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Topic: Spline interpolation

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	Numerical Interpolation
		The bottom line is, no matter how smooth the interpolation is and how close it is to the raw data, the problem is not completely solved unless the physical meaning behind the theme has been captured.
		Although the polynomial interpolation is probably the most widely used interpolating method, the rational function interpolation stands out when the data or function changes rapidly in some local regions, e.g., poles.
		The cubic spline interpolation uses third degree polynomials to connect the data points which often results in strikingly smooth curve fits.
	www.efunda.com /math/num_interpolation/num_interpolation.cfm (224 words)

	Encyclopedia article: Spline interpolation (Site not responding. Last check: 2007-10-24)
		Spline interpolation is preferred over polynomial interpolation (additional info and facts about polynomial interpolation) because the interpolation error (additional info and facts about interpolation error) can be made small even when using low degree polynomials for the spline.
		The spline of degree n which interpolates the same data set is not uniquely defined and we have to fill in n-1 additional degrees of freedom (additional info and facts about degrees of freedom) to construct a unique spline interpolant.
		The natural cubic spline is approximately the same curve as created by the spline device (additional info and facts about spline device).
	www.absoluteastronomy.com /encyclopedia/s/sp/spline_interpolation.htm (505 words)

	Interpolation - Wikipedia, the free encyclopedia
		Furthermore, the interpolant is a polynomial and thus infinitely differentiable.
		Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together.
		For instance, rational interpolation is interpolation by rational functions, and trigonometric interpolation is interpolation by trigonometric polynomials.
	en.wikipedia.org /wiki/Interpolation (1047 words)

	end conditions in univariate spline interpolation (Site not responding. Last check: 2007-10-24)
		Spline interpolation, at knots for even order $k$ and at `half'-knots (i.e., midpoints of knot intervals) for odd order, leaves an even number of degrees of freedom still to be chosen, and usually chosen by prescribing additional conditions at both ends.
		The not-a-knot condition in cubic spline interpolation is easily shown to be correct since, in effect, we are interpolating at $x_1,..., x_n$ by a cubic spline with simple interior knots at $x_3,..., x_{n-2}$, hence the Schoenberg-Whitney conditions are easily seen to hold.
		The explanation is for the simplest case, that of periodic cubic spline interpolation: The data $(x_,y_i)$, $i=1,...,N$ are to be matched by a cubic spline $f$ with period interval $I=[x_1..
	www.cs.wisc.edu /~deboor/toast/pages011.html (666 words)

	Smoothing spline interpolation (Site not responding. Last check: 2007-10-24)
		Spline interpolation consists of the approximation of a function by means of series of polynomials over adjacent intervals with continuous derivatives at the end-point of the intervals.
		The interpolation consists of the one-piece polynomial interpolation.
		Smoothing spline interpolation is designed to smooth data sets which are mildly contaminated with isolated errors.
	www.eso.org /projects/esomidas/doc/user/98NOV/volb/node154.html (171 words)

	Siemens Energy & Automation -- Spline Interpolation - A Primer
		Spline is aware of the curvature of the contour and constrains path speed along the curvature based on the machine axes’ acceleration and jerk capabilities.
		Spline mode will transition into and out of a circular, helical or linear span by interpolating smooth curves whose tangent vectors at the point of transition are equal to the tangent vectors of the circular, helical, or linear span.
		In this circumstance, the spline span is converted to a linear span.
	www.sea.siemens.com /machine/product/R4/mcR4spln.html (2297 words)

	Journal of Population Research: Spline interpolation for demographic variables: the monotonicity problem
		Splines are polynomials of specified degree (usually cubics) which are fitted to each of the intervals in the data, and forced to be smooth and continuous at the joining points or 'knots'.
		A typical application of spline interpolation in demography is the estimation of a single-year age distribution from five-yearly or other regularly or irregularly grouped data.
		A spline fitted to the cumulative distribution of deaths will often not be monotonic at this point, so the differences (the single year of age estimates) will be negative, clearly an impossible result.
	www.findarticles.com /p/articles/mi_m0PCG/is_1_21/ai_n6155266 (1111 words)

	Spline Interpolation in C
		If spline() successfully computes the coefficients, then the interpolation function seval() may be used.
		The end condition at either x or x may be either (1) the natural condition where the third derivative of the cubic spline at the end point equals that of the unique cubic that passes through the last four knots or (2) a user specified slope.
		Fit a spline to y = e for x = 0.0, 0.1,...1.0 and then evaluate the spline function, its derivative and its integral at a few sample points.
	www.mech.uq.edu.au /staff/jacobs/nm_lib/doc/spline.html (723 words)

	Cubic Spline Interpolation® (Site not responding. Last check: 2007-10-24)
		Often a polynomial cannot be found which interpolates all the data points.
		Splines are piece-wise defined functions whose individual curves meet at the points.
		Natural Splines…-the ends of the spline curve extend beyond the boundaries of the data and become linear.
	online.redwoods.edu /instruct/darnold/LAPROJ/Fall98/SkyMeg/splinepres (56 words)

	GNU Scientific Library -- Reference Manual - Interpolation (Site not responding. Last check: 2007-10-24)
		The interpolation is piecewise smooth, and its behavior at the end-points is determined by the type of interpolation used.
		The number of terms in the interpolating polynomial is equal to the number of points.
		It computes a cubic spline interpolation of the 10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i + \cos(i^2) for i = 0 \dots 9.
	www.gnu.org /software/gsl/manual/gsl-ref_26.html (595 words)

	What are OMDI Math Tools (Site not responding. Last check: 2007-10-24)
		Interpolation is "guessing" the value at some point on a line (or surface) given the surrounding points.
		Where surface averaging is analogous to linear interpolation, bi-cubic spline interpolation is curvatious, and operates by performing multiple spline interpolations on the rows and columns to estimate the desired point.
		For the first method, the XY interpolants is a selection of n-rows by 2 columns (where the first column is X and the second column is Y), and the XYZ data points are a selection of n-rows by 3 columns (where the columns are X, Y, Z respectively.) Z' is returned.
	www.octavian.com /Layout.jsp?product=xl_math&type=document (3375 words)

	ipedia.com: Linear interpolation Article (Site not responding. Last check: 2007-10-24)
		In numerical analysis a linear interpolation of certain points that are in reality values of some function f is typically used to approximate the function f.
		The basic linear interpolation operation is so commonly used in computer graphics that it is known as a lerp.
		Bilinear interpolation is often used as a crude anti-aliasing filter.
	www.ipedia.com /linear_interpolation.html (434 words)

	Interpolation Algorithms (Site not responding. Last check: 2007-10-24)
		A pixel interpolation algorithm must target the detail preservation versus smoothness dilemma; the all-purpose ideal situation, of course, is to achieve a good balance between both paradigms.
		Bicubic spline interpolation takes an additional parameter: the size in pixels of the block used to construct smooth cubic interpolation spline functions.
		Bilinear interpolation is functionally identical to bicubic interpolation, except that four source pixels and two linear interpolation functions are used.
	pleiades-astrophoto.com /pixinsight/help/18_geometric/interpolation/interpolation_algorithms.html (558 words)

	An Evaluation of Spatial Interpolation Methods on Air Temperature in Phoenix, AZ (Site not responding. Last check: 2007-10-24)
		The ultimate goal of STIM is to develop an interpolation method that incorporates environmental, anthropogenic, and geographical data with one application of the STIM to create air temperature surfaces at the detailed spatial resolution representing the heterogeneity of the urban area.
		Interpolation is a method or mathematical function that estimates the values at locations where no measured values are available.
		The input parameters for the spline function are the input sampled points, the interpolation attribute (i.e., temperature), the type (regularized or tension), the weight, number of points to consider for each new value and the output cell size.
	www.cobblestoneconcepts.com /ucgis2summer/anderson/anderson.htm (3476 words)

	SPLINE - Interpolation and Approximation of Data (Site not responding. Last check: 2007-10-24)
		SPLINE is a library of C++ routines, using double precision arithmetic, for constructing and evaluating spline functions.
		Also included are a set of routines that return the local "basis matrix", which allows the evaluation of the spline in terms of local function data.
		SPLINE is also available in a FORTRAN90 version and a MATLAB version.
	www.csit.fsu.edu /~burkardt/cpp_src/spline/spline.html (638 words)

	AMTH247 Lecture 13 Interpolation III
		One property of polynomial interpolation, which is undesirable in practical applications, is that the interpolating polynomial can show oscillations which are not present in the data.
		In cubic spline interpolation cubic polynomials are used on each interval with the additional requirement that the interpolating be twice continuously differentiable.
		Cubic spline interpolation, and piecewise polynomial interpolation in general, are instances of linear interpolation.
	turing.une.edu.au /~amth247/Lectures_2003/Lecture_13/lecture (887 words)

	Gridding and Interpolation (Site not responding. Last check: 2007-10-24)
		Unlike curve-fitting algorithms, interpolation requires that the interpolating function be an exact fit at each of the tabulated data points.
		Interpolation does not use any type of error analysis and its accuracy depends upon the behavior of the interpolating function between successive data points.
		Interpolates a surface from polar coordinates to rectangular coordinates.
	idlastro.gsfc.nasa.gov /idl_html_help/mathematics9.html (354 words)

	Description of Soil Water Characteristics Using Cubic Spline Interpolation -- Kastanek and Nielsen 65 (2): 279 -- Soil ...
		Measured soil water characteristic for Troup E3 soil (Dane and Puckett, 1992) approximated by simple cubic splines and the equations of van Genuchten (1980) and Brooks and Corey (1964).
		Measured soil water characteristic (Stephens and Rehfeldt, 1985) approximated by simple cubic splines and the equations of van Genuchten (1980) and Brooks and Corey (1964).
		Coefficients of the spline interpolation between the virtual data points shown in Fig.
	soil.scijournals.org /cgi/content/full/65/2/279 (2152 words)

	Efficient Image Magnification by Bicubic Spline Interpolation
		For high-resolution bicubic spline interpolation, the continuous image consists of a number of square regions, each region bordered on its corners by input pixels.
		The continuous image in each region is defined by high-resolution cubic spline interpolation in both the horizontal and vertical directions.
		For high-resolution bicubic spline interpolation, the index of the lowest indexed input pixel on which an intermediate pixel depends is _
	members.bellatlantic.net /~vze2vrva/design.html (715 words)

	Digital Photo Interpolation - what is the best image resizing / resampling method? Genuine Fractals, S-Spline, Bicubic, ...
		Photo interpolation is the process by which the number of pixels comprising an image is increased to allow printing enlargements that are of higher quality than photos that are not interpolated.
		Interpolation is commonly needed to make quality large prints from digital photos and film-scanned images.
		Interpolation of test photos was performed by the tool author using the method of his choosing.
	www.americaswonderlands.com /digital_photo_interpolation.htm (973 words)

	Sampling Theory and Spline Interpolation
		In the spline method, the signal is reconstructed using sample-weighted cardinal splines as opposed to sample-weighted sinc functions.
		From images figure 1 and figure 2, it may appear that the spline interpolation is smoother than the sinc interpolation.
		This is because the support of the cardinal splines is more compact than that of the sinc function.
	cnx.rice.edu /content/m11126/latest (464 words)

	The Origin Forum - B-spline interpolation not smooth?
		I've tried both spline interpolation procedures from the "After I plot a spline curve, how can I access the spline data?" item in the knowledge base and ended up with a curve that was not smooth if you zoom in enough.
		Zoom way in on the interpolated curve and you will see that it is made up of very small plateaus 10 to 20 points wide (if you set "Connect" for this curve to one of the spline modes it looks just fine).
		The interpolated curve is indeed very smooth (no plateaus), but differentiating the curve shows that it is composed of straight-line segments (plateaus in the 1st derivative).
	www.originlab.com /forum/topic.asp?TOPIC_ID=4085 (749 words)

	Spline interpolation (Site not responding. Last check: 2007-10-24)
		Spline method is used for interpolation of function.Spline S(x)
		Spline is defined and specifically determined by 4 coefficients at every subinterval together with two conditions at the both interval ends.Division d defines interpolation spline of function f(x) &isin C
		(3) if the interpolation function have a specified values on boundaries it is to be named given condition type of spline
	www.matf.bg.ac.yu /r3nm/NumericalMethods/Interpolation/Spline.html (130 words)

	Addix Software Consultancy - Spline library overview (Site not responding. Last check: 2007-10-24)
		Once the product is loaded via the appropriate Add-In/Reference Manager, all of the spline interpolation functions that it provides appear as native functions and can be invoked either from spreadsheet cell formulae or from Visual Basic code.
		Interpolation to provide a precise representation of both conic curves and surfaces.
		Access to all the other supported spline types is unrestricted during the evaluation period but is restricted to registered users once the evaluation period has expired.
	www.addix.com /products/spline/overview.html (172 words)

	Cubic Spline Interpolation
		The cubic spline interpolation is a piecewise continuous curve, passing through each of the values in the table.
		Cubic splines are popular because they are easy to implement and produce a curve that appears to be seamless.
		As we have seen, a straight polynomial interpolation of evenly spaced data tends to build in distortions near the edges of the table.
	www.physics.utah.edu /~detar/phycs6720/handouts/cubic_spline/cubic_spline/node1.html (258 words)

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		That is, interpolation using a polynomial of higher degree does not necessarily yield a better result.
		Learn how to use the golden search method combined with the cubic spline interpolation to find the period of a function when the function is known only on a set of data points.
		Use the cubic spline interpolation to evaluate the function f(x).
	www.cse.ucsc.edu /~hongwang/AMS147/Assignments/A06.html (470 words)

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