Continuous Curvelet Transform -: I.: Resolution of the wavefront set

We discuss a Continuous Curvelet Transform (CCT), a transform f -> Gamma(f) (a, b, theta) of functions f(x(1), x(2)) on R-2 into a transform domain with continuous scale a > 0, location b is an element of R-2, and orientation theta is an element of [0, 2 pi). Here Gamma(f) (a, b, theta) = < f, gamma(ab theta)> projects f onto analyzing elements called curvelets gamma(ab theta) which are smooth and of rapid decay away from an a by root a rectangle with minor axis pointing in direction theta. 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. Candes, F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Process. 82 (2002) 1519-1543; E.J. Candes, L. Demanet, Curvelets and Fourier integral operators, C. R. Acad. Sci. Paris, Ser. I (2003) 395-398; E.J. Candes, 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 (x(0), theta(0)), Gamma(f) (a, x(0), theta(0)) decays rapidly as a -> 0 if f is smooth near x(0), or if the singularity of f at x(0) is oriented in a different direction than theta(0). Generalizing these examples, we show that decay properties of Gamma(f) (a, x(0), theta(0)) for fixed (x(0), theta(0)), as a -> 0 can precisely identify the wavefront set and the H-m-wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x(0), theta(0)) near which Gamma(f) (a, x, theta) is not of rapid decay as a -> 0; the H-m-wavefront set is the closure of those points (x(0), theta(0)) where the 'directional parabolic square function' S-m(x, theta) = (f vertical bar Gamma(f) (a, x, theta)vertical bar(2) da/a(3+2m)) (1/2) form 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 CC T 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. (c) 2005 Elsevier Inc. All rights reserved.