Foundations of Data Science

11.8 Expressing a Function in Terms of Wavelets
Given a wavelet system with scale function φ and mother wavelet ψ we wish to express
a function f(x) in terms of an orthonormal basis of the wavelet system. First we will express
f(x) in terms of scale functions φ jk (x) = φ(2 j x − k). To do this we will build a tree
similar to that in Figure 11.2 for the Haar system only computing the coefficients will be
much more complex. Recall that the coefficients at a level in the tree are the coefficients
to represent f(x) using scale functions with the precision of the level.
Let f(x) = ∑ ∞
k=0 a jkφ j (x − k) where the a jk are the coefficients in the expansion of
f(x) using level j scale functions. Since the φ j (x − k) are orthogonal
a jk =
Expanding φ j in terms of φ j+1 yields
a jk =
=
∫ ∞
x=−∞
∑d−1
m=0
∫ ∞
x=−∞
f(x)
c m
∫ ∞
x=−∞
∑d−1
m=0
∑d−1
= c m a j+1,2k+m
m=0
Let n = 2k + m. Now m = n − 2k. Then
a jk =
f(x)φ j (x − k)dx.
c m φ j+1 (2x − 2k − m)dx
f(x)φ j+1 (2x − 2k − m)dx
∑d−1
n=2k
c n−2k a j+1,n (11.4)
In construction the tree similar to that in Figure 11.2, the values at the leaves are
the values of the function sampled in the intervals of size 2 −j . Equation 11.4 is used to
compute values as one moves up the tree. The coefficients in the tree could be used if we
wanted to represent f(x) using scale functions. However, we want to represent f(x) using
one scale function whose scale is the support of f(x) along with wavelets which gives us
an orthogonal set of basis functions. To do this we need to calculate the coefficients for
the wavelets.. The value at the root of the tree is the coefficient for the scale function.
We then move down the tree calculating the coefficients for the wavelets.
Finish by calculating wavelet coefficients
maybe add material on jpeg
Example: Add example using D 4 .
Maybe example using sinc
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