Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges
Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges
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发表时间：
2000
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通讯作者：
E. Candès;D. Donoho
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作者：
E. Candès;D. Donoho
摘要：人们普遍认为，为了有效地表示沿沿着边缘具有不连续性的平滑对象，必须使用自适应表示，该自适应表示在某种意义上“跟踪"不连续性集的形状。这种民间信仰--有些人会说民间定理--是不正确的。至少，这种适应可能带来的数量优势比通常认为的要小得多。我们最近构建了一个紧凑的curvelets框架，它提供了稳定的，高效的，和接近最佳的表示，否则光滑的物体具有不连续性沿着光滑曲线。通过将朴素阈值化应用于这样的对象的曲波变换，可以形成具有L（sup 2）近似的速率的m项近似，该速率与试图跟踪不连续集的复杂自适应方案可获得的速率相媲美。在这篇文章中，我们解释了有效的m项近似的基本问题，有效的自适应表示的建设，曲波框架的建设，和一个粗略的分析曲波方案的性能。
Abstract : 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(sup 2) 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.