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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=f⁢x P N x M f x , we find

f⁢x=∑k=0N a k ⁢xMk=∑k=0N a k Mk⁢xk=∑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 f⁢x f x is also a polynomial over xx with degree NN.

Splines

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

Dilate φ⁢x φ x by MM. φ⁢xM 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 ⁢x−n+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=0M−1 φ 0 ⁢x−k=∑k=0M−1 h M 0 ⁢k⁢ φ 0 ⁢x−k 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=0M−1z−k 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,k∈ZM−1 h M n ⁢k⁢ φ n ⁢x−k 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=1Mn⁢∑k=0M−1z−k 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 ⁢x−n+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 ⁢φ⁢2j⁢t−k 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 ⁢x−k 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 ⁢x−1 φ 1 x β 1 x 1 , the Hat function gives the 5/3 filter bank.
  • H⁢z=1+z−122 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+z−122⁢−1+4⁢z−1+z−22 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.

Figure 1: Spline 5/3 Biorthogonal Wavelets
Figure 1 (spline53.png)

IIR and Semiorthogonal

Basis φ⁢t−k φ t k for V 0 V 0 are non-orthogonal. We impose wavelets w⁢t−k w t k to be orthogonal to φ⁢t φ t .

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

G⁢z=(−z1−2⁢N)⁢H⁢−z-1⁢A⁢−z-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=2⁢z-1⁢H⁢z-1⁢A⁢z-1A⁢z−2 H ~ z 2 z H z A z A z 2
(8)
G ~ ⁢z=2⁢H⁢−zA⁢z−2 G ~ z 2 H z A z 2
(9)
The A⁢z−2 A z 2 in the denominator makes the analysis filters IIR, where A⁢z A z is the z-transform of the sequence, a⁢k=〈φ⁢t,φ⁢t−k〉 a k φ t φ t k .

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