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In my studies of wavelets, there appear to be 3 different families of them:

  1. The Continuous wavelet transform
  2. The Discrete wavelet transform
  3. The Redundant wavelet transform

They are all based on the same concept, but vary in how they are shifted/scaled, and/or decimated or not at every scale level.

My question is, where does each type find utility? For example, why would I want to use the Redundant Wavelet Transform over the Discrete Wavelet Transform, over the Continuous wavelet Transform in a particular application?

What advantages/disadvantages does one type of transform have over another, as far as its applicability is concerned?

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  • $\begingroup$ We had this discussion not long ago. I don't have time to answer in full at the moment, but take a look at the discussion here: dsp.stackexchange.com/questions/7626/… $\endgroup$ – user2718 Jan 28 '13 at 20:17
  • $\begingroup$ @BruceZenone In that question I was asking about terminology, here I am inquiring as to specific applicability of the above transforms. Example: Let us say we wanted to compress a signal. Why use the CWT over the DWT over the RWT? etc. This is a question about applicability of the transforms in specific applications. $\endgroup$ – Spacey Jan 28 '13 at 20:45
  • $\begingroup$ Got it. I'll let someone else chime in. I'd like to see more perspective on this question as well. $\endgroup$ – user2718 Jan 29 '13 at 0:01
  • $\begingroup$ You may like to read the book "The World According to Wavelets" by Barbara Hubbard for a more historical and application oriented treatment of the subject. $\endgroup$ – user2718 Jan 29 '13 at 13:09
  • $\begingroup$ @BruceZenone That looks like a very interesting book, Ill make sure to procure it! :-) $\endgroup$ – Spacey Jan 29 '13 at 13:44
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The denomination you propose (Continuous wavelet transform, Discrete wavelet transform, Redundant wavelet transform) might be not precise enough. Yet here are the major differences.

Continuous wavelet transform: per se, they are a pure mathematical tool. You can use almost any wavelet you wish, with very little requirements. There exist discretizations, and they could be very redundant (roughly $10-100\times$ the original signal per dimension). You mostly use it on 1D signals due to the redundancy, but it is sometimes used in 2D and 3D with limited scales. It is used when one looks as fine details of the signals: fractal aspects, measures of regularity, precise onset or rupture detection in the signal's shape (continuity, changes in derivatives), adaptive filtering. One can just observe (analysis) or detect on 2D scalograms, or select an inverse transform (synthesis,), for instance for adaptive filtering.

Discrete wavelet transform: one only have a limited number of wavelets available. Often used in orthogonal or biorthogonal fashion, edp' when critical sampling or orthogonality is important: compression, statistical analysis, or when memory or computation is too expensive. It may induce wavelet-related artifacts with low-amplitude signals. Requires some technical skills to be used in detection.

Redundant wavelet transform: a mix between the two above. Limited choice in the wavelet, yet intermediate redundancy (roughly $1,5-10\times$ per dimension) and more important some invariance: translation, rotation, shear invariance, that is useful for detection, and often more robust to noises. Often used in restoration (deblurring, deconvolution, denoising), feature analysis.

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