Continuous curvelet transform Ⅰ. Resolution of the wavefront set

摘要

We discuss a Continuous Curvelet Transform (CCT), a transform f → Γ_f (a, b, θ) of functions f (x_1, x_2) on R~2 into a transform domain with continuous scale a > 0, location b ∈R~2, and orientation θ ∈ [0, 2π). Here Γ_f (a, b, θ) = < f, γ_(abθ) > projects f onto analyzing elements called curvelets Yabe which are smooth and of rapid decay away from an a by a~(1/2) 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. Candes, F. Guo, New multi-scale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Process. 82 (2002) 1519-1543; EJ. Candes, L. Demanet, Curvelets and Fourier integral operators, C. R. Acad. Sci. Paris, Ser. I (2003) 395-398; EJ. 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, θ_0), Γ_f (a, x_0, θ_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 θ_0. Generalizing these examples, we show that decay properties of Γ_f (a, x_0, θ_0) for fixed (x_0, θ_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, θ_0) near which Γ_f f(a,x,θ) is not of rapid decay as a → 0; the H~m-wavefront set is the closure of those points (x_0, θ_0) where the 'directional parabolic square function' S~m(x, θ) = (∫ ∣Γ_f (a, x, θ)∣~2 da/a~(3 + 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.