Continuous curvelet transform

Abstract

We discuss a Continuous Curvelet Transform (CCT), a transform fΓf(a,b,θ) of functions f(x1,x2) on R2 into a transform domain with continuous scale a>0, location bR2, and orientation θ0,2π). Here Γf(a,b,θ)=f,γabθ projects 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 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 a0 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 a0 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 a0; 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)2 da/ a3+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. 2005 Elsevier Inc. All rights reserved.