Continuous Curvelet Transform: I. Resolution of the Wavefront Set
Emmanuel J. Candès, David L. Donoho · CaltechAUTHORS (California Institute of Technology) · 2003
We discuss a Continuous Curvelet Transform (CCT), a transform f ↦ → Γf (a, b, θ) of functions f(x1,x2) onR 2,into atransform domain with continuous scale a>0, location b ∈ R 2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) =〈f,γabθ 〉 where the inner products project f onto analyzing elements called curvelets γabθ which are smooth and of rapid decay away from an a by √ a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track ’ the behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0,θ0), Γf (a, x0,θ0) decays rapidly as a → 0iff is smooth near x0, orifthe singularity of f at x0 is oriented in a different direction than θ0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0,θ0) for fixed (x0,θ0), as a → 0 can precisely identify the wavefront set and the H m-