Functional optimization properties of median filtering

In this letter, we use a new approach for studying the properties of median filtering. Specifically, using threshold decomposition, it is shown that median filtering operation minimizes a two-term cost function of the output state of the median filter. The first term of the cost function measures the smoothness between the median filter output and its neighbor points within the operation window, and the second term measures the discrepancy between the filter output and its original signal. The results from this study have provided us with a new tool to analyze and understand some of the properties of the median filtering operation, including weighted median filtering.<>