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An image corresponds to a very long vector, with one component for each pixel (three components for a color image). By a change of basis the long vector is concentrated into a much smaller number of components, ready for compression. We study the block Toeplitz matrix that produces a new basis from a bank of two filters -- lowpass and highpass. The filter coefficients determine the success of the compression. They also determine whether iteration of the lowpass filter (with rescaling) will lead to a useful wavelet basis for function spaces.
Thus the construction of wavelets comes from a problem in
matrix analysis. Actual compression uses 4--5 iterations of
the basis change.
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