Curvelets – A Surprisingly Effective Nonadaptive Representation For Objects with Edges

It is widely believed that to efficiently represent an otherwise smooth object with discontinuities along edges, one must use an adaptive representation that in some sense ‘tracks’ the shape of the discontinuity set. This folk-belief — some would say folk-theorem — is incorrect. At the very least, the possible quantitative advantage of such adaptation is vastly smaller than commonly believed. We have recently constructed a tight frame of curvelets which provides stable, efficient, and near-optimal representation of otherwise smooth objects having discontinuities along smooth curves. By applying naive thresholding to the curvelet transform of such an object, one can form m-term approximations with rate of L approximation rivaling the rate obtainable by complex adaptive schemes which attempt to ‘track’ the discontinuity set. In this article we explain the basic issues of efficient m-term approximation, the construction of efficient adaptive representation, the construction of the curvelet frame, and a crude analysis of the performance of curvelet schemes.