Wavelets

Each basic form in an ordinary Walsh or Fourier transform has nonzero elements spread throughout. With wavelets the elements are more localized. As noted in the late 1980s basic forms can be set up by scaling and translating just a single appropriately chosen underlying shape. The (a) Haar and (b) Daubechies wavelets Ψ[x] shown below both have the property that the basic forms 2^{m/2} Ψ[2^{m} x - n] (whose 2D analogs are shown as on page 573) are orthogonal for every different m and n.

The pictures below show images built up by keeping successively more of these basic forms. Sharp edges have fewer wiggles than with Fourier transforms.