As we gradually transit from first-generation to second-generation wavelets, the style of thinking varies. In Daubechie’s construction, we heavely used algebraic tricks and motivations such as the polynomial of some special property.Despite the elegance of such construction, it turns out that there are alternative approaches which rely on the language of geometry. Geometry is usually more intuitive (at least to some people including myself) and easier to be connected to objects in the real world (nevertheless we live in a geometric world while algebra is more abstract). Lifting-based construction enjoys remarkable simplicity and generality (it is highly recommended you read through the tutorial article athome.pdf posted in the course homepage – only high-school mathematics is required).
The fascinating aspect of science is that many seemingly different things are in fact connected at a fundamental level. The algebraic and geometric aspect of wavelets represent two seemingly different routes but are intrinsically related (Daubechies and Sweldens have shown any known wavelet can be factorized into a finite of lifting steps by a variation of Euclid’s algorithm). Such unification of two different languages is analogous to the unification of two descriptions (matrix vs. wave) of quantum mechanics. Probably the most intriguing (and still open) unification is the language of determistic and stochastic – as put by A. Einstein, does God play dice or not? It is plausible that both statistical and determinstic descriptions can reveal the truth though it will take another Eintein to show us how they are equivalent (or at least connected in some way).
