Wavelets in Clifford Analysis

Clifford analysis is a higher dimensional functions theory for the Dirac operator and builds a bridge between complex function theory and harmonic analysis. The construction of wavelets is done in three different ways. Firstly, a monogenic mother wavelet is obtained from monogenic extensions (Cauchy–Kovalevskaya extensions) of special functions like Hermite and Laguerre polynomials. Based on the kernel function, Cauchy wavelets are also monogenic but not square integrable in the usual sense. On the other hand, these wavelets and their kernels are connected to the Cauchy–Riemann equations in the upper half space as well as to Bergman and Hardy spaces. Secondly, a group theoretical approach is used to construct wavelets. This approach considers pure dilations and rotations as group actions on the unit sphere. It can be generalized by using the action of the Spin group because the Spin group is a double cover of the rotation group, whereas dilations arise from Möbius transformations. Here, Clifford analysis gives the tools to construct wavelets. Finally, an application to image processing based on monogenic wavelets is considered. Here, the starting point are scalar-valued functions and the resulting Clifford wavelets are boundary values of monogenic functions in the upper half space. One proceeds in two steps. First choose a real- or complex-valued primary wavelet and then construct from that using the Riesz transform = Hilbert transform Clifford wavelets and Clifford wavelet frames.