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By Shakuntala Devi

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Then ψ is called a classical shearlet. Thus, a classical shearlet ψ is a function which is wavelet-like along one axis and bump-like along another one. The frequency support of a classical shearlet is illustrated in Fig. 3a. Notice that there exist several choices of ψ1 and ψ2 satisfying conditions (10) and (11). One possible choice is to set ψ1 to be a Lemari`e–Meyer wavelet and ψˆ 2 to be a spline (cf. [22, 31]). Introduction to Shearlets 19 a b Support of the Fourier transform of a classical shearlet.

Labate Definition 3. For ψ ∈ L2 (R2 ), the Continuous Shearlet Transform of f ∈ L2 (R2 ) is the mapping L2 (R2 ) f → SH ψ f (a, s,t) = f , σ (a, s,t)ψ , (a, s,t) ∈ S. Thus, SH ψ maps the function f to the coefficients SH ψ f (a, s,t) associated with the scale variable a > 0, the orientation variable s ∈ R, and the location variable t ∈ R2 . Of particular importance are the conditions on ψ under which the Continuous Shearlet Transform is an isometry, since this is automatically associated with a reconstruction formula.

We start our discussion by examining sufficient conditions for the existence of cone-adapted discrete shearlet systems which are compactly supported and form a frame for L2 (R2 ). These conditions can be derived by extending the classical tq equations from the theory of wavelets to this situation (cf. [47]). Before stating the main result, let us first introduce the following notation. Introduction to Shearlets 29 For functions φ , ψ , ψ˜ ∈ L2 (R2 ), we define Θ : R2 × R2 → R by Θ (ξ , ω ) = |φˆ (ξ )||φˆ (ξ + ω )| + Θ1(ξ , ω ) + Θ2(ξ , ω ), where Θ1 (ξ , ω ) = ∑ ∑ ψˆ (SkT A2− j ξ ) ψˆ (Sk T A2− j ξ + ω ) ∑ ∑ ψˆ˜ (Sk A˜ 2− j ξ ) ψˆ˜ (Sk A˜ 2− j ξ + ω ) .

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