Continuous Curvelet Transform: I. Resolution of the Wavefront Set

We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R , into a transform domain with continuous scale a > 0, location b ∈ R, 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 → 0 if f is smooth near x0, or if the 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 Hwavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S(x, θ) = (∫ |Γf (a, x, θ)| da a3+2m )1/2 is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (FourierBros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set.