Abstract. Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

We define a set of operators that localise a radial image in radial space and radial frequency simultaneously. We find the eigenfunctions of this operator and thus define a non-separable orthogonal set of radial wavelet functions that may be considered optimally concentrated over a region of radial space and radial scale space, defined via a doublet of parameters. We give analytic forms to their energy concentration over this region. We show how the radial function localisation operator can be generalised to an operator, localising any L 2 (R 2) function. We show that the latter operator, with an appropriate choice of localisation region, approximately has the same eigenfunctions as the radial operator. Based on the radial wavelets we define a set of quaternionic valued wavelet functions that can extract local orientation for discontinuous signals and both orientation and phase structure for oscillatory signals. The full set of quaternionic wavelet functions are component wise orthogonal; hence their statistical properties are tractable, and we give forms for the variability of the estimates of the local phase and orientation, as well as the local energy of the image. By averaging estimates across wavelets, a substantial reduction in the variance is achieved.

Abstract. This study is concerned with the uncertainty principles which are related to the Weyl-Heisenberg, the SIM(2) and the Affine groups. A general theorem which associates an uncertainty principle to a pair of self-adjoint operators was previously used in finding the minimizers of the uncertainty principles related to various groups, e.g., the one and twodimensional Weyl-Heisenberg groups, the one-dimensional Affine group, and the two-dimensional similitude group of IR 2, SIM(2) = IR 2 ×(IR + × SO(2)). In this study the relationship between the affine group in two dimensions and the SIM(2) group is investigated in terms of the uncertainty minimizers. Moreover, we present scale space properties of a minimizer of the SIM(2) group. 1

Mathematical methods for time series analysis and digital image processing