Median Filters

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

Gallagher, N. C., Jr., and Wise, G. L., A theoretical analysis of the properties of median filters, IEEE Trans. Acoustics, Speech, Signal Proc. 29, 1136 (1981). 

Erode applied to a mask peels off a layer from every white island in the mask. Dilate does (almost) the inverse if erode has not managed to delete a white island completely, it is almost restored. The median operator reduces noise in the image without broadening the peaks. The operator is frequently addressed median filter . 

In similar manner, spikes in the image from cosmic rays are extinguished by simple application of the median filter with a small submatrix size (3 or 5). 

If the illuminant is constant, we can remove the influence of the illuminant by computing the first derivative. This works if two neighboring pixels have the same illuminant. If two neighboring pixels have a different illuminant, then we probably have a shadow edge between the two pixels. Weiss suggests simply applying a median filter to the first derivatives to eliminate the influence of the illuminant. 

Perform median filtering with selective criteria... 

The aim of the signal analysis in the MErKoFer project was to determine significant deviations of the various signals (i.e., process and other recorded values) from their stable state. Of course, care had to be taken of noise, oscillations, and similar issues. Several methods where therefore applied and evaluated. Partially, Gauss or median filters were used for preprocessing. 

Figure 6.12. The actual test signal, the test signal with noise and its denoised estimates using the moving median filter with different window lengths.

The performance of denoising methods tends to deteriorate in the presence of outliers. Doymaz et al. [59] proposed a robust filtering strategy that uses a median filter (MM) (see Section 6.2.1) in tandem with the coefficient denoising method [7]. Here, this strategy is briefly reviewed and its key benefits for denoising are pointed out. 

Figure 14. Thresholding process a) input image, b) median filtering, c) background, d) subtracting the background from the image, e) intensity adjustment and f) binary image produced by thresholding.

FMHfiltering. A FIR-Median Hybrid Filter (FMH) is a median filter which has a preprocessed input from M linear FIR filters [3, 21]. Thus, the FMH filter output is the median of only M values, which are the outputs of M FIR filters applied to the original data. For an FMH filter of length 21 + 1 with three FIR substructures (M = 3), the data are split into three parts on which the FIR filters are applied. Then, the median operator is applied on the outputs of all FIR filters to obtain the output of the FMH filter. For this particular example, the three FIR filters used are ... 

Standard median and FMH filters have been widely used in non-Gaussian error elimination [21,27]. Standard median filters simply use the middle observation from data in a moving window, whereas FMH filters preprocess the data with FIR filters as discussed earlier. FMH filters are superior to the standard median filters due to their improved ability to preserve temporally localized features, while eliminating errors. However, proper selection of the FIR filters requires knowledge about the maximum duration of the outliers. When such knowledge is available, the length of the FIR filters used can be... 

Fig. 15 (a) A bumps signal contaminated with white noise of variance 0.5 and outlier patches of length 3. (b) robust OLMS filtering, median filter length = 9, Haar wavelet, scale depth = 2 (MSE = 0.8366). 

As in the case of any filtering or compression method, multiscale filtering relies on some information about the data and the nature of the errors to tune its filter parameters, which include the threshold, decomposition depth, the wavelet filter, and the size of the median filter for the robust techniques. Hints for selecting these tuning parameters in off-line and on-line modes are discussed below. 

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