Abstract
A wavelet transform reduces the dimensionality of a signal by breaking it down into a family of small waves, which are expanded and translated. The wavelet transform calculates the correlation between a signal and these wavelets, at all scales and positions. This lecture introduces the continuous wavelet transform and the dyadic transform, where scales are limited to powers of 2, as well as orthonormal wavelet bases. The main properties of these three types of multiscale representations for signals and images are summarized. The Littlewood-Paley condition guarantees that the dyadic wavelet transform is a unitary operator and therefore invertible. Orthogonal wavelet bases are constructed using a structure of nested vector spaces known as multiresolutions. These multiresolutions are linked to the existence of filters that govern transitions from one scale to another, and enable the implementation of the fast wavelet transform algorithm.
The nonlinear approximation of a signal in an orthogonal wavelet basis is obtained by keeping only the large wavelet coefficients. This means building an approximation that adapts to the local regularity of the signal. The decay of a signal's approximation error is related to its Lipchitzian regularity. Applications are shown for signal and image denoising, by thresholding wavelet coefficients. This makes it possible to restore regular image areas and contours.
