Interpolation - Wikipedia, the free encyclopedia

Interpolation - Wikipedia, the free encyclopedia: "Interpolation
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For other uses, see Interpolation (disambiguation).

In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

In engineering and science one often has a number of data points, as obtained by sampling or experimentation, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points.

A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose we know the function but it is too complex to evaluate efficiently. Then we could pick a few known data points from the complicated function, creating a lookup table, and try to interpolate those data points to construct a simpler function. Of course, when using the simple function to calculate new data points we usually do not receive the same result as when using the original function, but depending on the problem domain and the interpolation method used the gain in simplicity might offset the error.

It should be mentioned that there is another very different kind of interpolation in mathematics, namely the 'interpolation of operators'. The classical results about interpolation of operators are the Riesz-Thorin theorem and the Marcinkiewicz theorem. There also are many other subsequent results.
An interpolation of a finite set of points on an epitrochoid. Points through which curve is splined are red; the blue curve connecting them is interpolation.
Contents
[hide]

* 1 Example
* 2 Piecewise constant interpolation
* 3 Linear interpolation
* 4 Polynomial interpolation
* 5 Spline interpolation
* 6 Interpolation via Gaussian processes
* 7 Other forms of interpolation
* 8 Related concepts
* 9 References
* 10 External links

[edit] Example

For example, suppose we have a table like this, which gives some values of an unknown function f.
Plot of the data points as given in the table.
x f(x)
0 0
1 0 . 8415
2 0 . 9093
3 0 . 1411
4 −0 . 7568
5 −0 . 9589
6 −0 . 2794

Interpolation provides a means of estimating the function at intermediate points, such as x = 2.5.

There are many different interpolation methods, some of which are described below. Some of the concerns to take into account when choosing an appropriate algorithm are: How accurate is the method? How expensive is it? How smooth is the interpolant? How many data points are needed?

[edit] Piecewise constant interpolation
Piecewise constant interpolation, or nearest-neighbor interpolation.
For more details on this topic, see Nearest-neighbor interpolation.

The simplest interpolation method is to locate the nearest data value, and assign the same value. In one dimension, there are seldom good reasons to choose this one over linear interpolation, which is almost as cheap, but in higher dimensions, in multivariate interpolation, this can be a favourable choice for its speed and simplicity.

[edit] Linear interpolation
Plot of the data with linear interpolation superimposed
Main article: Linear interpolation

One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of determining f(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take f(2.5) midway between f(2) = 0.9093 and f(3) = 0.1411, which yields 0.5252.

Generally, linear interpolation takes two data points, say (xa,ya) and (xb,yb), and the interpolant is given by:

y = y_a + (x-x_a)\frac{(y_b-y_a)}{(x_b-x_a)} at the point (x,y)

Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not differentiable at the point xk.

The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by g, and suppose that x lies between xa and xb and that g is twice continuously differentiable. Then the linear interpolation error is

|f(x)-g(x)| \le C(x_b-x_a)^2 \quad\mbox{where}\quad C = \frac18 \max_{y\in[x_a,x_b]} |g''(y)|.

In words, the error is proportional to the square of the distance between the data points. The error of some other methods, including polynomial interpolation and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.

[edit] Polynomial interpolation
Plot of the data with polynomial interpolation applied
Main article: Polynomial interpolation

Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a linear function. We now replace this interpolant by a polynomial of higher degree.

Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:

f(x) = − 0.0001521x6 − 0.003130x5 + 0.07321x4 − 0.3577x3 + 0.2255x2 + 0.9038x.

Substituting x = 2.5, we find that f(2.5) = 0.5965.

Generally, if we have n data points, there is exactly one polynomial of degree at most n−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power n. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation solves all the problems of linear interpolation.

However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. Furthermore, polynomial interpolation may not be so exact after all, especially at the end points (see Runge's phenomenon). These disadvantages can be avoided by using spline interpolation.

[edit] Spline interpolation
Plot of the data with Spline interpolation applied
Main article: Spline interpolation

Remember that linear interpolation uses a linear function for each of intervals [xk,xk+1]. Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.

For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by

f(x) = \left\{ \begin{matrix} -0.1522 x^3 + 0.9937 x, & \mbox{if } x \in [0,1], \\ -0.01258 x^3 - 0.4189 x^2 + 1.4126 x - 0.1396, & \mbox{if } x \in [1,2], \\ 0.1403 x^3 - 1.3359 x^2 + 3.2467 x - 1.3623, & \mbox{if } x \in [2,3], \\ 0.1579 x^3 - 1.4945 x^2 + 3.7225 x - 1.8381, & \mbox{if } x \in [3,4], \\ 0.05375 x^3 -0.2450 x^2 - 1.2756 x + 4.8259, & \mbox{if } x \in [4,5], \\ -0.1871 x^3 + 3.3673 x^2 - 19.3370 x + 34.9282, & \mbox{if } x \in [5,6]. \\ \end{matrix} \right.

In this case we get f(2.5)=0.5972.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother. However, the interpolant is easier to evaluate than the high-degree polynomials used in polynomial interpolation. It also does not suffer from Runge's phenomenon.

[edit] Interpolation via Gaussian processes

Gaussian process is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression, i.e. for fitting a curve through noisy data. In the geostatistics community Gaussian process regression is also known as Kriging.

[edit] Other forms of interpolation

Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials. The discrete Fourier transform is a special case of trigonometric interpolation. Another possibility is to use wavelets.

The Whittaker–Shannon interpolation formula can be used if the number of data points is infinite.

Multivariate interpolation is the interpolation of functions of more than one variable. Methods include bilinear interpolation and bicubic interpolation in two dimensions, and trilinear interpolation in three dimensions.

Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to Hermite interpolation problems.

[edit] Related concepts

The term extrapolation is used if we want to find data points outside the range of known data points.

In curve fitting problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the error. In the simplest case this leads to least squares approximation.

Approximation theory studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function."

MRPT : Rao-Blackwelized Particle Filter (RBPF) approach to map building (SLAM).

The MRPT project: mrpt::slam::CMetricMapBuilderRBPF Class Reference:

Visit link for more details


"mrpt::slam::CMetricMapBuilderRBPF Class Reference
#include <mrpt/slam/CMetricMapBuilderRBPF.h>

Inheritance diagram for mrpt::slam::CMetricMapBuilderRBPF:

List of all members.
Detailed Description
This class implements a Rao-Blackwelized Particle Filter (RBPF) approach to map building (SLAM).

Internally, the list of particles, each containing a hypothesis for the robot path plus its associated metric map, is stored in an object of class CMultiMetricMapPDF.

This class processes robot actions and observations sequentially (through the method CMetricMapBuilderRBPF::processActionObservation) and exploits the generic design of metric map classes in MRPT to deal with any number and combination of maps simultaneously: the likelihood of observations is the product of the likelihood in the different maps, etc.

A number of particle filter methods are implemented as well, by selecting the appropriate values in TConstructionOptions::PF_options. Not all the PF algorithms are implemented for all kinds of maps.

For an example of usage, check the application 'rbpf-slam', in 'apps/RBPF-SLAM'. See also the wiki page.

Note:
Since MRPT 0.7.1 the semantics of the parameters 'insertionLinDistance' and 'insertionAngDistance' changes: the entire RBFP is now NOT updated unless odometry increments surpass the threshold (previously, only the map was NOT updated). This is done to gain efficiency.

Since MRPT 0.6.2 this class implements full 6D SLAM. Previous versions worked in 2D + heading only.

See also:
CMetricMap

Definition at line 62 of file CMetricMapBuilderRBPF.h."

SOM ( System on Module )

Accelerated System Design using FPGA Based System-On-Modules (SOM) - Wikipedia, the free encyclopedia: "System-On-Module (SOM)

For companies designing or redesigning an Embedded Systems product there is the tendency to utilize as many third-party components as possible for functions that are well-understood and readily available in the marketplace[citation needed]. This is a direct result of the increasing commoditization of advanced functionalities in the electronic component market. This paradigm is what makes companies use prepackaged power-modules for example, thereby avoiding the complexity and cost of designing 'in house'. A plethora of vendors supply ready built units with all the functionality required (except for specific or niche applications). Thus, it makes no sense for a company to design its own hardware when it has the choice of buying fully-tested, reliable products at lowere cost.[neutrality disputed]"

Loading other than shell

TS-7400 System on Module w/ Ultra-Fast Linux Bootup: "Since the TS-7400 actually boots to an initrd with a read-only mounted filesystem, it is possible to have something other than a shell prompt running after bootup by editing the /linuxrc shell script on the initrd. Additional TS-7400 software features include:"

(initrd)

Linux initial RAM disk (initrd) overview: "What's an initial RAM disk?

The initial RAM disk (initrd) is an initial root file system that is mounted prior to when the real root file system is available. The initrd is bound to the kernel and loaded as part of the kernel boot procedure. The kernel then mounts this initrd as part of the two-stage boot process to load the modules to make the real file systems available and get at the real root file system.

The initrd contains a minimal set of directories and executables to achieve this, such as the insmod tool to install kernel modules into the kernel.

In the case of desktop or server Linux systems, the initrd is a transient file system. Its lifetime is short, only serving as a bridge to the real root file system. In embedded systems with no mutable storage, the initrd is the permanent root file system. This article explores both of these contexts."

MIPS Pipeline Stages

File:MIPS Architecture (Pipelined).svg - Wikipedia, the free encyclopedia

MIP V ( 3D Graphics )

MIPS V is the fifth version of the architecture, announced on 21 October 1996 at the Microprocessor Forum 1996.[12] MIPS V was designed to improve the performance of 3D graphics applications. In the mid-1990s, a major use of non-embedded MIPS microprocessors were graphics workstations from SGI. MIPS V was complemented by the integer-only MIPS Digital Media Extensions (MDMX) multimedia extensions, which were announced on the same date as MIPS"