Smoothing
Encyclopedia
In statistics
and image processing
, to smooth a data set
is to create an approximating function
that attempts to capture important pattern
s in the data, while leaving out noise
or other fine-scale structures/rapid phenomena. Many different algorithm
s are used in smoothing. One of the most common algorithms is the "moving average", often used to try to capture important trends in repeated statistical survey
s. In image processing
and computer vision
, smoothing ideas are used in scale-space representations.
Smoothing may be distinguished from the related and partially overlapping concept of curve fitting
in the following ways:
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".
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
.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
, to smooth a data set
Data set
A data set is a collection of data, usually presented in tabular form. Each column represents a particular variable. Each row corresponds to a given member of the data set in question. Its values for each of the variables, such as height and weight of an object or values of random numbers. Each...
is to create an approximating function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
that attempts to capture important pattern
Pattern
A pattern, from the French patron, is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.These elements repeat in a predictable manner...
s in the data, while leaving out noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
or other fine-scale structures/rapid phenomena. Many different algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s are used in smoothing. One of the most common algorithms is the "moving average", often used to try to capture important trends in repeated statistical survey
Statistical survey
Survey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....
s. In image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
and computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
, smoothing ideas are used in scale-space representations.
Smoothing may be distinguished from the related and partially overlapping concept of curve fitting
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...
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.
However, the terminology used across applications is mixed. For example, use of an interpolating spline
Spline (mathematics)
In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...
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 transformationLinear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
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
Hat matrix
In statistics, the hat matrix, H, maps the vector of observed values to the vector of fitted values. It describes the influence each observed value has on each fitted value...
.
Specific smoothing and filter types
- Additive smoothingAdditive smoothingIn statistics, additive smoothing, also called Laplace smoothing , or Lidstone smoothing, is a technique used to smooth categorical data...
- Butterworth filterButterworth filterThe Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
- Digital filterDigital filterIn electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
- Kalman filterKalman filterIn statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...
- Kernel smootherKernel smootherA kernel smoother is a statistical technique for estimating a real valued function f\,\,\left by using its noisy observations, when no parametric model for this function is known...
- Laplacian smoothingLaplacian smoothingLaplacian smoothing is an algorithm to smooth a polygonal mesh. For each vertex in a mesh, a new position is chosen based on local information and the vertex is moved there...
- Low-pass filterLow-pass filterA low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
- Recursive filterRecursive filterIn signal processing, a recursive filter is a type of filter which re-uses one or more of its outputs as an input. This feedback typically results in an unending impulse response , characterised by either exponentially growing, decaying, or sinusoidal signal output components.However, a recursive...
- Savitzky–Golay smoothing filterSavitzky–Golay smoothing filterThe Savitzky–Golay smoothing filter is a type of filter first described in 1964 by Abraham Savitzky and Marcel J. E. Golay.The Savitzky–Golay method essentially performs a local polynomial regression on a series of values to determine the smoothed value for each point...
- Local regressionLocal regressionLOESS, or LOWESS , is one of many "modern" modeling methods that build on "classical" methods, such as linear and nonlinear least squares regression. Modern regression methods are designed to address situations in which the classical procedures do not perform well or cannot be effectively applied...
also known as "loess" or "lowess" - Smoothing splineSmoothing splineThe smoothing spline is a method of smoothing using a spline function.-Definition:Let ;x_1...
- Ramer–Douglas–Peucker algorithm
- Moving average
- Kolmogorov–Zurbenko filterKolmogorov–Zurbenko filterKolmogorov–Zurbenko Filter was first proposed by Kolmogorov and formally defined by Zurbenko. It is k time iterations of a moving average filter of m points and belongs to the class of low pass filter. KZ filter has two parameters, the length of the moving average window and the number of...
Other
- ConvolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
- Curve fittingCurve fittingCurve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...
- Edge preserving smoothing
- Graph cuts in computer visionGraph cuts in computer visionAs applied in the field of computer vision, graph cuts can be employed to efficiently solve a wide variety of low-level computer vision problems , such as image smoothing, the stereo correspondence problem, and many other computer vision problems that can be formulated in terms of energy minimization...
- Numerical smoothing and differentiationNumerical smoothing and differentiationAn experimental datum value can be conceptually described as the sum of a signal and some noise, but in practice the two contributions cannot be separated. The purpose of smoothing is to increase the Signal-to-noise ratio without greatly distorting the signal...
- Scale spaceScale spaceScale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision...
- Statistical signal processingStatistical signal processingStatistical signal processing is an area of Applied Mathematics and Signal Processing that treats signals as stochastic processes, dealing with their statistical properties...
- Window functionWindow functionIn signal processing, a window function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation...
External links
- Hastie, T.J. and Tibshirani, R.J. (1990), Generalized Additive Models, New York: Chapman and Hall.
- Chapter on data smoothing from the instruction manual for Wolfram Research's MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...