本文探讨了SVD（奇异值分解）和小波变换在信号近似方面的核心区别，强调了小波变换在处理具有高度局部特征或空间变异性的信号时的优越性。
T
he main difference between the two is that wavelet transforms use a wavelet basis while SVD uses an
eigenfunction
basis like the Fou
rier basis. They both offer the same functionality i.e. approximation of signals, and hence appear to resemble each other. But their overall properties are different.
A wavelet basis is desirable, when the signal you are trying to decompose has highly localized features or spatial variability such as edges in images or point singularities in a time series. It is well known that linear approximations are sub-optimal for representin g signals that aren't smooth, i.e. don't lie in ellipsoids. As Laurent Duval pointed out, wavelets form a non-linear approximation of a signal and are hence outperform linear approximations for such signals. The wavelet-vagueluette decomposition is an SVD like decomposition thats uses the wavelet basis and offers better approximation of the signal given localized features. [1] offers a better introduction to the usefulness of wavelets and wavelet-vaguelette decompositions as well as additional references.
A wavelet basis is desirable, when the signal you are trying to decompose has highly localized features or spatial variability such as edges in images or point singularities in a time series. It is well known that linear approximations are sub-optimal for representin g signals that aren't smooth, i.e. don't lie in ellipsoids. As Laurent Duval pointed out, wavelets form a non-linear approximation of a signal and are hence outperform linear approximations for such signals. The wavelet-vagueluette decomposition is an SVD like decomposition thats uses the wavelet basis and offers better approximation of the signal given localized features. [1] offers a better introduction to the usefulness of wavelets and wavelet-vaguelette decompositions as well as additional references.
[1] Wavelets and W.V.D 10 minute tour http://www.stanford.edu/~
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