Skip to content Skip to navigation

Connexions

You are here: Home » Content » Multiresolution Properties of Splines

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Multiresolution Properties of Splines

Module by: Sujesh Sreedharan. E-mail the author

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 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=0k∈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 φ2jt−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.
  • Hz=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+4z-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 φt−k φ t k for V 0 V 0 are non-orthogonal. We impose wavelets wt−k 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. wt−k 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=-z1−2NH-z-1A-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=2z-1Hz-1Az-1Az-2 H ~ z 2 z H z A z A z 2 (8)
G ~ z=2H-zAz-2 G ~ z 2 H z A z 2 (9)
The Az-2 A z 2 in the denominator makes the analysis filters IIR, where Az A z is the z-transform of the sequence, ak=<φt,φt−k> a k φ t φ t k .

Content actions

Give Feedback:

E-mail the module author | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks