Wavelets and Edge Detection CS698 Final Project
Submitted To: Professor Richard Mann Submitted By: Steve Hanov Course: CS698 Date: April 10, 2006
Wavelets have had a relatively short and troubled history. They seem to be forever confined to footnotes in textbooks on Fourier theory. It seems that there is little that can be done with wavelets that cannot be done with traditional Fourier analysis. Stephane Mallat was not the father of wavelet theory, but he is certainly an evangelist. His textbook on the subject, A Wavelet Tour of Signal Processing , contains proofs about the theory of wavelets, and a summation about what is known about them with applications to signal ...view middle of the document...
For most signals, this is not the case. Consider music, which is continuously varying in pitch. Fourier analysis done on the entire song tells you which frequencies exist, but not where they are. The short time Fourier transform (STFT) is often used when the frequencies of the signal vary greatly with time.  In the JPEG image encoding standard, for example, the image is first broken up into small windows with similar characteristics. The Fourier transform is not applied to the entire image, but only to these small blocks. The disadvantage of this technique can be seen at high compression ratios, when the outlines of the blocks are clearly visible artifacts. A second disadvantage is in resolution of analysis. When larger windows are used, lower frequencies can be detected, but their position in time is less certain. With a smaller window, the position can be determined with greater accuracy, but lower frequencies will not be detected. The wavelet transform helps solve this problem. Once applied to a function f(t), it provides a set of functions Wsf(t). Each function describes the strength of a wavelet scaled by factor s at time t. The wavelet extends for only a short period, so its effects are limited to the area immediately surrounding t. The wavelet transform will give information about the strengths of the frequencies of a signal at time t. In the first pages of his treatise , Mallat defines a wavelet as a function of zero average,
THEORY It is best to describe wavelets by showing how they differ from Fourier methods. A signal in the time domain is described by a function f(t), where t is usually a moment in time. When we apply the Fourier transform to the signal, we obtain a function F(ω) that takes as input a frequency, and outputs a complex number describing the strength of that frequency in the original signal. The real part is the strength of the cosine of that frequency, and the imaginary part is the strength of the sine. One way to obtain the Fourier transform of a signal is to repeatedly correlate the sine and cosine
ψ (t )dt = 0
which is dilated with scale parameter s, and translated by u:
ψ u , s (t ) =
Unlike the sine and cosine functions, wavelets move toward quickly zero as their limits approach to +/-∞. In , Mallat notes that the derivative of a smoothing function is a wavelet with good properties. Such a wavelet is shown in Figure 1.
1 0.9 0.8
0.2 0.4 0.3
0.7 0.6 0.5 0.4 0.3
-0.2 0.1 0 -0.1
0.2 0.1 0
Figure 1: A smoothing function, and its corresponding wavelet.
By correlating the signal with this function at all possible translations and scales, we obtain the continuous wavelet transform. The transformation also increases the dimension of the function by one. Since we have both a scaling and position parameter, a 1-D signal will have a 2D wavelet transform. As an...