Abstract
This note is a very basic introduction to wavelets. It starts with
an orthogonal basis of piecewise constant functions, constructed by dilation
and translation. 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 to compute with them — always
indirectly and recursively.
We comment informally on the contest between these transforms in signal
processing, especially for video and image compression (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
already competitive, and they are ahead for fingerprints. We present a sample
of this developing theory.
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