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# Multiresolution Properties of Splines

Module by: Sujesh Sreedharan. E-mail the author

Summary: B-splines form a simple set of scaling functions satisfying the dilation equation with binomial filter coefficients. However B-Splines other than the zeroth order B-spline (the Haar function) are not orthogonal to its own shifts. Hence to form a perfect reconstruction(PR) filter bank system, biorthogonal scaling functions are used. Also semiorthogonal filter banks can be used to form a PR filter bank system.

## Polynomials

Polynomials P N x P N x have an interesting scaling property that if we scale the domain variable xx by a real number MM, the resulting function is still a polynomial with the same degree. A polynomial of degree NN over real numbers is defined as follows:

P N x=k=0N a k xk P N x k 0 N a k x k
(1)
The coefficients a k a k of the monomial xk x k are real for real polynomials. In general they could be complex numbers and P N x P N x can be defined over the complex domain.

Consider P N xM=fx P N x M f x , we find

fx=k=0N a k xMk=k=0N a k Mkxk=k=0N b k xk f x k 0 N a k x M k k 0 N a k M k x k k 0 N b k x k
(2)
So fx f x is also a polynomial over xx with degree NN.

## Splines

We know that splines are smoothly connected pieces of polynomials. Let φx φ x be a spline. φx φ x is a piecewise polynomial over integer intervals.

Dilate φx φ x by MM. φxM φ x M is also a piecewise polynomial over integer intervals with same degree. So it can be represented by using B-Spline Bases.

In particular integer dilates of B-splines can be represented by the unscaled B-spline bases themselves which gives us a scaling equation for B-Splines.

### M-Scale Relation

Consider shifted causal B-splines φnx= β n xn+12 φ x n β n x n 1 2 . These are obtained by the ( n+1 n 1 )-fold convolution of φ 0 x φ 0 x , the Haar scaling function.

φ 0 xM=k=0M1 φ 0 xk=k=0M1 h M 0 k φ 0 xk φ 0 x M k 0 M 1 φ 0 x k k 0 M 1 h M 0 k φ 0 x k
(3)
where h M 0 k h M 0 k is the filter with z-transform H M 0 z=k=0M1zk H M 0 z k 0 M 1 z k (discrete pulse of length MM). Convolve φ 0 xM φ 0 x M with itself n+1 n 1 times. This gives us the following result:
φ n xM=k=0,kZM1 h M n k φ n xk φ n x M k 0 M 1 k h M n k φ n x k
(4)
where
H M n z=1Mn H M 0 n+1zn1=1Mnk=0M1zk H M n z 1 M n H M 0 z n 1 1 M n k 0 M 1 z k
(5)
Thus we have an MM-scale relation for all integers MM. If M=2 M 2 , H 2 n z H 2 n z is a binomial filter whose coefficients form the Pascal's triangle.

## Spline Wavelets

We saw that shifted B-splines φ n x= β n xn+12 φ n x β n x n 1 2 are scaling functions.

We can form a nested set of subspaces V j V j , such that

j , V j V j + 1 : V j = Span k φ2jtk j V j V j + 1 V j Span k φ 2 j t k
(6)
And they form a multiresolution analysis system that is dense in L 2 L 2 . How do we find the spline wavelets? φ n xk φ n x k are not orthogonal so we need to find either biorthogonal or semi-orthogonal wavelets. The biorthogonal spline wavelets are compact and give FIR filters for the wavelet filter bank. The semi-orthogonal spline wavelets generate IIR filter bank structures. There are also other spline wavelets such as Stromberg's one sided orthogonal splines and the Battle-Lemarie Wavelets.

### Example 1: BIORTHOGONAL SPLINE WAVELETS

These are the wavelets obtained using biorthogonality of the filters on synthesis and analysis sides of the filter bank. For further details refer to BIORTHOGONAL WAVELETS.

• φ 1 x= β 1 x1 φ 1 x β 1 x 1 , the Hat function gives the 5/3 filter bank.
• Hz=1+z122 H z 1 z 1 2 2 is the scaling filter for φ 1 x φ 1 x .
• The biorthogonal scaling filter is H ^ z=1+z1221+4z1+z22 H ^ z 1 z 1 2 2 1 4 z 1 z 2 2 .
The scaling filters and wavelets generated are plotted in Figure 1.

### IIR and Semiorthogonal

Basis φtk φ t k for V 0 V 0 are non-orthogonal. We impose wavelets wtk w t k to be orthogonal to φt φ t .

V 0 W 0 V 0 W 0 and V 0 W 0 V 0 W 0 as in orthogonal scaling functions. wtk w t k need not be an orthogonal set. The analysis filters are IIR. The high pass synthesis filter Gz G z is given by

Gz=(z12N)Hz-1Az-1 G z z 1 2 N H z A z
(7)
H ~ z H ~ z and G ~ z G ~ z are the low pass and high pass analysis filters respectively.
H ~ z=2z-1Hz-1Az-1Az2 H ~ z 2 z H z A z A z 2
(8)
G ~ z=2HzAz2 G ~ z 2 H z A z 2
(9)
The Az2 A z 2 in the denominator makes the analysis filters IIR, where Az A z is the z-transform of the sequence, ak=φt,φtk a k φ t φ t k .

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