Answer 0: ask yourself if you really need wavelets

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. For the first one, the analysis wavelet is less regular, but its moments nicely concentrate piecewise regular part 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.

Answer 0: ask yourself if you really need wavelets

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.

Answer 0: ask yourself if you really need wavelets

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. For the first one, the analysis wavelet is less regular, but its moments nicely concentrate piecewise regular part 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.

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 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.

Answer 0: ask yourself if you really need wavelets

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. For the first one, the analysis wavelet is less regular, but its moments nicely concentrate piecewise regular part 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.