I’ve never actually tried wavelet transform analysis before, so I’m trying to understand it better, but there’s one aspect that’s puzzling me:

The Morlet wavelet has two parameters, eta and sigma. Eta controls how fast it wiggles, and sigma how quickly the function dies out. (Essentially, it’s like the basis function for a Fourier transform multiplied by a Gaussian envelope.)

In wavelet analysis we pick a parent function, which is often a Morlet wavelet with particular values of eta and sigma that we have chosen. 

We then create a set of child functions from the parent function by shifting and dilating the function in k-space.

Each child function is then used as a basis function for a transform of chi(k) calculated by integrating over a range of k-values, much like what is done to calculate a Fourier transform. Since the Morlet wavelet is localized in k-space, and the child functions are shifted to focus on different regions of k-space, our wavelet transform produces plots which are a function of k. But since dilating the parent function by different amounts yields child functions with different frequencies, the plots are also a function of R. Therefore the result is a two-dimensional (k and R) contour plot.

So far, so good. 

But what I’m wondering about is the effect of the initial choice of eta and sigma for the parent function.

The dimensionless product of eta and sigma has a clear effect. If (eta)(sigma) is small, the parent function will not have very many oscillations of significant size; it it’s large, it will have a lot of oscillations. So if we want good resolution in R (and thus relatively poor resolution in k), we want a big value of (eta)(sigma). In the limit as (eta)(sigma) becomes arbitrarily large, we recover the Fourier transform. If we want good resolution in k, we use a small value for (eta)(sigma). In the limit as (eta)(sigma) becomes arbitrarily small we recover chi(k).

That still makes sense to me!

But what difference do sigma and eta make individually? In other words, how is sigma = 0.5 and eta = 10.0 different from sigma = 1.0 and eta = 5.0? Since we end up creating child functions that dilate the parent function anyway, I can’t see that it should have any effect at all. 

And if those two parameters don’t make an independent effect, why do we pretend there are two independent-looking parameters? 

I’ve seen at least one early paper that suggests choosing eta so that it’s close to the path length you’re most interested in probing, but I can’t see how that actually makes any difference, given the dilation.

I expect there’s a good chance I’m missing something obvious, or have a fundamental misunderstanding of part of the process, and look forward to learning more!

Best,

Scott Calvin

Lehman College of the City University of New York