In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ3, wavelets in ℝn (n > 3), and wavelets on the 2-sphere S2. Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis.

In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform.

Three-dimensional wavelets

Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].