Spline wavelet

Wang are based on a certain spline interpolation formula.

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.

A spline of order n with a set of knots {xr} is a function S(x) in Cn such that, for each r, the restriction of S(x) to the interval [xr, xr+1) coincides with a polynomial with real coefficients of degree at most n in x.

If the separation xr+1 - xr, where r is any integer, between the successive knots in the set of knots is a constant, the spline is called a cardinal spline.

is a standard choice for the set of knots of a cardinal spline.

For any positive integer m the cardinal B-spline of order m, denoted by Nm(x), is defined recursively as follows.

Concrete expressions for the cardinal B-splines of all orders up to 5 and their graphs are given later in this article.

The cardinal B-spline of order m satisfies the following two-scale relation: The cardinal B-spline of order m satisfies the following property, known as the Riesz property: There exists two positive real numbers

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.

It is given by The two-scale relation for this wavelet is The B-spline of order 3, namely

That these define a multi-resolution analysis follows from the following: Let m be a fixed positive integer and

is a basic wavelet relative to the cardinal B-spline function

of real numbers such that is known as the cardinal spline interpolation problem.

defined by is the fundamental cardinal spline interpolation problem.

is now the solution of the following system of equations: The fundamental cardinal interpolatory spline

Splitting this expression into partial fractions and expanding each term in powers of z in an annular region the values of

defined by is a basic wavelet relative to the cardinal B-spline of order

The wavelet of order 2 using interpolatory spline is given by The expression for

can be put in the following form: The following piecewise linear function is the approximation to

Compactly supported B-spline wavelets were discovered by Charles K. Chui and Jian-zhong Wang and published in 1991.

of order m discovered by Chui and Wong and denoted by

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 The closed form expression for compactly supported B-spline wavelet of order 1 is The two-scale relation for the compactly supported B-spline wavelet of order 2 is The closed form expression for compactly supported B-spline wavelet of order 2 is The two-scale relation for the compactly supported B-spline wavelet of order 3 is The closed form expression for compactly supported B-spline wavelet of order 3 is The two-scale relation for the compactly supported B-spline wavelet of order 4 is The closed form expression for compactly supported B-spline wavelet of order 4 is The two-scale relation for the compactly supported B-spline wavelet of order 5 is The closed form expression for compactly supported B-spline wavelet of order 5 is 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.

be the cardinal B-spline of order m. The Fourier transform of

for the m-th order Battle-Lemarie wavelet is that function whose Fourier transform is The m-th order Battle-Lemarie wavelet is the function