Biomedical Imaging Group (BIG), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland.
IEEE Trans Image Process. 2010 Mar;19(3):636-52. doi: 10.1109/TIP.2009.2038832. Epub 2009 Dec 22.
Our main goal in this paper is to set the foundations of a general continuous-domain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions ( d >or= 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner-product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of [Formula: see text] into another "steerable" wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli's steerable pyramid whose construction was primarily based on filterbank design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially behave like the N th-order partial derivatives of an isotropic Gaussian kernel.
我们在本文中的主要目标是为设计可引导的、可逆的信号变换（也称为帧）在多维（d>=2）的通用连续域框架奠定基础。为此，我们引入了 Riesz 变换的自可逆 N 阶扩展。我们证明了这个广义变换具有以下显著性质：平移不变性、比例不变性、内积保持性和可引导性。令人欣慰的是，变换将任何[公式：见文本]的主小波帧（或基）映射到另一个“可引导”的小波帧，同时保持帧边界。该概念提供了与 Simoncelli 的可引导金字塔的功能对应物，其构建主要基于滤波器组设计。所提出的机制允许在任意数量的维度中指定任何阶数的可引导小波；它还产生了一个完美重建滤波器组算法。我们用设计一种新的多维 Riesz-Laplace 小波族来说明该方法，该方法本质上类似于各向同性高斯核的 N 阶偏导数。
