Linearized Riesz transform and quasi-monogenic shearlets

The only quadrature operator of order two on L2(R) 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 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.