ABSTRACT
In the language of wavelets, the scaling function is dilated to make an image of the signal at half resolution; in the language of signal processing, a low-pass filter is applied to the signal, and the result is subsampled. (Flouting etymology, signal processing texts speak of "decimating" the signal by a factor of two: taking one sample out of two.)
Since at each step we take only half as many samples as before, it doesn't take long before the signal is reduced to nothing, or almost nothing. The trick is that we don't lose anything: the information encoded by the wavelets is precisely the information subtracted from the signal when we "decimate" it. We can retrace our steps and find the original signal, adding to the picture of the signal at one resolution the information encoded by the wavelets at that resolution, in order to get the image of the signal at a resolution twice as fine.
