I wish all assignments were this nice. Particularly for a non-mathematician guy like me, this text provides a very useful description on why wavelet theory is important and why I'm exchanging my free time to learn about it. You may want take a look on the full paper here.
The author goes by several interesting examples in order to show a concise history of the wavelet theory and applications. The main highlight of her paper is the comparison of the wavelet analysis with the Human perception of its surroundings. Revisiting the author's example, the Human perception allows us to go from a wide-scale overview perspective up to a very fine-grained detailed perspective. When we are overlooking a forest from a jet flying over 2000 feet above the ground, we can barely see any trees, we just see a very long green "carpet". As we get closer, we realize that such "carpet" is made of tress and if we go even closer we will see branches, leaves, etc. This means that the Human perception allows us to visit whatever granularity we need. However, cameras and/or computers do not have such capability. Digital images only represent one perspective at a time. Getting closer to a digital picture will not give you more than a blurred view of the picture.
According to the author, wavelet is one attempt of representing such different levels of granularities on mathematical models. This is the kind of theory that has been so useful that people all around the world have been using variations of it for a very long time way before it got formalized and known as wavelets.
It is very likely that you have been using wavelets for quite a long time without even knowing it. The jpeg2000 (very popular picture format on many digital camera brands) uses wavelets as one of the steps to compress the image files. On the specific case of jpeg2000, wavelets have the capability of splitting what we consider to be the objects on the scene (face, car, etc) of the background information (usually blur or fuzzy image). By doing that, it generates two stream of numbers (pixel values and variation) that can be compressed in up to 200:1 scale.
As a relatively new concept, the wavelet theories are growing in number and complexity in a very fast pace. Several different areas are still evaluating the best way to take advantage of the wavelet features. The most important is that in the case of wavelets, the theory and the practical applications seems to be evolving together. Not only evolving the mathematical foundations but also the several applications that certainly will bring major breakthroughs to several areas of science.