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   -> Volume 4, Issue 5

Answer: Derivatives of wavelets (WD 4.4 #14)
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Mitch Oslick

PostPosted: Tue Dec 03, 2002 11:34 am    
Subject: Answer: Derivatives of wavelets (WD 4.4 #14)
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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: "Two-Scale Difference
Equations I. Existence and Global Regularity of Solutions" and "Two-Scale
Difference Equations II. Local Regularity, Infinite Products of Matrices and
Fractals" (SIAM J. Math. Anal., 22(5):1388-1410 and 23(4):1031-79, respective-
ly). Those papers discuss the solution of two-scale 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 l-times
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 two-scale 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 l-times 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_{1-n} phi(2x - n), psi is l-times
differentiable whenever phi is, and we have

psi^(l) (x) = sum_{n=0}^{N} 2^l (-1)^n c_{1-n} phi^(l) (2x - n).

In particular, the papers discuss how to compute solutions to two-scale
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 half-integers, 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
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