Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas.
The main motivation behind the development; of wavelets and the many related ideas (see Figure 1.1) was the search (or fast algorithms to compute compact representations of functions and data sets. How can such compact representations be achieved? There are many approaches, some more computationally intensive, others less, but they all amount to exploiting structure in the data or underlying functions. Depending on the application area this goes by different names such as the exploitation of "structure," "smoothness," "coherence," or "correlation." Of course, for purely random signals or data no compact; representations can be found. But most of the time we are interested in realistic data and functions which do exhibit; some smoothness or coherence. In these cases wavelets and the fast wavelet transform turn out to be very useful tools.