Computing the convolution between a 2D signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing (e.g. ridge/edge detection) to high level computer vision (e.g. pattern recognition). Through decades of research, there still lacks an efficient method for solving this problem. In this paper, we investigate this problem from the perspective of approximation by considering the following problem: what is the optimal basis for approximating all rotated versions of a given bivariate function? Surprisingly, solely minimising the $L^{2}$ -approximation-error leads to a rotation-covariant linear expansion, which we name Fourier-Argand representation. This representation presents two major advantages: 1) rotation-covariance of the basis, which implies a “strong steerability” — rotating by an angle $\alpha $ corresponds to multiplying each basis function by a complex scalar $e^{-ik\alpha }$ ; 2) optimality of the Fourier-Argand basis, which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction-selective filters. We show the relation between the Fourier-Argand representation and the Radon transform, leading to an efficient implementation of the decomposition for digital filters. We also show how to retrieve accurate orientation of local structures/patterns using a fast frequency estimation algorithm.

中文翻译：

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傅里叶-阿甘德表示：可操纵模式的最佳基础。

计算2D信号与具有可变方向的相应滤波器之间的卷积是一个基本问题，它出现在从低级图像处理（例如，脊/边缘检测）到高级计算机视觉（例如，模式识别）的各种任务中。通过数十年的研究，仍然缺乏解决该问题的有效方法。在本文中，我们通过考虑以下问题从逼近角度研究此问题：逼近给定双变量函数的所有旋转形式的最佳基础是什么？令人惊讶的是，仅将 $ L ^ {2} $ -近似误差导致旋转协变线性展开，我们将其命名为Fourier-Argand表示。这种表示具有两个主要优点：1）基础的旋转协方差，这意味着“强大的可操纵性”-旋转一个角度 $ \ alpha $ 对应于将每个基函数乘以复标量 $ e ^ {-ik \ alpha} $ ; 2）傅里叶-阿尔甘德（Fourier-Argand）基的最佳性，它确保了一些基函数足以准确地逼近复杂的模式和高度方向选择性的滤波器。我们展示了傅里叶-阿尔甘德表示形式与Radon变换之间的关系，从而实现了数字滤波器分解的高效实现。我们还展示了如何使用快速频率估算算法来检索局部结构/图案的准确方向。