Continuous Curvelet Transform -: II.: Discretization and frames
Continuous Curvelet Transform -: II.: Discretization and frames
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DOI：
10.1016/j.acha.2005.02.004
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发表时间：
2005-09-01
影响因子：
2.5
通讯作者：
Donoho, DL
中科院分区：
文献类型：
--
作者：
Candès, EJ;Donoho, DL
我们开发了一个统一的角度来看，几个分解表现出方向抛物线标度。在每个分解中，单个原子在精细尺度上是高度各向异性的，有效支持服从抛物线标度原理宽度近似于长度（2）。我们的比较允许扩展定理已知的一个分解到其他人。我们从R-2上函数f（x（1），x（2））的连续曲波变换f -> Gamma（f）（a，B，theta）开始，参数空间由尺度a > 0索引，位置B是R-2的元素，方向是theta。该变换将f投影到曲波gamma（ab theta）上，产生系数Gamma（f）（a，B，theta）= < f，gamma（ab theta）>;相应的曲波gamma a（B theta）由极频域坐标中的抛物线膨胀定义。我们建立了一个再生公式和Parseval关系的转换，这些曲波提供了一个连续的紧框架。CCT与哈特·史密斯在他的傅里叶积分算子研究中开创的连续变换密切相关。史密斯变换是基于一个单一的母小波的真正的仿射抛物尺度，而CCT只能被视为真正的仿射抛物尺度在欧几里德坐标，采取略有不同的母小波在每个尺度。与CCT不同，Smith变换不提供连续的紧框架。我们表明，与正确的基础小波在史密斯变换，分析元素的两个变换变得越来越相似，越来越精细的尺度。我们推导出一个离散的紧框架，基本上通过采样的CCT在二进间隔在规模a（j）= 2（-j），在方向上，θ（j，l）= 2 π 2（-j/2）l，和均匀间隔采样的旋转各向异性网格上的空间。该框架是[E. J. Candes，F.]中“Curvelets 2002”框架的复杂化。郭，信号处理。82（2002）1519-1543; E.J. Candes，L.德马内角R. Acad. Sci.巴黎，系列1336（2003）395-398; E. J. Candes，D.L. Donoho，Comm. Pure Appl. Math. LVII（2004）219-266]。我们比较这个离散帧与复合系统，在粗尺度上是相同的帧，但在细尺度上是基于采样史密斯变换，而不是CCT。我们能够显示出一个非常接近的两个系统在精细尺度，在一个强大的运营商规范意义。史密斯连续变换是用来形成傅里叶积分算子（FIO）的分子分解。我们的研究结果表明，曲波框架的复合框架，使用真正的仿射抛物尺度在精细尺度的密切近似，使我们能够交叉应用史密斯的结果，证明离散曲波变换给出稀疏表示的FIO的顺序为0。这产生了另一种证明最近的结果Candes和Demanet的稀疏性FIO表示离散曲波帧。(c)2005年爱思唯尔公司All rights reserved.
We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle width approximate to length(2). Our comparisons allow to extend theorems known for one decomposition to others. We start from a continuous curvelet transform f -> Gamma(f) (a, b, theta) of functions f (x(1), x(2)) on R-2, with parameter space indexed by scale a > 0, location b is an element of R-2, and orientation theta. The transform projects f onto a curvelet gamma(ab theta), yielding coefficient Gamma(f) (a, b, theta) = < f, gamma(ab theta)>; the corresponding curvelet gamma a(b theta) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform pioneered by Hart Smith in his study of Fourier Integral Operators. Smith's transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in Euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith's transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith's transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales. We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a(j) = 2(-j), at equispaced intervals in direction, theta(j,l) = 2 pi 2(-j/2)l, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the 'Curvelets 2002' frame in [E.J. Candes, F. Guo, Signal Process. 82 (2002) 1519-1543; E.J. Candes, L. Demanet, C. R. Acad. Sci. Paris, Ser. 1336 (2003) 395-398; E.J. Candes, D.L. Donoho, Comm. Pure Appl. Math. LVII (2004) 219-266]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but at fine scales is based on sampling Smith's transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith's continuous transform was intended for use in forming molecular decompositions of Fourier Integral Operators (FIOs). Our results showing close approximation of the curvelet frame by a composite frame using true affine parabolic scaling at fine scales allow us to cross-apply Smith's results, proving that the discrete curvelet transform gives sparse representations of FIOs of order 0. This yields an alternate proof of a recent result of Candes and Demanet about the sparsity of FIO representations in discrete curvelet frames. (c) 2005 Elsevier Inc. All rights reserved.