Curvelets and curvilinear integrals

Let delta (t): I --> R-2 be a simple closed unit-speed C-2 curve in R-2 with normal (n) over right arrow (t). The curve delta generates a distribution Gamma which acts on vector fields (v) over right arrow (x(1), x(2)): R-2 --> R-2 by line integration according to Gamma((v) over right arrow) = integral(v) over right arrow(delta (t)) (.) (n) over right arrow (t)dt. We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Phi = (<(<phi>)over right arrow>(mu)) with dual Phi* = <(<phi>)over right arrow>*(mu); then an m-term approximation to Gamma can be formed by selecting in terms (mu (i)': 1 less than or equal to i less than or equal to m) and taking <(<Gamma>)over tilde>(m)((v) over right arrow) = Sigma (m)(i=1) Gamma(<(<phi>)over right arrow>*(mui))[(v) over right arrow,<(<phi>)over right arrow>(mui)]. Here the mu (i) can be chosen adaptively based on the curve delta. We are interested in finding a vector basis or frame for which the above scheme yields the highest-quality m-term approximations. Here performance is measured by considering worst-case error on vector fields which are smooth in an L-2 Sobolev sense: Err(Gamma,<(<Gamma>)over tilde>(m)) = supp{\ Gamma((v) over right arrow)-<(<Gamma>)over tilde>(m)((v) over right arrow)\: parallel to Div((v) over right arrow)parallel to (2) less than or equal to 1}. We establish an isometry between this problem and the, problem of approximating objects with edges in L-2 norm. Starting from the recently-introduced tight frames of scalar curvelets, we construct a vector frame, of curvelets for this problem. Invoking results on the near-optimality of scalar curvelets in representing objects with edges, we argue that vector curvelets provide near-optimal quality in-term approximations. We show that they dramatically outperform both wavelet and Fourier-based representations in terms of in-term approximation error.