Sparse Image and Signal Processing: The Ridgelet and Curvelet Transforms
Sparse Image and Signal Processing: The Ridgelet and Curvelet Transforms
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稀疏图像和信号处理：脊波和曲波变换
DOI：
10.1017/cbo9780511730344.006
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
J. Fadili
中科院分区：
文献类型：
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作者：
Jean;F. Murtagh;J. Fadili
前言脊波和曲波变换是对小波变换的推广。首先，它们合并了角度对齐信息，然后，此外，对齐的长度也被覆盖。与所有这些变换一样，支持多个比例。这些转换的动机是从与边缘相关的构建块构建图像。此外，与前面的章节一样，计算这些变换的效率是一个重要的实用方面。在这一章中，我们考虑脊波变换及其实现的一些算法。然后，我们讨论了曲波变换及其算法。背景和示例小波依赖于在所有尺度和位置上出现的大致各向同性元素的字典。它们不能很好地描述高度各向异性的元素，并且只包含固定数量的方向元素，与尺度无关。尽管它们在图像处理中有广泛的影响，但它们不能有效地表示具有高度各向异性元素的对象，例如直线或曲线结构(例如，边缘)。这是因为小波是非几何的，没有利用边缘曲线的正则性。根据这一推理，提出了新的结构，如脊线(Candes和Donoho 1999)和曲线(Candes和Donoho 2001,2002；Starck等人)。2002年)。脊波和曲波是多尺度方向选择变换家族中的特殊成员，最近在计算和应用调和分析领域引起了一系列的研究活动。
INTRODUCTION The ridgelet and curvelet transforms generalize the wavelet transform. First, they incorporate angular alignment information, and then, in addition, the length of the alignment is covered. As with all of these transforms, multiple scales are supported. The motivation for these transforms is to build up an image from edge-related building blocks. Furthermore, as in previous chapters, the efficiency of computing these transforms is an important practical aspect. In this chapter, we consider the ridgelet transform and a number of algorithms for its implementation. Then we proceed to the curvelet transform and algorithms for it. BACKGROUND AND EXAMPLE Wavelets rely on a dictionary of roughly isotropic elements occurring at all scales and locations. They do not describe well highly anisotropic elements and contain only a fixed number of directional elements, independent of scale. Despite the fact that they have had wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g., edges). The reason is that wavelets are nongeometrical and do not exploit the regularity of the edge curve. Following this reasoning, new constructions have been proposed such as ridgelets (Candes and Donoho 1999) and curvelets (Candes and Donoho 2001, 2002; Starck et al. 2002). Ridgelets and curvelets are special members of the family of multiscale orientation-selective transforms, which have recently led to a flurry of research activity in the field of computational and applied harmonic analysis.