Soon after they were introduced, it was realized that wavelets offered representations of signals and images of interest that are far more sparse than those offered by more classical representations; for instance, Fourier series.
Owing to their increased spatial localization at finer scales, wavelets prove to be better adapted to represent signals with discontinuities or transient phenomena because only a few wavelets actually interact with those discontinuities. It turns out that sparsity has extremely important consequences and this lecture will briefly discuss three vignettes.
- First, enhanced sparsity yields the same quality of approximation with fewer terms, a feat which has implications for lossy image compression since it roughly says that fewer bits are needed to achieve the same distortion.
- Second, enhanced sparsity yields superior statistical accuracy since there are fewer degrees of freedom or parameters to estimate. This gives scientists better methods to tease apart the signal from the noise.
- Third, enhanced sparsity has important consequences for data acquisition itself: a new technique known as compressed sensing is turning a few fields a bit upside down for it effectively says that to make a high-resolution image we need to collect far fewer samples than were thought necessary.