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> Volume 4, Issue 5
Answer: Derivatives of wavelets (WD 4.4 #14)

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Mitch Oslick Guest

Posted: Tue Dec 03, 2002 11:34 am Subject: Answer: Derivatives of wavelets (WD 4.4 #14)




Answer: Derivatives of wavelets (WD 4.4 #14)
In volume 4, no. 4 of the Wavelet Digest, Tony Cai asked about computing the
derivatives of scaling and wavelet functions associated with multiresolution
analyses (specifically, Daubechies' orthonormal wavelets and coiflets). The
answer is found in two papers by Daubechies and Lagarias: "TwoScale Difference
Equations I. Existence and Global Regularity of Solutions" and "TwoScale
Difference Equations II. Local Regularity, Infinite Products of Matrices and
Fractals" (SIAM J. Math. Anal., 22(5):13881410 and 23(4):103179, respective
ly). Those papers discuss the solution of twoscale difference equations
f(x) = sum_{n=0}^{N} c_n f(kx  n)
(which, for k = 2, is of course the fundamental equation governing multireso
lution analysis scaling functions). If a solution f exists and is ltimes
differentiable, simple differentiation of the above equation shows that
f^(l) (x) = sum_{n=0}^{N} k^l c_n f^(l) (kx  n),
i.e., f^(l) satisfies a twoscale difference equation with coefficients k^l c_n
rather than simply c_n. So any method which computes the scaling function phi
from the dilation coefficients c_n can be used to compute phi^(l), assuming
that phi is ltimes differentiable and that the method converges; simply use
2^l c_n in place of c_n. And since the wavelet psi can be computed from phi,
specifically, psi(x) = sum_{n=0}^{N} (1)^n c_{1n} phi(2x  n), psi is ltimes
differentiable whenever phi is, and we have
psi^(l) (x) = sum_{n=0}^{N} 2^l (1)^n c_{1n} phi^(l) (2x  n).
In particular, the papers discuss how to compute solutions to twoscale
difference equations exactly on the dyadic rationals (i.e., x = 2^{j} m,
j and m integers; we are considering the multiresolution case with k = 2).
The method comes from the observation that if we know the exact values of phi
on the integers, then the difference equations allow us to compute exactly its
values on the halfintegers, and from those the exact values on the quarter
integers, and so on, and that the exact values of phi on the integers can be
found by solving for the eigenvector with eigenvalue 1 of a particular matrix
(and normalizing appropriately). This observation holds essentially true when
solving for a derivative of phi, except that the values of phi^(l) on the
integers correspond to the eigenvector with eigenvalue 2^{l}. The eigenvector
normalization is straightforward when l = 0 (i.e., when solving for phi) but
a little involved when l >= 1. For details, see the papers themselves or send
me email.
Good luck,
Mitch Oslick
mho@nova.stanford.edu 





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