Restoration of a signal contaminated by additive white Gaussian noise is a classical problem in signal processing which dated back as early as Wiener’s filter solution. Despite the optimality of Wiener filtering for stationary Gaussian processes, most signals in real worlds are characterized by nonstationarity and non-Gaussianity. That is why signal modeling plays an essential role in any denoising algorithms.
Wavelets provide a powerful tool for characterizing the class of transient signals. Transient properties of signals give rise to singularities – the opposite to smoothness and continuity. In mathematics, Sobolov-space functions are often used to model functions satisfying strong smoothness constraint. However, the smoothness constraint of Sobolov space is so strong that it does not allow any exception – e.g., sudden change of intensity values (discontinuities). Besov-space functions are a more suitable model because they allow the exceptions to some degree (e.g., following an exponential decay). This is in contrast to non-smooth functions (e.g., AWGN) which generate exceptions almost everywhere. It turns out wavelet bases are the most appropriate choices for approximating Besov-space functions, which of course is due to their good localization property in both time and frequency. Such popular view of how wavelet decomposition distinguishes signals from noise has been prevelant in the past decade.
In my own opnion, the biggest “oversight” by wavelets is that it focuses on variation (or transience). Invariant property is also fundamental to out understanding of signals. Imagine Donoho’s famous toy example (blocks), step-wave functions indeed are the worst for describing the transience under Fourier bases (remember the celebrated Gibbs phenomenon?). However, when you look at all transiences as a whole set – some kind of global invariance appears as a striking phenomenon. Step-wave functions can have arbitrary contrast and positions, but they are all self-similar to each other. If one wants to identify the fundamental principles underlying natural signals, self-organization contributes to both transient and invariant properties – the boundary between these two could be rather vague. What we need is a theory that can handle both (wavelet clearly can only handle the former not the latter).
