Non-linear filters compute the filtered value from adjacent pixels, but the algorithm is not the linear combination of adjacent pixel values.

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Non-linear filters are procedures that calculate the filtered value from the neighbours of a given pixel, but not as the linear combination of the neighbour pixel values. We can no longer talk about kernel matrixes, only kernels or kernel windows.

The basic idea behind rank filters is to use the intensity values of pixels under a kernel window, namely sort them in order, then to select new intensity value for the filtering pixel by this order. One of the most widely used rank filters is the median filter, where the middle value is chosen in the sorted intensity list as the filtering pixel value (Figure 72).

The result of filtering is a smoothened image, but transfer function cannot be assigned to it. Local noises are filtered very efficiently. "Salt and pepper" errors (small value, point-based fluctuations) are successfully removed, because by ordering the intensity values the highly different values (dark or light) pixels are sorted to the edges. Figure 73 describes a median filter applied on a noisy curve.

Figure 72. A noisy curve (with dashed lines) and median filtered variation of if (continuous curve). Length of the kernel is "a".

An important attribute of the median filter having a kernel size of *2k+1* is that it eliminates those lines from the digital image that are thinner than k. It is useful when emphasizing large areas. Unfortunately, it may shift the edges, round off corners, however the algorithm may be altered to eliminate these defects.

Figure 73. From left to right: Original image, "salt and pepper" errors. Conservatively smoothened image removing errors. 5x5 median filtered image. 11x11 median filtered image.

As in the case of the median filter, high intensity values are considered noise. Certain sports are following the scoring system of the olympics. Values are sorted under the kernel window, and then all those candidates are eliminated that are greatly different from the median. It can be parameterized that how many elements shall be ignored from the largest and smallest elements.

The conservative smoothening is a noise filtering method, mostly suitable for eliminating "salt and pepper" noises. Its strategy is to sort all the pixels within the kernel window in increasing order except the actual pixel. We get a [min..max] interval, then check if the actual pixel is inside this interval or not (Figure 73). If it is inside the interval, then the intensity value of the actual pixel is not changed; if it is greater than the maximum, then its value will be the max. value, just like in the minimal case.

The Kuwahara filter is a noise and smoothening filter. One of its most important attributes is that it does not shift or blur the edges away. Let us split up the kernel window into four overlapping (k+1)x(k+1) sized squares, starting from four corners (Figure 74). The filtered pixel's value will be the average intensity value of the current square, having the smallest standard deviation.

Figure 74. Window arrangement of a Kuwahara filter and the behaviour of the kernel over a corner.

Letters are identifying the square to which a pixel belongs. The middle pixel is contained by every square. After sorting all the pixels to the right squares, the average pixel value and the empirical distribution values are calculated for each square. After that, the average value is given to the pixel having the smallest standard deviation (Figure 75).

Figure 75. From left to right: Original image. A 5x5 Kuwahara filtered image. A 11x11 Kuwahara filtered image. The filtered image re-filtered with a 5x5 Kuwahara filter.

A Társadalominformatika: moduláris tananyagok, interdiszciplináris tartalom- és tudásmenedzsment rendszerek fejlesztése az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával, az ELTE TÁMOP 4.1.2.A/1-11/1-2011-0056 projekt keretében valósult meg.

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