New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

Abstract

This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 − j , each element has an envelope that is aligned along a "ridge" of length 2 − j/2 and width 2 − j .We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeysThis rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n −1 as n → ∞, which is considerably worse than the optimal behavior.

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

“…Different geometrical methods were recently developed to design efficient image approximation schemes, where correlations along curves are essentially taken into account to locally capture the geometry of the given image data. Curvelets [6,7] and shearlets [21,22] are prominent examples for non-adaptive redundant function frames with strong anisotropic directional selectivity.…”

“…For piecewise Hölder continuous functions f of second order with discontinuities along C 2 -curves, Candès and Donoho [7] proved that a best approximation f N to f with N curvelets satisfies the asymptotic decay rate…”

“…Up to the (log 2 N ) 3 factor, this curvelet N -term approximation rate is asymptotically optimal (see [7,Section 1.3]). Similar decay rates were proven by Guo and Labate [21] for shearlet frames.…”

The power of adaption for approximating functions with singularities

“…A given function f (image, signal) is represented by a few of the largest coefficients of its expansion in a suitable wavelet basis (or curvelet basis in 2-D); see, e.g., [5,8,9,15]. It is also believed that adaptive methods, that rely on adjusting the choice of the basis to the underlying function and its discontinuities (edges), are much more efficient than nonadaptive methods where the basis is fixed for all f 's.…”

Image denoising based on the edge-process model

“…The details can be found in [24][25][26][27]. Examples of bidimensional multiresolution transforms in image denoising applications can be found in [28,29].…”