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   -> Volume 3, Issue 5


Question: New wavelet ?
 
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Richard K Barrett (richard@astro.gla.ac.uk)
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PostPosted: Mon Dec 02, 2002 12:55 pm    
Subject: Question: New wavelet ?
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Question: New wavelet ?

Dear Waveleteers,

I have recently been introduced to the mystic art of wavelet transforms
thanks to the wavelets section in the new edition of Numerical Recipes.
As a complete novice, I have a question for the more
experienced reader of the Wavelet Digest. I apologize in advance for the
length of this question, but I don't know how to make it short *and*
intelligible.

In Numerical Recipes it says that the wavelet filter, H(w) (w=frequency),
which is the fourier transform of the sequence of wavelet filter
coefficients, c(1), c(2), ..., c(N), must satisfy :

H(w) is periodic, with period 2*pi, (1)
|H(0)|^2 = 2 (|H| denotes the modulus of H), (2)
|H(w)|^2 + |H(w+pi)|^2 = 2 (3)

and must satisfy an `approximation condition of order p', which corresponds
to
H, and its first p-1 derivatives are zero at w = pi. (4)
I can derive a wavelet filter satisfying an approximation condition of
infinite order (`p=infinity') using the famous infinitely differentiable
but not analytic function given by
E(x) = exp{1/(x^2 - 1)}, for -1<x<1, and E(x)=0 otherwise,
which is only non-zero for -1<x<1.
A simple linear transformation of the x co-ordinate can give a function
with a `hump' between any two points a and b, say, and the value zero
elsewhere. Therefore define
E(a,b,s;x) = exp{s^2/[(x-a)(x-b)]}, for a<x<b, and E(a,b,s;x) = 0 otherwise.
(Here s is simply a (non-zero) parameter, which may, for the moment, be taken
to be 1.) Note that all derivatives of E with respect to x exist and
are continuous, and all derivatives are zero for x<=a and x>=b. It is this
last property that gives an approximation condition of infinite order.

It is (fairly) easy to verify that the following definition of H(w) satisfies
(1), (2), (3) and (4) with p=infinity :

Put H(n,a,b,s;w) = sqrt(2) Cos {(2n+1)F(a,b,s;w)},
where n is any integer and a, b, and s are the parameters introduced above,
and we require 0<=a<b<=pi (to simplify things, imagine that a=0, b=pi,
s=1, and n=0). It is the form of the function F(a,b,s;w) that ensures (4)
holds with p=infinity. It can be seen that if all derivatives of F (with
respect to w) are zero at w=0 and w=pi then H(w) will have all derivatives
zero at w=pi (note the role of (3) here). Furthermore, forcing F(w=pi)=pi/2
guarantees that H(w=pi) = 0, giving (4) with p=infinity. In terms of this
function F, conditions (1) - (4) become, therefore :

although not essential, we may assume that F is 2*pi periodic in w, (1')

F(w=0) = 0, (2')
F(w+pi) = (pi/2) - F(w) for 0<=w<=pi, (3')
F(w=pi)=pi/2, all derivatives of F at w=0 and w=pi are zero. (4')

In addition to these conditions, we might as well demand that F is
infinitely differentiable w.r.t. w (this will ensure that, while the
wavelet is not compact, the filter coefficients, c(i), will fall of faster
than any polynomial). One solution to these conditions is the function
satisfying the differential equation ( ' = d/dw)

F'(a,b,s;w) = k*E(a,b,s;w) for 0<=w<=pi and
F'(a,b,s;w) = -k*E(a,b,s;w) for pi<=w<=2*pi

with initial condition F(w=0)=0, and where k is a renormalization constant
chosen to ensure F(w=pi) = pi/2.

The properties of these wavelets are (stop me if I'm boring you) :

1) In general the filter coeffs. compare well with the Lemarie wavelet.
s may be used to control the rate at which the filter coefficients fall off.
2) The actual wavelets are smooth for s=8 and get `sharper' as s decreases.
3) For some parameter values the wavelet functions have a fractal appearance
(some of them actually look kind of `hairy'), although I suspect that this
is due to numerical error in the calculation of the wavelet coefficients
and the inverse wavelet transform.

One further point worth noting is that it is possible to multiply the function
F by another function, g(w) say, and still get a wavelet with the same
properties -- the only constraint is that we must have g(b)=1, and the
derivatives of g(w) must all be zero at w=b to ensure that the derivatives
of g*F are all zero there.

At last, after all this nonsense, I arrive at my question.
Is the wavelet derived a: interesting
b: original

I look forward to your replies.

Richard Barrett.

P.S. Wim Sweldens has told me that this wavelet is related to the Meyer Wavelet.
I would like to know what the practical applications of such wavelets are.
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