Continuous curvelet transform: I. Resolution of the wavefront set

Abstract

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 , 2 π ) . Here Γ f ( a , b , θ ) = 〈 f , γ a b θ 〉 projects f onto analyzing elements called curvelets γ a b θ 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 ( 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 ( 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 d a a 3 + 2 m ) 1 / 2 is not locally integrable. The C C T 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 C C 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.