N = 6 scaling functions and mother wavelets

introduction

The impetus for this post is figure 1.4 on page 6 of Burrus et al. (Since it’s been a while, that’s Burrus, C. Sidney; Gopinath, Ramesh A.; Guo, Haitao. Introduction to Wavelets and Wavelet Transforms, A Primer. Prentice Hall, 1998.ISBN 0 13 489600 9.)

They offered the figure as an illustration of four different scaling functions; in addition, these scaling functions were parameterized by two angles.

Now I know how to produce the drawings of those scaling functions — and I also know how to produce drawings of the corresponding mother wavelets, which they did not show on page 6. It also turns out that they have interchanged the legends for figures (a) and (b) — and when we can produce the drawings ourselves, that mistake becomes almost irrelevant.

This post also serves to reiterate the calculation sequence for plotting a scaling function and its corresponding mother wavelet.

Mother wavelet from scaling function: D4 and Haar

D4 dyadic wavelet

We can compute the mother wavelet from the scaling function. Let me show you how to do this for Daubechies’s D4.

First, we need the scaling function.

the D4 scaling function again, quickly

This is review. There is a matrix M0 which has an eigenvector with eigenvalue 1, and that eigenvector gives me the values of the scaling function at the integers. Once I have initial values, I can compute by recursion (and because of the specific form of the recursion, only at points whose denominator is a power of 2).

I start by constructing the matrix M0 in principle…

The dyadic expansion of Daubechies D4 scaling function

introduction

edit jun 13: corrected one of the 4 equations at the end.

I have shown you one way to compute the D4 scaling function. I am not entirely comfortable with the method, but I don’t need to be: I can do better.

The method I’ve shown you begins with a terrible approximation — a constant function or signal — and improves it by repeatedly applying a fixed filter to it.

The primer by Burrus et al. also provides MATLAB® code for a quite different method. It turns out that we can find the exact values of the D4 scaling function at the integers. The dilation equation can then be used to find the exact values at the halves… then the exact values at the quarters… and so on. It’s called a dyadic expansion because it works for points whose denominator is a power of 2.

Their code, however, was challenging to figure out.

But I don’t need to figure it out. I can do better.

I can do so much better that it must be illegal, immoral, and fattening.

Wavelets: The D4 scaling function

Introduction

I am going to do something a little risky. I’m going to show you how to approximate a particular function – the Daubechies D4 scaling function – even though I am a little shaky on the justification.

If you have ever looked at wavelets, you have probably seen this picture. It is a quintessential “father wavelet” – more commonly, I think, called a “scaling function”.

I want to compute it; this is not relevant to computing wavelet coefficients, because just as this is computed from its definition, so wavelet coefficients which depend on it and its associated wavelets are computed from its definition rather than from its values.

Talking about this is much like talking about one of one’s favorite movies: what makes something one of my favorite movies is deeply personal, and it doesn’t usually affect my friends that way. To them, at best, it’s just another movie; at worst, they think it is a dud.