Abstract
The lecture introduces the construction of orthogonal wavelet bases and the fast algorithm for calculating wavelet decomposition coefficients. A wavelet basis is obtained from a multiresolution by calculating the orthogonal complement Wjof Vjin Vj-1. Each space Wjadmits an orthonormal basis by dilating a parent wavelet by 2jand translating it. The union of these bases at all 2j scalesdefines an orthonormal wavelet base of L2 space. These wavelets are constructed from the low-pass filter h that specifies each multiresolution, by introducing a new band-pass filter g that depends on h. These filters satisfy a quadrature condition that is sufficient to generate orthonormal wavelet bases. The fast algorithm calculates wavelet coefficients across scales by iterating convolutions with h and g followed by subsampling.
In multiple dimensions, wavelet bases are obtained with a tensor product of wavelets of a single variable. In two dimensions for images, the basis is obtained with 3 wavelets that capture the variations of the image in different directions, at each scale 2j.
