"We present three recent developments in wavelets and subdivision: wavelet-type transforms that map integers to integers, with an application to lossless coding for images; rate-distortion bounds that realize the compression given by nonlinear approximation theorems for a model where wavelet compression outperforms the Karhunen-Loeve approach; and smoothness results for irregularly spaced subdivision schemes, related to wavelet compression for irregularly spaced data."
Baroness Ingrid Daubechies is a Belgian-American physicist and mathematician at Duke University. She is best known for her work with wavelets in image compression.
"The development of wavelets is an example where ideas from many different fields combined to merge into a whole that is more than the sum of its parts. The subject area of wavelets, developed mostly over the last 15 years, is connected to older ideas in many other fields, including pure and applied mathematics, physics, computer science, and engineering."
"Mathematicians have various ways of judging the merits of new theorems and constructions. One very important criterion is esthetic — some developments just “feel” right, fitting, and beautiful. Just as in other venues where beauty or esthetics are discussed, taste plays an important role in this, but I think I am not alone in being especially excited when apparently different fields suddenly meet in a new concept, a new understanding. It is often of the sparks of such encounters that our esthetic enjoyment of mathematics is born. Another important criterion for according merit to some particular piece of mathematics is the extent to which it can be useful in applications; this is the criterion almost exclusively used by nonmathematicians."
"... In their mathematical aspect, wavelets are rooted in the use of dilations and convolutions in Calderón-Zygmund theory in harmonic analysis. ... Algorithmically, wavelets are related to subband filtering in electrical engineering. Subband filtering was developed from the 70-s on; exact reconstruction procedures were discovered in the early 80-s. These were obviously fast algorithms, meant as a front-end processing step before encoding or compressing information in various types of signals. A lot of effort went into optimizing the filters for various applications, and this subfield of electrical engineering is now quite mature. ... Another algorithmic ancestor of wavelets are the multiple algorithms in numerical analysis, closer to mathematics, but still ad hoc."
"The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale. Forerunners of this technique were invented independently in pure mathematics (Calderóns resolution of the identity in harmonic analysis—see e.g., Calderón (1964), physics (coherent states for the (ax + b)-group in quantum mechanics, first constructed by Aslaksen and Klauder (1968), and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering (QMF filters by Esteban and Galland (1977), and later QMF filters with exact reconstruction property by Smith and Barnwell (1986), Vetterli (1986) in electrical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet (1983)). The last five years have seen a synthesis between all these different approaches, which has been very fertile for all the fields concerned."