脊波理论:从脊波变换到Curvelet变换

Ridgelet Theory: from Ridgelet Transform to Curvelet

本文旨较系统地评述继小波理论后,新近发展起来的具有变革意义的脊波理论的发展沿革、研究现状、应用前景和存在的问题。在信号处理、数据压缩、模式识别、统计估值等领域,获得对某些函数类的高的非线性逼近能力是至关重要的。由一维小波张成的二维小波虽然能有效表示含“点奇异”的二维函数,但对于含“线奇异”的二维函数,却不能获得最优的甚至哪怕是“近似最优”的非线性逼近阶。Candes提出的脊波变换巧妙地将二维函数中的“直线奇异”转化为“点奇异”,再用小波进行处理,能获得对含“直线奇异”的二维或高维函数最优的非线性逼近阶。正交脊波,则延续了脊波变换将“直线奇异”转化为“点奇异”进行处理的思想,并且构成一组L2(R2)上的标准正交基。单尺度脊波和Curvelet变换由脊波变换发展而来,分别利用了函数局部化和频带剖分的思想,将脊波理论发展到了一个更高的阶段,这两种变换都能“近似最优”的表示直线和曲线奇异,因而具有更好的应用前景。

This paper aims to review the newly-developed ridgelet theory with focus on its development history, application potential and existent problems. It is of extreme importance to obtain as high as possible nonlinear approximation ability for certain function classes in application fields such as signal processing, data compression, pattern recognition and statistic estimation. The commonly used 2-D wavelet is the tensor products of 1-D wavelet. It can effectively represent functions smooth away from ＂Point Singularity＂ but fails to deal with those smooth away from ＂Linear Singularity＂. Ridgelet transform proposed by Candes can obtain the optimal nonlinear approximation rate for functions smooth away from ＂Linear Singularity＂ by first converting ＂Linear Singularity＂ into ＂Point Singularity＂ and then dealing the resulting ＂Point Singularity＂ with wavelet systems. Orthonormal ridgelet inherits such key idea of ridgelet transform, and furthermore constitutes a normalized orthonormal system. Based on a localization principle and subband decomposition, monoscale ridgelet and curvelet were proposed, both of which were derived from ridgelet transform and can efficiently deal with smooth images with smooth edges including straight and curve singularity. They, hence, are of better application potention.