Smoothing (mathematics) – Wikipedia

Smooth In the mathematical context, it means transferring a curve into a curve with a lower curvature, which at the same time deviates as little as possible from the original. In this sense, approximation polynomial low order meet the requirements of the smoothing very well. Smoothing is often used synonymously to the word filter. In contrast to smoothing, filters in mathematics means removing certain components or characteristics of a curve, usually frequency shares or noise. Many, but not all filters also have the property of the smoothing.

The procedure that fulfills the property of the smoothing is most strictly the Whitaker-Henderson method. ^{[first] [2]} Here the optimum between smoothness (minimal medium-sized drainage) and accuracy (minimal error square to the original) is calculated. The ratio of both sizes is specified as a freely selectable parameter.

Compensation bill

Finds a given amount of data and a given model the best -papalproximizing parameters

Regressionsanalyse

Finds relationships between the given data

Local regression

Regression analysis with local (mostly bell -shaped) weighting of the surrounding values

LTI-Filter [ Edit | Edit the source text ]

Fourier analysis forms the theoretical basis for LTI filters. It disassembles a function into a number of sinus functions of different frequency. From this Frequency spectrum can then selectively delete high frequencies.

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 folding of the signal with a filter core (often only Filter called). Example: folding with the rectangle filter. It simply consists of replacing the value at each point of the signal with the average of your neighbors. More complex filters are characterized by the fact that they weighted Represent medium values.

In the context of one -dimensional signals, such as sound or tension runs, filters that suppress high frequencies, Tiefpass-Filter called. In the context of two -dimensional signals such as pictures, one speaks of Bleat . Various such filters are available. They differ in the weight with which neighboring values ​​go into the mean. Some known filters are:

Rectangle filter

Its use can lead to artifacts because it regularly shifts frequencies by half a period length.

Sinc-Filter

represents the ideal low pass, i.e. that is, it completely deletes frequencies above the desired barrier – everyone else remains untouched.

Gauss-filter

weakens frequencies the higher they are.

Exponential smoothing and sliding average

are often used in time series. The weighting of the values ​​falls exponentially with age. The most recent data are the highest weight.

Non -linear filter [ Edit | Edit the source text ]

Since the flat -rate suppression of high frequencies also “blured” edges, there are other procedures that try to maintain them:

Ranking filter

In contrast to the rectangle filter, do not use the mean, but, for example, the median or the maximum.

Sigmafilter

Reduce the noise of pictures without falsifying the edges.

1. ↑ Whittaker, E. T.: On a new method of graduation. In: Proceedings of the Edinburgh Mathematical Society 41 (1923), S. 63–75, doi: 10.1017/S0013091500077853 .
2. ↑ The Whittaker Henderson method is also known as a Hodrick-Prescott filter in economics and, according to this reference, goes back to the astronomer Schiaparelli (1867). R. J. HODRICK, E.C. Prescott: Postwar Us Business Cycles: An Empirical Investigation. In: Journal of Money, Credit & Banking 29 (1997), Feb, No. 1, pp. 1–16, JSTOR: 2953682 .