文档介绍：APPEARED IN BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 28, Number 2, April 1993, Pages 288-305
WAVELET TRANSFORMS VERSUS FOURIER TRANSFORMS
Gilbert Strang
ct.
Abstra This note is a very basic introduction to wavelets. It starts with an
orthogonal basis of piecewise constant functions, constructed by dilation and trans-
lation. The “wavelet transform” maps each f(x) to its coefficients with respect to
this basis. The mathematics is simple and the transform is fast (faster than the
Fast Fourier Transform, which we briefly explain), but approximation by piecewise
constants is poor. To improve this first wavelet, we are led to dilation equations and
their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly
quick pute with them — always indirectly and recursively.
ment informally on the contest between these transforms in signal process-
ing, especially for video and pression (including high-definition television).
So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine
Transform — is often chosen. But wavelets are petitive, and they are
ahead for fingerprints. We present a sample of this developing theory.
1. The Haar wavelet
To explain wavelets we start with an example. It has every property we hope
for, except one. If that one defect is accepted, the construction is simple and
putations are fast. By trying to remove the defect, we are led to dilation
equations and recursively defined functions and a small world of fascinating new
problems — many still unsolved. A sensible person would stop after the first
wavelet, but fortunately mathematics goes on.
The basic example is easier to draw than to describe:
Figure 1. Scaling function φ(x), wavelet W (x), and the next level of
detail.
1991 Mathematics Subject Classification. Primary 42A06, 41A05, 65D05.
Key words and phrases. Wavelets, Fourier transform, dilation, orthogonal basis.
I am grateful to the National S