# What are the known use cases for different wavelet families?

There are numerous wavelet families that differ by a number of parameters (like the number of vanishing moments, symmetry, etc.).

• What are the known use cases where one particular family is known to be more suitable and provide better results?

For example, Daubechies 9/7 are used in the JPEG2000 image format. As far as I understand Daubechies wavelets were designed specially to have a high number of vanishing moments so that makes them more suitable for signal compression tasks. What are the known use cases for other families of wavelets?

And as second part:

• How does one decide which one is the most suitable for it's task?
• Are there any known heuristics, common knowledge, etc?
• Hi Alexander, welcome. Your question is IMHO too broad. That's like going to a mechanical engineering site and asking what the different use cases for different kinds of mechanical joints are. There's far too many to give you a complete listing, and for those that are common, we'd just point you to textbook examples. So maybe you'd want to explain why you're asking for "known use cases"; this feels like you're writing some report as homework, and we'll probably be more happy to help you find good ressources than to give you a list that you can drag&drop into your document. – Marcus Müller Feb 17 '17 at 9:22
• I have been long gathering good practices and caveats on using wavelets. Still, I cannot tell which could be a best. One cannot easily separate the transformation by itself from the subsequent processing. – Laurent Duval Feb 17 '17 at 9:28
• The issue of "What is the best wavelet" has been coming up frequently and in various forms with lots of great respones, on this SE site, you might want to have a look at those too. – A_A Feb 17 '17 at 9:57

Say yes. Let us concentrate on 2-band real discrete wavelet first. JPEG 2000 is a special case, where CDF9/7 and 5/3 biorthogonal wavelets are used. For the first one, the analysis wavelet is moderately regular, but its moments nicely concentrate piecewise regular parts of the data. The synthesis wavelet is smoother, and visually compensates quantization artifacts. The 5/3 works similarly, with the addition of exact computations on integers. There are almost symmetric, which is somehow better for edge location. The paper Mathematical Properties of the JPEG2000 Wavelet Filters gathers some of the most notable.

Answer 0.5. apart from that example, there is very few "fit-for-all signals" optimality results.

The quality of wavelets depends a lot on the nature of the data: signals, images, on the task, on the resource constraints. But the most important is the mastery of the wavelets you use. Because most often, you are trying to enhance or restore data, but the target is unknown, and quality metrics are often imperfect, so you can only get informed guesses on simulated datasets. Theoretical properties are guides, but you need models for signals or noise at least (eg spline wavelets with a fractional Brownian noise).

Orthogonal wavelets, says Daubechies', are widely used because orthogonality is useful to prove stuff, and their "shortest" support interval for given moments is nice. On natural signals, a rule of thumb is that it is rare to observe polynomial parts above degree 3. Hence short DB should suffice. However, longer ones are often used to compensates for the stronger asymmetry of the shortest.

The (overall) symmetry of the wavelet is also observed: anti-symmetric to act like derivatives, symmetric to mimic Laplacians.

There is a no-go theorem of discrete 1D 2-band wavelets: except for Haar, they cannot be real, orthogonal with symmetry and finite support. Haar seems crude, but it is very fast, and since you can almost compute everything (delays, etc.) it can do a marvelous job.

But if you go deeper, you can find that $M$-band wavelets $M=4$ for instance, can have all the required properties, with more degrees of freedom. And they have better aliasing cancelation between channels, which makes them better at textures, for denoising.

Answer 1: a good knowledge of wavelets properties restricts the candidate list, and a little trial-and-error is often unavoidable

But you can also use a set of different wavelets to enhance the result of a single one: perform the processing with each wavelet, and (wisely) combine the result. One of the most efficient technique is based on a pair of Hilbert-related wavelets: take any wavelet basis you like (for its properties), and build a second basis with an (approximate) Hilbert transform relationship. The resulting primal and dual wavelets share the same moment property, have an opposite symmetry, and form together a dual-tree wavelet transform, that provides almost shift invariance at little cost. But you can also use different shifted versions (time-invariant wavelets), symmetrized combinations, etc.

Answer 2: a union of wavelet bases is often a powerful combination.