Construction of regular and irregular shearlet frames

In this paper, we study the construction of irregular shearlet systems, i.e., systems of the form , where ψ ∈ L²(ℝ²), Λ is an arbitrary sequence in ℝ₊ × ℝ × ℝ², A_a is a parabolic scaling matrix and S_s a shear matrix. These systems are obtained by appropriately sampling the Continuous Shearlet Transform. We derive sufficient conditions for such a discrete system to form a frame for L²(ℝ²), and provide explicit estimates for the frame bounds. Among the examples of such discrete systems, one is the Parseval frame of shearlets previously introduced by the authors, which is optimal in approximating 2-D smooth functions with discontinuities along C²-curves. This study provides the framework for the construction of a variety of discrete directional multiscale systems with the ability to detect orientations inherited from the Continuous Shearlet Transform.