Wavelets

A wavelet is a square integrable function whose translations and dilations form an orthonormal basis for the Hilbert space L₂ (ℜ^N). To be more precise, let us consider the wavelet functions in Euclidean space ℜ. A wavelet function f has such a property letting f_{m, n}(x) = 2^m/2f(2^mx-n) for all integers m, n, f is an orthonormal basis for L₂(ℜ). That is, a wavelet function is one function such that it can generate an orthonormal basis for L₂(ℜ). That is one of the reasons wavelets are interesting and useful for computation. Wavelets have several extremely importune applications in signal and image processing. For example, edge detection. The following are examples of images and "edges".

Click here to see how well we can compress images using the wavelets we constructed.
To know more about wavelets, there are several webpages available, check them out: Wavelet Digest and Wavelet Idr.