IX - An approximation tour

Abstract

In signal processing, orthogonal bases are of interest because they can efficiently approximate certain types of signals with just a few vectors. Two examples of such applications are image compression and the estimation of noisy signals. Approximation theory studies the error produced by different approximation schemes in an orthonormal basis. A linear approximation projects the signal over M vectors chosen a priori. In Fourier or wavelet bases, this linear approximation is particularly precise for uniformly regular signals. However, better approximations are obtained by choosing the M basis vectors depending on the signal. Signals with isolated singularities are well approximated in a wavelet basis with this non-linear procedure. A further degree of freedom is introduced by choosing the basis adaptively, depending on the signal properties. From families of wavelet packet bases and local cosine bases, a fast dynamical programming algorithm is used to select the best basis that minimizes a Schur concave cost function. The approximation vectors chosen from this best basis outline the important signal structures, and characterize their time-frequency properties. Pursuit algorithms generalize these adaptive approximations by selecting the approximation vectors from redundant dictionaries of time-frequency atoms, with no orthogonality constraint.