Smoothing
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other finescale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal. Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust.[1] Many different algorithms are used in smoothing.
Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways:
 curve fitting often involves the use of an explicit function form for the result, whereas the immediate results from smoothing are the "smoothed" values with no later use made of a functional form if there is one;
 the aim of smoothing is to give a general idea of relatively slow changes of value with little attention paid to the close matching of data values, while curve fitting concentrates on achieving as close a match as possible.
 smoothing methods often have an associated tuning parameter which is used to control the extent of smoothing. Curve fitting will adjust any number of parameters of the function to obtain the 'best' fit.
However, the terminology used across applications is mixed. For example, use of an interpolating spline fits a smooth curve exactly through the given data points and is sometimes called "smoothing".
Linear smoothers
In the case that the smoothed values can be written as a linear transformation of the observed values, the smoothing operation is known as a linear smoother; the matrix representing the transformation is known as a smoother matrix or hat matrix.
The operation of applying such a matrix transformation is called convolution. Thus the matrix is also called convolution matrix or a convolution kernel. In the case of simple series of data points (rather than a multidimensional image), the convolution kernel is a onedimensional vector.
Algorithms
One of the most common algorithms is the "moving average", often used to try to capture important trends in repeated statistical surveys. In image processing and computer vision, smoothing ideas are used in scale space representations. The simplest smoothing algorithm is the "rectangular" or "unweighted slidingaverage smooth". This method replaces each point in the signal with the average of "m" adjacent points, where "m" is a positive integer called the "smooth width". Usually m is an odd number. The triangular smooth is like the rectangular smooth except that it implements a weighted smoothing function.[2]
Some specific smoothing and filter types, with their respective uses, pros and cons are:
Algorithm  Overview and uses  Pros  Cons 

Additive smoothing  used to smooth categorical data.  
Butterworth filter  Slower rolloff than a Chebyshev Type I/Type II filter or an elliptic filter 


Chebyshev filter  Has a steeper rolloff and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. 


Digital filter  Used on a sampled, discretetime signal to reduce or enhance certain aspects of that signal  
Elliptic filter  
Exponential smoothing 


Kalman filter 

Estimates of unknown variables it produces tend to be more accurate than those based on a single measurement alone  
Kernel smoother 

The estimated function is smooth, and the level of smoothness is set by a single parameter.  
Kolmogorov–Zurbenko filter 



Laplacian smoothing  algorithm to smooth a polygonal mesh.[4][5]  
Local regression also known as "loess" or "lowess"  a generalization of moving average and polynomial regression. 


Lowpass filter 


Moving average 



Ramer–Douglas–Peucker algorithm  decimates a curve composed of line segments to a similar curve with fewer points.  
Savitzky–Golay smoothing filter 


Smoothing spline  
Stretched grid method 

See also
 Convolution
 Curve fitting
 Discretization
 Edge preserving smoothing
 Filtering (signal processing)
 Graph cuts in computer vision
 Numerical smoothing and differentiation
 Scale space
 Scatterplot smoothing
 Smoothness
 Statistical signal processing
 Subdivision surface, used in computer graphics
 Window function
References
 Simonoff, Jeffrey S. (1998) Smoothing Methods in Statistics, 2nd edition. Springer ISBN 9780387947167
 O'Haver, T. (January 2012). "Smoothing". terpconnect.umd.edu.
 Easton, V. J.; & McColl, J. H. (1997)"Time series", STEPS Statistics Glossary
 Herrmann, Leonard R. (1976), "Laplacianisoparametric grid generation scheme", Journal of the Engineering Mechanics Division, 102 (5): 749–756.
 Sorkine, O., CohenOr, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.P. (2004). "Laplacian Surface Editing". Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. SGP '04. Nice, France: ACM. pp. 175–184. doi:10.1145/1057432.1057456. ISBN 3905673134.CS1 maint: multiple names: authors list (link)
Further reading
 Hastie, T.J. and Tibshirani, R.J. (1990), Generalized Additive Models, New York: Chapman and Hall.