There is a wide range of applications for function systems on $S^3$ , among them X-Ray diffraction tomography whose mathematical model is given by the spherical X-ray transform. Here, one needs to reconstruct the so-called orientation density function which is well-localized function on $S^3$ . To approximate such a function we discuss the construction of Wavelet and Gabor frames on the three-sphere. In both cases we use the representation of the corresponding group (Lorentz group in the case of the wavelet transform and Euclidean group in the case of the Gabor transform). While this provides us with the continuous transforms in reality we need a discrete version. To this end we will use co-orbit space theory to construct wavelet and Gabor frames. Here we will show the difference in the construction of both cases. To illustrate the applicability of our frames we present an algorithm for the inversion of the spherical X-Ray transform