Edge analysis and identification using the continuous shearlet transform
Edge analysis and identification using the continuous shearlet transform
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DOI：
10.1016/j.acha.2008.10.004
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发表时间：
2009-07-01
影响因子：
2.5
通讯作者：
Lim, Wang-Q
中科院分区：
文献类型：
--
作者：
Guo, Kanghui;Labate, Demetrio;Lim, Wang-Q
众所周知，连续小波变换能够识别函数或分布 f 的奇点集。最近的研究表明，小波变换的某些多维推广对于捕获有关 f 奇点几何形状的附加信息非常有用。在本文中，我们考虑连续剪切波变换，其映射 f 是 L-2(R-2) --> SH(psi)f (a, s, t) = < f, psi(ast)> 的元素，其中分析元素 psi(ast) 在连续尺度 a > 0 上形成良好定域函数的仿射系统，位置 t 是 R-2 的元素，沿斜率 S 的线定向是频域中 R 的元素。我们证明，连续剪切波变换可以准确识别平面物体边缘的位置和方向。特别是，如果 f = Sigma(N)(n=1) f(n)chi Omega(n)，其中函数 f(n) 是平滑的并且集合 Omega(n) 具有平滑边界，则可以使用 SH(psi)f (a, s, t) 的渐近衰减作为 a --> 0（精细尺度）来精确表征边界偏导数 Omega(n) 的位置和方向。这改进了最近在文献中获得的类似结果，并为边缘检测和分析的改进算法的开发提供了理论背景。 (C) 2008 Elsevier Inc. 保留所有权利。
It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution f. It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities of f. In this paper, we consider the continuous shearlet transform, which is the mapping f is an element of L-2(R-2) --> SH(psi)f (a, s, t) = < f, psi(ast)>, where the analyzing elements psi(ast) form an affine system of well localized functions at continuous scales a > 0, locations t is an element of R-2, and oriented along lines of slope S is an element of R in the frequency domain. We show that the continuous shearlet transform allows one to exactly identify the location and orientation of the edges of planar objects. In particular, if f = Sigma(N)(n=1) f(n)chi Omega(n) where the functions f(n) are smooth and the sets Omega(n) have smooth boundaries, then one can use the asymptotic decay of SH(psi)f (a, s, t), as a --> 0 (fine scales), to exactly characterize the location and orientation of the boundaries partial derivative Omega(n). This improves similar results recently obtained in the literature and provides the theoretical background for the development of improved algorithms for edge detection and analysis. (C) 2008 Elsevier Inc. All rights reserved.