Median filtering is among the most utilized tools for smoothing real-valued data, as it is robust, edge-preserving, value-preserving, and yet can be computed efficiently. For data living on the unit circle, such as phase data or orientation data, a filter with similar properties is desirable. For these data, there is no unique means to define a median; so we discuss various possibilities. The arc distance median turns out to be the only variant which leads to robust, edge-preserving and value-preserving smoothing. However, there are no efficient algorithms for filtering based on the arc distance median. Here, we propose fast algorithms for filtering of signals and images with values on the unit circle based on the arc distance median. For non-quantized data, we develop an algorithm that scales linearly with the filter size. The runtime of our reference implementation is only moderately higher than the Matlab implementation of the classical median filter for real-valued data. For quantized data, we obtain an algorithm of constant complexity w.r.t. the filter size. We demonstrate the performance of our algorithms for real life data sets: phase images from interferometric synthetic aperture radar, planar flow fields from optical flow, and time series of wind directions.

中文翻译：

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相位或方向数据的快速中值滤波


中值滤波是最常用的平滑实值数据的工具之一，因为它具有鲁棒性、边缘保留、价值保留，并且可以高效计算。对于单位圆上的数据，例如相位数据或方向数据，需要具有类似属性的滤波器。对于这些数据，没有唯一的方法来定义中位数；所以我们讨论各种可能性。弧距中值被证明是唯一能够实现鲁棒、边缘保留和价值保留平滑的变体。然而，目前还没有有效的基于弧距中值的过滤算法。在这里，我们提出了基于弧距离中值对单位圆上的值进行信号和图像滤波的快速算法。对于非量化数据，我们开发了一种随滤波器大小线性缩放的算法。我们的参考实现的运行时间仅略高于实值数据的经典中值滤波器的 Matlab 实现。对于量化数据，我们获得了与滤波器大小相关的恒定复杂度的算法。我们展示了我们的算法在现实数据集上的性能：干涉合成孔径雷达的相位图像、光流的平面流场以及风向的时间序列。