Continuous curvelet transform

We discuss a Continuous Curvelet Transform (CCT), a transform f↦Γf(a,b,θ) of functions f(x1,x2) on R2 into a transform domain with continuous scale a>0, location b∈R2, and orientation θ∈[0,2π). Here Γf(a,b,θ)=〈f,γabθ〉 projects f onto analyzing elements called curveletsγabθ 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 [E.J. Candès, F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Process. 82 (2002) 1519-1543; E.J. Candès, L. Demanet, Curvelets and Fourier integral operators, C. R. Acad. Sci. Paris, Sér. I (2003) 395-398; E.J. Candès, D.L. Donoho, Curvelets: a surprisingly effective nonadaptive representation of objects with edges, in: A. Cohen, C. Rabut, L.L. Schumaker (Eds.), Curve and Surface Fitting: Saint-Malo 1999, Vanderbilt Univ. Press, Nashville, TN, 2000]. 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 show that decay properties of Γf(a,x0,θ0) for fixed (x0,θ0), as a→0 can precisely identify the wavefront set and the Hm-wavefront 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 Hm-wavefront set is the closure of those points (x0,θ0) where the 'directional parabolic square function' Sm(x,θ)=(∫|Γf(a,x,θ)|2daa3+2m)1/2 is not locally integrable. The CCT is closely related to a continuous transform pioneered 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 (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set.