Vert \ \right\Vert^2 \le \left \Vert \sum_^\infty c_k N_m(x-k) \right\Vert^2 \le B \left\Vert\\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)=\begin1 & 0\le x < 1 \\ 0 &\text\end The two-scale relation for this B-spline is :N_1(x)=N_1(2x)+N_1(2x-1)


Linear B-spline

The B-spline of order 2, namely N_2(x), is the linear B-spline. It is given by :N_2(x)=\beginx & 0\le x < 1 \\ -x+2 & 1\le x<2 \\ 0 &\text\end The two-scale relation for this wavelet is :N_2(x)=\fracN_2(2x)+N_2(2x-1)+\fracN_2(2x-2)


Quadratic B-spline

The B-spline of order 3, namely N_3(x), is the quadratic B-spline. It is given by : N_3(x)= \begin \fracx^2 & 0\le x < 1 \\ -x^2 +3x-\frac & 1\le x<2 \\ \fracx^2 -3x + \frac & 2\le x<3 \\ 0 &\text\end The two-scale relation for this wavelet is :N_3(x)=\fracN_3(2x)+\fracN_3(2x-1)+\fracN_3(2x-2)+\fracN_3(2x-3)


Cubic B-spline

The cubic B-spline is the cardinal B-spline of order 4, denoted by N_4(x). It is given by the following expressions: : N_4(x)= \begin \fracx^3 & 0\le x < 1 \\ -\fracx^3+2x^2-2x+\frac & 1\le x < 2 \\ \fracx^3-4x^2+10x-\frac & 2\le x< 3 \\ - \fracx^3 +2x^2 -8x +\frac & 3\le x < 4 \\ 0 & \text \end The two-scale relation for the cubic B-spline is : N_4(x)=\fracN_4(2x)+\fracN_4(2x-1)+\fracN_4(2x-2)+\fracN_4(2x-3)+\fracN_4(2x-4)


Bi-quadratic B-spline

The bi-quadratic B-spline is the cardinal B-spline of order 5 denoted by N_5(x). It is given by : N_5(x)= \begin \fracx^4 & 0 \le x < 1 \\ -\fracx^4+\fracx^3-\fracx^2 +\fracx-\frac & 1\le x < 2 \\ \fracx^4 -\frac x^3 +\fracx^2 -\fracx +\frac & 2\le x < 3 \\ -\fracx^4 +\fracx^3 -\fracx^2 +\fracx -\frac & 3 \le x < 4 \\ \fracx^4 - \fracx^3 + \fracx^2 - \frac x + \frac & 4 \le x < 5 \\ 0 & \text \end The two-scale relation is : N_5(x)=\fracN_5(2x)+\fracN_5(2x-1)+\fracN_5(2x-2)+\fracN_5(2x-3)+\fracN_5(2x-4)+\fracN_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 \fracx^5 & 0\le x < 1 \\ -\fracx^5+\fracx^4 -\fracx^3 +\fracx^2 - \fracx +\frac & 1 \le x < 2 \\ \fracx^5 - x^4 +\frac x^3 -\fracx^2 +\fracx -\frac & 2 \le x < 3 \\ -\fracx^5 +\fracx^4 - \fracx^3 +\fracx^2 -\fracx+\frac & 3 \le x < 4 \\ \fracx^5 -x^4 +\fracx^3 - \fracx^2 +\fracx -\frac & 4 \le x < 5 \\ -\fracx^5 +\fracx^4 -3x^3 +18x^2 -54 x +\frac & 5 \le x < 6 \\ 0 & \text \end


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_(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 In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and ...
of the set \. That these define a multi-resolution analysis follows from the following: # The spaces V_k satisfy the property: \cdots \subset V_\subset V_\subset V_0 \subset V_1\subset V_2 \subset \cdots. # The closure in L^2(R) of the union of all the subspaces V_k is the whole space L^2(R). # The intersection of all the subspaces V_k is the singleton set containing only the zero function. # For each integer k the set \ is an unconditional basis for V_k. (A sequence in a Banach space ''X'' is an unconditional basis for the space ''X'' if every permutation of the sequence is also a basis for the same space ''X''.)


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 \ (this closure is denoted by W_0) is the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...
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 \ of real numbers, the problem of finding a sequence \ of real numbers such that :\sum_^\infty c_ N_m\left(j+\frac-k\right) = f_j for all j, is known as the ''cardinal spline interpolation problem''. The special case of this problem where the sequence \ is the sequence \delta_, where \delta_ is the Kronecker delta function \delta_ defined by :\delta_=\begin1,&\text i=j \\ 0, & \text i\ne j \end, 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_^\infty c_ N_m\left(x+\frac-k\right) where the sequence \ is now the solution of the following system of equations: :\sum_^\infty c_ N_m\left(j+\frac-k\right) = \delta_


Procedure to find the fundamental cardinal interpolatory spline

The fundamental cardinal interpolatory spline L_m(x) can be determined using
Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a dis ...
s. Using the following notations :A(z)=\sum_^\infty \delta_z^k =1, :B_m(z)=\sum_^\infty N_m\left(k+\frac\right)z^k, :C_m(z)=\sum_^\infty c_ z^k, it can be seen from the equations defining the sequence c_ that :B_m(z)C_m(z)=A(z) from which we get :C_m(z)=\frac. This can be used to obtain concrete expressions for c_.


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_^\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, N_4(2) = \frac, N_4(3)=\frac. Thus B_4(z) reduces to :B_4(z)=\fracz^+\fracz^0+\fracz^1=\frac This yields the following expression for C_4(z). :C_4(z)=\frac Splitting this expression into partial fractions and expanding each term in powers of ''z'' in an annular region the values of c_ can be computed. These values are then substituted in the expression for L_4(x) to yield :L_4(x)= \sum_^\infty (-1)^k \sqrt(2-\sqrt)^N_4(x+2-k)


Wavelet using fundamental interpolatory spline

For a positive integer ''m'', the function \psi_m(x) defined by :\psi_(x)=\fracL_(2x-1) is a basic wavelet relative to the cardinal B-spline of order N_m(x). The subscript ''I'' in \psi_ 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_(x)=\fracL_4(2x-1) The expression for L_4(x) now yields the following formula: :\psi_(x)=\frac\sum_^\infty (-1)^k \sqrt(2-\sqrt)^N_4(2x+1-k) Now, using the expression for the derivative of N_m(x) in terms of N_(x) the function \psi_2(x) can be put in the following form: :\psi_(x)=\sum_^\infty (-1)^k 4 \sqrt(2-\sqrt)^\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_(x)\approx \begin 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


Two-scale relation

The two-scale relation for the wavelet function \psi_m(x) is given by :\psi_(x)=\sum_^\infty q_nN_m(2x-n) where q_n= \sum_^m (-1)^jc_.


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. The compactly supported B-spline wavelet relative to the cardinal B-spline N_m(x) of order ''m'' discovered by Chui and Wang and denoted by \psi_(x), has as its support the interval , 2m-1/math>. 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_(x)=\frac\sum_^ (-1)^j N_(j+1)\fracN_(2x-j) This is an ''m''-th order spline. As a special case, the compactly supported B-spline wavelet of order 1 is :\psi_(x)=N_2(1)\fracN_2(2x) = \begin1 & 0\le x < \frac \\ -1 & \frac \le x < 1 \\ 0 & \text\end which is the well-known
Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repr ...
.


Properties

# The support of \psi_(x) is the closed interval , 2m-1/math>. # The wavelet \psi_(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_(x-n_0) for some nonzero constant c_0 and for some integer n_0. # \psi_(x) is symmetric for even ''m'' and antisymmetric for odd ''m''.


Two-scale relation

\psi_m(x) satisfies the two-scale relation: :\psi_(x)=\sum_^q_nN_m(2x-n) where q_n=\frac\sum_^m N_(n-j+1).


Decomposition relation

The decomposition relation for the compactly supported B-spline wavelet has the following form: :N_m(2x-l) = \sum_^ \left a_N_m(x-k) + b_\psi_(x-k)\right where the coefficients a_ and b_ are given by : a_= - \frac\sum_^\infty q_c_, : b_= \frac\sum_^\infty p_c_. Here the sequence c_ 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_(x)= N_1(2x)-N_1(2x-1) The closed form expression for compactly supported B-spline wavelet of order 1 is : \psi_(x)= \begin 1 & 0\le x < \frac \\ -1 & \frac \le x < 1\\ 0 & \text \end


Compactly supported B-spline wavelet of order 2

The two-scale relation for the compactly supported B-spline wavelet of order 2 is :\psi_(x)= \frac\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_(x)= \begin \fracx & 0\le x < \frac\\ -\fracx + \frac & \frac \le x < 1\\ \fracx - \frac & 1 \le x < \frac\\ -\frac x + \frac & \frac \le x < 2 \\ \frac x- \frac & 2 \le x < \frac\\ -\frac x + \frac & \frac \le x < 3 \\ 0 & \text \end


Compactly supported B-spline wavelet of order 3

The two-scale relation for the compactly supported B-spline wavelet of order 3 is :\psi_(x)= \frac\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/math> The closed form expression for compactly supported B-spline wavelet of order 3 is : \psi_(x)= \begin \fracx^2 & 0\le x < \frac\\ - \fracx^2+ \fracx- \frac & \frac \le x < 1\\ \fracx^2- \fracx + \frac & 1 \le x < \frac\\ -\frac x^2+ \fracx- \frac & \frac \le x < 2 \\ \frac x^2 - \frac x + \frac & 2 \le x < \frac\\ -\frac x^2 + \frac x- \frac & \frac \le x < 3 \\ \fracx^2-\fracx+ \frac & 3 \le x < \frac \\ -\fracx^2 +\fracx- \frac & \frac \le x < 4 \\ \fracx^2-\fracx+\frac & 4 \le x < \frac \\ -\fracx^2+\fracx-\frac & \frac \le x < 5 \\ 0 & \text \end


Compactly supported B-spline wavelet of order 4

The two-scale relation for the compactly supported B-spline wavelet of order 4 is :\psi_(x)= \frac\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/math> The closed form expression for compactly supported B-spline wavelet of order 4 is : \psi_(x)= \begin \fracx^3 & 0\le x < \frac\\ -\fracx^3+\fracx^2-\fracx+\frac & \frac \le x < 1\\ \fracx^3-\fracx^2+\fracx-\frac & 1 \le x < \frac\\ -\fracx^3+\fracx^2-\fracx+\frac & \frac \le x < 2 \\ \fracx^3-\fracx^2+\fracx-\frac & 2 \le x < \frac\\ -\fracx^3+\frac x^2- \fracx+ \frac & \frac \le x < 3 \\ \fracx^3- \fracx^2+ \fracx-\frac & 3 \le x < \frac \\ -\fracx^3+\fracx^2-\fracx+\frac & \frac \le x < 4 \\ \fracx^3-\fracx^2+\fracx- \frac & 4 \le x < \frac \\ -\fracx^3+\fracx^2-\fracx+ \frac & \frac \le x < 5 \\ \fracx^3-\fracx^2+\fracx- \frac & 5 \le x < \frac \\ -\fracx^3+\fracx^2-\fracx+\frac & \frac \le x < 6 \\ \fracx^3-\fracx^2+\fracx-\frac & 6 \le x < \frac \\ -\fracx^3+\fracx^2-\fracx+ \frac & \frac \le x < 7 \\ 0 & \text \end


Compactly supported B-spline wavelet of order 5

The two-scale relation for the compactly supported B-spline wavelet of order 5 is : \psi_(x)= \frac\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_(x)= \begin \fracx^4 & 0\le x < \frac \\ - \fracx^4+\fracx^3-\fracx^2+\fracx-\frac & \frac \le x < 1 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 1\le x < \frac \\ -\fracx^4+\fracx^3-\fracx^2+\fracx-\frac & \frac \le x < 2 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 2\le x < \frac \\ -\fracx^4+\fracx^3-\fracx^2+\fracx-\frac & \frac\le x < 3 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 3\le x < \frac \\ - \fracx^4+\fracx^3-\fracx^2+\fracx-\frac & \frac\le x < 4 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 4\le x < \frac \\ -\fracx^4+\fracx^3-\fracx^2+ \fracx- \frac &\frac\le x < 5 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 5\le x < \frac \\ -\fracx^4+\fracx^3- \fracx^2+ \fracx- \frac & \frac\le x < 6 \\ \fracx^4- \fracx^3+ \fracx^2-\fracx+ \frac & 6\le x < \frac \\ -\fracx^4+\fracx^3- \fracx^2+ \fracx-\frac & \frac \le x < 7 \\ \fracx^4-\fracx^3+\fracx^2-\fracx+\frac & 7\le x < \frac \\ -\fracx^4+\fracx^3-\fracx^2+\fracx-\frac & \frac \le x < 8 \\ \fracx^4-\fracx^3+ \fracx^2-\fracx+ \frac & 8\le x < \frac \\ -\fracx^4+ \fracx^3- \fracx^2+\fracx-\frac & \frac \le x < 9\\ 0 & \text \end


Images of compactly supported B-spline wavelets


Battle-Lemarie wavelets

The Battle-Lemarie wavelets form a class of orthonormal wavelets constructed using the class of cardinal B-splines. The expressions for these wavelets are given in the frequency domain; that is, they are defined by specifying their Fourier transforms. The Fourier transform of a function of ''t'', say, F(t), is denoted by \hat(\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_m(\omega). The scaling function \phi_m(t) for the ''m''-th order Battle-Lemarie wavelet is that function whose Fourier transform is :\hat_m(\omega) = \frac. The ''m''-th order Battle-Lemarie wavelet is the function \psi_(t) whose Fourier transform is :\hat_(\omega) = - \frac


References


Further reading

* *{{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 Wavelets Continuous wavelets Splines (mathematics)