Reisert Marco, Burkhardt Hans
Institut für Informatik, University of Freiburg, Freiburg, Germany.
IEEE Trans Image Process. 2008 Dec;17(12):2265-74. doi: 10.1109/TIP.2008.2006601.
Steerable filters are a valuable tool for various low-level vision tasks. In this paper, we argue for the use of complex analysis in the context of 2-D steerable filters. In particular, we recommend the use of complex partial derivatives as a computational basis. Complex derivatives have a major advantage in comparison to real derivatives: they show a canonical rotation behavior, namely a rotation affects the derivative just by a multiplication with a complex unit number. So, the complex derivatives can be steered in a more elegant way and above that they are less expensive to compute. We present several analytical formulas for common and new filter kernels in terms of complex derivatives. Further we relate the complex derivatives of a Gaussian with the Gauss-Laguerre transform and show that the Gauss-Laguerre functions provide an optimal signal representation for local and smooth images. We discuss various finite difference schemes for the realization of the derivatives and use them in practice. In a first experiment, we use a newly introduced filter kernel for anisotropic blurring. The complex formalism offers an elegant way to locally adapt the shape and orientation of the kernel. Second, we use the proposed filters as matched filters to detect vessels in retinal images.
可操纵滤波器是用于各种低级视觉任务的宝贵工具。在本文中，我们主张在二维可操纵滤波器的背景下使用复分析。特别是，我们建议使用复偏导数作为计算基础。与实导数相比，复导数有一个主要优点：它们表现出一种规范的旋转行为，即旋转仅仅通过与一个复数单位相乘就会影响导数。因此，复导数可以以一种更优雅的方式进行操纵，而且计算成本更低。我们根据复导数给出了常见和新的滤波器核的几个解析公式。此外，我们将高斯函数的复导数与高斯 - 拉盖尔变换联系起来，并表明高斯 - 拉盖尔函数为局部平滑图像提供了最优的信号表示。我们讨论了用于实现导数的各种有限差分方案，并在实践中使用它们。在第一个实验中，我们使用新引入的滤波器核进行各向异性模糊处理。复形式主义提供了一种优雅的方式来局部调整核的形状和方向。其次，我们将所提出的滤波器用作匹配滤波器来检测视网膜图像中的血管。
