III.98 Wavelets

If you wish to send a black and white picture from one computer to another, then an obvious way of doing it is to encode it pixel by pixel: 0 for black and 1 for white. However, for certain pictures this would obviously be extremely inefficient. For instance, if the picture is a square, the left half of which is entirely white and the right half of which is entirely black, then it is clearly much better to send a set of instructions for reconstructing the picture than a list of every single pixel. Furthermore, the precise details of the pixels usually do not matter: if you want a patch of gray, then it is enough to put in black and white pixels in the right proportion and make sure that they are evenly distributed.

However, finding a good way of encoding pictures is difficult, and an important area of research in engineering. A picture can be thought of as a function from a rectangle to . The set of all such functions forms a VECTOR SPACE [I.3 §2.3], and a natural way to try to come up with a good encoding is to find a good basis for this space. Here “good” means that the functions one is interested in (that is, ones that correspond to the kinds of pictorial representations one might wish to send) are determined by just a few of their coefficients, apart from minor variations that are not detectable by the human eye.

Wavelets are a particularly good basis for many purposes. In some ways they are like FOURIER TRANSFORMS [III.27], but they are much better suited to encoding details such as sharp boundaries, and patterns that are “localized,” rather than spread throughout the picture. For more details, see WAVELETS AND APPLICATIONS [VII.3].