6 – WAVELET TRANSFORMS

Publisher Summary This chapter discusses wavelet transforms. The wavelet transform is a projection of a signal onto a series of basic functions called the wavelet basis. Many of the signals that one deals with in computer graphics are nonstationary. They possess a lot of high-frequency information in some places, and little of it in others. This is the reason why the techniques of adaptive super sampling have been developed and work so well. In regions of an image with a lot of edges, textures, and shadows, one needs to sample densely to account for the high frequencies in that region. If the image has a single flat color for a background, then one can sample sparsely. Wavelets form a 2D family of functions that are derived from an original function, ν called the scaling function. From the scaling function, one mother wavelet is created, and all the other wavelets spring from scaled, dilated, and shifted versions of that mother wavelet. Wavelets can form an orthonormal basis and can be implemented quickly using a fast wavelet transform analogous to the fast Fourier transform. Wavelets are proving to have great value in computer graphics where one often encounter signals that are mostly smooth, but contain important regions of high-frequency content.