中值滤波真的比线性滤波更好地保留边缘吗？
Does median filtering truly preserve edges better than linear filtering?
摘要：
图像处理研究人员通常断言，“中值滤波在去除存在边缘的噪声方面优于线性滤波。“使用一个简单的大- $n$决策理论框架，这个民间定理通常被认为是错误的。我们表明，中值滤波和线性滤波具有相似的渐近最坏情况下的均方误差（MSE）时，信噪比（SNR）是1阶，这对应于恒定的情况下，每个像素的噪声水平的数字信号。为了在渐近设置中看到中值平滑的显著益处，每像素噪声水平应当趋于零（即，SNR应该增长得非常大）。我们表明，使用两个非常不同的窗口宽度的两阶段中值滤波可以显着优于传统的线性和中值滤波的基础对象的边缘设置。在这个两阶段的过程中，第一遍，在一个精细的尺度，旨在增加SNR。第二遍，在一个粗略的尺度，正确地利用了中位数的非线性。基于非线性偏微分方程（PDE）的图像处理方法通常被认为是对存在边缘的线性滤波的改进。这样的方法似乎很难在决策理论框架中进行严格的分析。一个流行的例子是中曲率运动（MCM），其形式上是一种迭代中值滤波。我们的迭代中值滤波的结果表明，一些基于偏微分方程的方法是候选人严格优于线性滤波的渐近框架。
英文摘要：
Image processing researchers commonly assert that "median filtering is better than linear filtering for removing noise in the presence of edges." Using a straightforward large-$n$ decision-theory framework, this folk-theorem is seen to be false in general. We show that median filtering and linear filtering have similar asymptotic worst-case mean-squared error (MSE) when the signal-to-noise ratio (SNR) is of order 1, which corresponds to the case of constant per-pixel noise level in a digital signal. To see dramatic benefits of median smoothing in an asymptotic setting, the per-pixel noise level should tend to zero (i.e., SNR should grow very large). We show that a two-stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges. In this two-stage procedure, the first pass, at a fine scale, aims at increasing the SNR. The second pass, at a coarser scale, correctly exploits the nonlinearity of the median. Image processing methods based on nonlinear partial differential equations (PDEs) are often said to improve on linear filtering in the presence of edges. Such methods seem difficult to analyze rigorously in a decision-theoretic framework. A popular example is mean curvature motion (MCM), which is formally a kind of iterated median filtering. Our results on iterated median filtering suggest that some PDE-based methods are candidates to rigorously outperform linear filtering in an asymptotic framework.
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