Smoothing

Smoothing means in a mathematical context, to convert a curve into a curve with a lower curvature that is different at the same time as little as possible from the original. In this sense Näherungspolynome low order meet the requirements of smoothing very good. Smoothing filter is often used synonymously for the word. In contrast to the smoothing filter means in mathematics, to remove certain components or characteristics of a curve, usually frequency components or noise. Many, but not all of the filters have the property of smoothing.

The method that satisfies the most severe characteristic of the smoothing, the Henderson - Whittacker method. Here the optimum between smoothness ( minimum mean square n- th derivative ) and accuracy is calculated (minimum square error for the original). The ratio of these two quantities is given as free parameter.

Smoothing methods in statistics

Smoothing method of image and signal processing

LTI filter

Fourier analysis is the theoretical basis for LTI filter. It breaks down a function in a series of sine functions of different frequency. In this frequency spectrum of high frequencies can be then selectively removed.

However, it is not absolutely necessary to actually calculate the spectrum, because there is an equivalent method to filter out frequencies from a signal: the so-called convolution of the signal with a filter kernel (often just called filters). Example: convolution with the rectangular filter. It is simply to be replaced at any position of the signal value in each case by the average of their neighbors. More complex filters are characterized by the fact that they represent weighted averages.

In the context of one-dimensional signals, such as sound or voltage characteristics, low-pass filters are filters that suppress high frequencies, called. In the context of two-dimensional signals such as images is called Blur. Several such filters are available. They differ in weight shows received adjacent values ​​in the mean. Some well-known filters are:

Nonlinear filter

Since " blurred " edges, the flat-rate reduction of high frequencies, there are other methods that attempt to get these: