If I’m understanding your comment correctly, wavelets are a kind of discrete and/or finite quantization of the “full” infinite Fourier transform, by way of using more complex “basis vectors” than pure sine waves?
Very cool and thanks for the reading recommendation!
PM_ME_VINTAGE_30S@lemmy.sdf.org 5 days ago
The second part is basically correct, but the first part needs a little bit of explanation.
So depending on what you need to do, you can actually use continuous-time (or continuous-space) wavelet transforms, or discrete-time (or discrete-space) wavelet transforms. The continuous-time wavelet transform is, practically, “just as exact” as the continuous-time Fourier transform. So instead of being “better” or “lesser” than the Fourier transform, it’s really a “different perspective” on the same space of signals by choosing a different set of “basis” vectors [1].
Also, wavelets often are “more complicated” than sine waves, but not necessarily. In fact, one of the first wavelets discovered was this, the Haar wavelet:
Image
To be completely clear: this waveform is defined for any real number; it’s not sampled, and it is not a quantized version of some “better” wave! It is just 1 for any inputs between 0 and 1/2, -1 for any inputs between 1/2 and 1, and 0 everywhere else [2]. Although this wavelet happens to have a finite range (so not even countably infinite, you get {-1,0,1} and you don’t get upset), if you slap enough of these things together (possibly infinity of them), you can get back any “reasonable” waveform, where “reasonable” is precisely defined in Mallat’s book (it’s L^2® if you’ve been exposed to L^p spaces).
Hope you enjoy the book as much as I have!
[1] “Basis” is in quotes because it depends on what you mean by “basis”. Typically, a “basis” in linear algebra means that you need to be able to exactly recover any element in the vector space with a weighted finite sum of the bases — a Hamel basis. In signal processing, in particular in Mallat’s book, we typically extend the notion of basis to allow for infinite linear combinations with limits. This means that we need to choose a topology, which is absolutely a reasonable requirement in signal processing, but not necessarily in “pure” linear algebra. I believe that the definition for “(orthonormal) basis” in Mallat’s book (in Appendix A) is called a(n orthonormal) Schauder basis in other parts of applied math.
By contrast, “vectors” is not in quotes above because, using the “generic” definition of a vector space, the basis elements are indeed vectors, i.e. members of a space that you “cannot leave” by scaling or adding finite numbers of the basis elements.
Lastly, “signals” and “vectors” are mostly interchangeable within the signal processing discipline. In signal processing, we typically assume that signals have been given an inner product (“correlation”, “dot product”), therefore a norm (“length”) from the inner product, and therefore a topology (“abstract geometry”) from the norm. I.e., “signals” in signal processing usually have more structure than “vectors” in applied math.
[2] The values of the Haar wavelet at exactly {0,1/2,1} are in the plot. However, since continuous wavelet transforms are integral transforms, the values at the points {0,1/2,1} can be changed to whatever you want as long as it’s finite. Rigorously, the Lebesgue integrals in the wavelet transform definitions are “blind to” a “small” (measure zero) set like {0,1/2,1}. From a signal processing perspective, changing the signal only on “small” sets like {0,1/2,1} is not enough to change the signal energy.)
I think there is a reason why the choice of values at {0,1/2,1} given in the plot makes sense. Mathematically, the choice makes the Haar mother wavelet right-continuous and upper semi-continuous, but I can’t remember off the top of my head why this is helpful for applications.