Linearized Riesz transform and quasi-monogenic shearlets

S. Häuser, B. Heise, G. Steidl
2014 International Journal of Wavelets, Multiresolution and Information Processing  
The only quadrature operator of order two on L 2 (R 2 ) which covaries with orthogonal transforms, in particular rotations is (up to the sign) the Riesz transform. This property was used for the construction of monogenic wavelets and curvelets. Recently, shearlets were applied for various signal processing tasks. Unfortunately, the Riesz transform does not correspond with the shear operation. In this paper we propose a novel quadrature operator called linearized Riesz transform which is related
more » ... to the shear operator. We prove properties of this transform and analyze its performance versus the usual Riesz transform numerically. Furthermore, we demonstrate the relation between the corresponding optical filters. Based on the linearized Riesz transform we introduce finite discrete quasimonogenic shearlets and prove that they form a tight frame. Numerical experiments show the good fit of the directional information given by the shearlets and the orientation obtained from the quasi-monogenic shearlet coefficients. Finally we provide experiments on the directional analysis of textures using our quasi-monogenic shearlets. Figure 1 : (left to right) Tilings of the frequency plane by the support of the (i) second generation curvelets [3] (4 · 2 j/2 wedges at scale 2 −j ), (ii) discrete curvelets [4] with pseudopolar support, (iii) cone-adapted shearlets [11, 13] . Recently the monogenic curvelet transform was introduced by Storath [26]. He proved that this transform behaves at very fine scales like the usual curvelet transform and at coarse scales like the monogenic wavelet transform. Since the Riesz transform covaries with rotations, the monogenic curvelet amplitude is invariant to the rotation group operation of the curvelet transform. This is useful as long as the curvelet transform is defined with respect to curvelets which supports tile the frequency plane into concentric circle segments as originally proposed for second generation curvelets by Candés and Donoho in [3], see Fig. 1 left. However, in implementations Cartesian arrays are prefered over the polar tiling. The fast discrete curvelet transform [4] replaces the polar tiling of the frequency plane by Cartesian coronae based on concentric squares (instead of circles) and shears [4], see Fig. 1 middle. Of course, these curvelets cannot be obtained by rotating a mother curvelet on the respective scale. For a review on curvelets see [20] . An approach which takes Cartesian arrays from the beginning into account are shearlets which were introduced in [19]. Instead of rotations the group of shear matrices is involved into the construction. In this paper we focus on the cone adapted version of shearlets given in [11] . The tiling of the frequency plane by the support of these shearlets is shown in Fig. 1 right. Shearlets have the same nice approximation properties as curvelets and have found applications, e.g., for edge detection in [12, 17] . Moreover, shearlets as wavelets are directly related with square integrable group representations [5] and function spaces [6] . Of course the Riesz transform does not covaries with the shear operation.
doi:10.1142/s0219691314500271 fatcat:tsmaqj2pbjhxtdyyaclx6r75xy