In mathematical logic , Craig s interpolation theorem is a result about the relationship between different logical theory mathematical logic theories . Roughly stated, the theorem says that if a well formed formula formula &phi implies a formula &psi then there is a third formula &rho , called an interpolant ... occurrence in some formula of T . Proof of Craig s interpolation theorem We present here a constructive proof of the Craiginterpolation theorem for propositional logic . ref Harrison pgs ... theorem compactness , negation and conjunction, Robinson s joint consistency theorem and Craiginterpolation are equivalent. proof theory proof theoretically , via a Sequent calculus . If cut elimination is possible and as a result the subformula property holds, then Craiginterpolation is provable ... the logic. via translation to other logics enjoying Craiginterpolation. Applications Craiginterpolation has many applications, among them consistency proof s, model checking , proofs in Modularity ... Mathematical logic Category Lemmas de CraigInterpolation ko ja pl Twierdzenie ... , and &rho implies &psi . The theorem was first proved for first order logic by William Craig logician William Craig in 1957. Variants of the theorem hold for other logics, such as propositional logic . A stronger form of Craig s theorem for first order logic was proved by Roger Lyndon in 1959 the overall ... that P &or R is an interpolant for the implication &phi &rarr &psi . Lyndon s interpolation ... logic K, intuitionistic logic and mu calculus calculus , with similar complexity measures. Craiginterpolation can be proved by other methods as well. However, these proofs are generally non constructive ... 81262 0 cite book author Dov M. Gabbay and Larisa Maksimova title Interpolation and Definability Modal ... year 2006 isbn 978 0198511748 Eva Hoogland, Definability and Interpolation. Model theoretic investigations . PhD thesis, Amsterdam 2001. W. Craig, Three uses of the Herbrand Gentzen theorem in relating ... more details
Other uses In the mathematics mathematical field of numerical analysis , interpolation is a method of constructing ... analysis . A different problem which is closely related to interpolation is the approximation of a complicated ... domain and the interpolation method used the gain in simplicity might offset the error. There is also 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 ... thumb An interpolation of a finite set of points on an epitrochoid . Points through which curve is spline mathematics splined are red the blue curve connecting them is interpolation. Example For example ... Interpolation Data.svg right thumb 230px Plot of the data points as given in the table. cellpadding ... 0 . 9589 6 align right &minus 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 ...? How many data points are needed? br clear all Piecewise constant interpolation Image Piecewise constant.svg thumb right Piecewise constant interpolation, or nearest neighbor interpolation . details Nearest neighbor interpolation The simplest interpolation method is to locate the nearest ... interpolation see below is almost as easy, but in higher dimensional multivariate interpolation , this could be a favourable choice for its speed and simplicity. br clear all Linear interpolation Image Interpolation example linear.svg right thumb 230px Plot of the data with linear interpolation superimposed Main Linear interpolation One of the simplest methods is linear interpolation sometimes .... Generally, linear interpolation takes two data points, say x sub a sub , y sub a sub and x sub b sub ... x , y Linear interpolation is quick and easy, but it is not very precise. Another disadvantage ... error estimate shows that linear interpolation is not very precise. Denote the function which we want ... more details
distinguish interpellation disambiguation Interpolation may refer to Interpolation in mathematics, specifically numerical analysis also Interpolation of operators, as in the Riesz Thorin theorem and Marcinkiewicz interpolation theorem Craiginterpolation in mathematical logic Interpolation theory biology Interpolation music Interpolation manuscripts Variable interpolation in programming Image scaling image interpolation Tweening video interpolation disambig bs Interpolacija vor de Interpolation es Interpolaci n desambiguaci n fr Interpolation nds Interpolatschoon pl Interpolacja uk ... more details
Craiginterpolation in logic Marcinkiewicz interpolation theorem about non linear operators Riesz Thorin interpolation theorem about linear operators mathdab ... more details
Unreferenced date June 2008 Context date October 2009 In image processing , stairstep interpolation is a general method for interpolating the pixels after enlarging an image. The key idea is to interpolate multiple times in small increments using any interpolation algorithm that is better than nearest neighbor interpolation , such as bilinear interpolation , and bicubic interpolation . A common scenario is to interpolate an image by using a bicubic interpolation which increases the image size by no more than 10 110 of the original size at a time until the desired size is reached. See also Anti aliasing B zier surface Cubic Hermite spline , the one dimensional analogue of bicubic spline Lanczos resampling Sinc filter Spline interpolation External links http www.imphotography.com downloads ssimagesize.htm Photoshop plugin to achieve stairstep interpolation Category Image processing ... more details
Trilinear interpolation is a method of multivariate interpolation on a Three dimensional space 3 dimensional ..., unstructured grid unstructured mesh as used in finite element analysis , other methods of interpolation ... provide a straightforward procedure. Trilinear interpolation is frequently used in numerical analysis , data analysis , and computer graphics . Compared to linear and bilinear interpolation Trilinear interpolation is the extension of linear interpolation , which operates in spaces with dimension math D 1 math , and bilinear interpolation , which operates with dimension math D 2 math , to dimension math D 3 math . The order of accuracy is 1 for all these interpolation schemes, and it requires math 1 n D 8 math adjacent pre defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, it is equivalent to 3 dimensional tensor B spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators Method On a periodic and cubic lattice with spacing 1, let math x d math , math ... thumb Eight corner points on a cube surrounding the interpolation point C Image 3D interpolation2.svg right thumb Depiction of 3D interpolation First we interpolate along math x math imagine we are pushing .... The result of trilinear interpolation is independent of the order of the interpolation ... have the values C000, C100, C010, C110, C001, C101, C011, C111. Next, we perform linear interpolation ... to find C10. Now we do interpolation between C00 and C10 to find C0, C01 and C11 to find C1. Finally, we calculate the value C via linear interpolation of C0 and C1 In practice, a trilinear interpolation is identical to three successive linear interpolation s, or two bilinear interpolation s combined with a linear interpolation math C approx l b C 000 , C 010 , C 100 , C 110 , b C 001 , C 011 , C 101 , C 111 math See also Linear interpolation Bilinear interpolation Tricubic interpolation ... more details
In numerical analysis , transfinite interpolation is a means to construct Function mathematics functions over a planar domain in such a way that they match a given function on the boundary. This method is applied in geometric model ling and in the field of finite element method . References Dyken, C., Floater, M. Transfinite mean value interpolation , Computer Aided Geometric Design , Volume 26, Issue 1, January 2009, Pages 117&ndash 134 Category Interpolation mathapplied stub ... more details
Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation , which can also be used as a verb harv Raymond 2003 . Linear interpolation ... interpolation. If the two known points are given by the coordinates math scriptstyle x 0,y 0 math ... from the figure on the right. It is a special case of Polynomial interpolation Constructing the interpolation polynomial polynomial interpolation with n 1. Solving this equation ... 0 y 1 x x 0 y 0 x 1 x 0 math which is the formula for linear interpolation in the interval math scriptstyle ... are normalized distances between the unknown point and each of the end points. Interpolation of a data set Image Interpolation example linear.svg thumb right 300px Linear interpolation on a data set red points consists of pieces of linear interpolants blue lines . Linear interpolation on a set of data ... class math C 0 math . Linear interpolation as approximation Linear interpolation is often ... p denotes the linear interpolation polynomial defined above math p x f x 0 frac f x 1 f x 0 x 1 ... is, the worse the approximations made with simple linear interpolation. Applications Linear interpolation .... Linear interpolation is an easy way to do this. The basic operation of linear interpolation between ... operations for example, a bilinear interpolation can be accomplished in two lerps. Because this operation ... is known be smoother than C sup 0 sup , it is common to replace linear interpolation with spline interpolation , or even polynomial interpolation in some cases. Multivariate Linear interpolation as described ... of linear interpolation is called bilinear interpolation , and in three dimensions, trilinear interpolation ... of bilinear interpolation in the figure below. Other extensions of linear interpolation can be applied ... below . Image Bilininterp.png right thumb Example of bilinear interpolation on the unit square ... top and the convex polytopes on which it is linear bottom . History Linear interpolation has been ... more details
In numerical analysis , multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable. The function to be interpolated is known at given points math x i, y i, z i, dots math and the interpolation problem consist of yielding values at arbitrary points math x,y,z, dots math . Regular grid For function values known on a regular grid having predetermined, not necessarily uniform, spacing , the following methods are available. Any dimension Nearest neighbor interpolation 2 dimensions Barnes interpolation Bilinear interpolation Bicubic interpolation B zier surface Lanczos resampling Delaunay triangulation Inverse distance weighting Kriging Natural neighbor Spline interpolation Resampling bitmap Bitmap resampling is the application of 2D multivariate interpolation in image processing . Three of the methods applied on the same dataset, from 16 values located at the black dots. The colours represent the interpolated values. gallery Image Nearest2DInterpolExample.png Nearest neighbor Image BilinearInterpolExample.png Bilinear Image BicubicInterpolationExample.png Bicubic gallery See also Padua points , for polynomial interpolation in two variables. 3 dimensions Trilinear interpolation Tricubic interpolation See also Resampling bitmap bitmap resampling . Tensor product splines for N dimensions Catmull Rom splines can be easily generalized to any number of dimensions. The cubic Hermite spline article will remind you that math mathrm CINT x f 1 ... interpolation article. However, the fact remains that if there are math n math terms in the 1 dimensional ... interpolation Triangulated irregular network based natural neighbor Triangulated irregular network based linear interpolation a type of piecewise linear function Inverse distance weighting Kriging ... papers multidim.pdf Multi dimensional Hermite Interpolation and Approximation , Prof. Chandrajit Bajaja, Purdue University Category Interpolation Category Multivariate interpolation Mathapplied stub es ... more details
In mathematics , bilinear interpolation is an extension of linear interpolation for interpolation interpolating ... is the bilinear form of math x math and math y math . The key idea is to perform linear interpolation ... values and in the position, the interpolation as a whole is not linear but rather quadratic ... right thumb Example of bilinear interpolation on the unit square with the z values 0, 1 ... y sub 2 sub . We first do linear interpolation in the x direction. This yields math f R 1 approx ..., 1 , then the interpolation formula simplifies to math f x,y approx f 0,0 , 1 x 1 y f 1,0 ... if the interpolation is not linear in the position x and y , it is linear in the amplitude, as it is apparent ... of the function f , . The result of bilinear interpolation is independent of the order order here meaning which axis is interpolated first and which second of interpolation. If we had first performed the linear interpolation in the y direction and then in the x direction, the resulting approximation would be the same. The obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation . Application in image processing In computer vision and image processing , bilinear interpolation is one of the basic resampling techniques. In texture mapping , it is also known ... Bilinear interpolation definition at www.pcmag.com ref When an image needs to be scaled up, each ... RGB or grayscale values so that the output image does not have non valued pixels. Bilinear interpolation ... can calculate and assign appropriate intensity values to pixels. Unlike other interpolation techniques such as nearest neighbor interpolation and bicubic interpolation , bilinear interpolation uses only ... the appropriate color intensity values of that pixel. Bilinear interpolation considers the closest ... Interpolation ref File Bilin3.png thumb left alt Bilinear interpolation Bilinear interpolation in grayscale ... by resizing an image to a non integral zoom factor, as opposed to nearest neighbor interpolation, which ... more details
Multiple issues lead missing December 2010 confusing December 2010 In mathematics , Birkhoff interpolation is an extension of polynomial interpolation . It refers to the problem finding a polynomial p of degree d such that math p n i x i y i qquad mbox for i 1, ldots,d, math where the data points math x i,y i math and the nonnegative integers math n i math are given. It differs from Hermite interpolation in that it is possible to specify derivatives of p at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff , who first studied the problem in harvtxt Birkhoff 1906 . In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution. For instance, there is no quadratic ... interpolation problem where the values of p &prime &minus 1 , p 0 and p &prime 1 are given always has a unique solution harv Passow 1983 . An important problem in the theory of Birkhoff interpolation ... as follows. Let d denote the number of conditions as above and let k be the number of interpolation ..., the incidence matrices for the interpolation problems mentioned in the previous paragraph are math ... 1 & 0 & 0 0 & 1 & 0 end bmatrix . math The question now becomes does a Birkhoff interpolation problem with a given incidence matrix have a unique solution for any choice of the interpolation points? The case with k 2 interpolation points was tackled by harvtxt P lya 1931 . Let S sub m sub denote the sum ... . math Then the Birkhoff interpolation problem with k 2 has a unique solution if and only if S sub ... first1 Eli title Book Review Birkhoff interpolation by G. G. Lorentz, K. Jetter and S. D. Riemenschneider ... P lya first1 George author1 link George P lya title Bemerkung zur Interpolation und zur Naherungstheorie ... Schoenberg title On Hermite Birkhoff interpolation doi 10.1016 0022 247X 66 90160 0 year 1966 journal ... Interpolation eo Interpolo de Birkhoff it Interpolazione di Birkhoff ... more details
In computer programming , variable interpolation also variable substitution or variable expansion is the process of evaluating an expression or string literal containing one or more variables, yielding a result in which the variables are replaced with their corresponding values in memory. It is a specialized instance of concatenation . Languages that support variable interpolation include Perl , PHP , Ruby programming language Ruby , Tcl , and most Unix shell s. In these languages, variable interpolation only occurs when the string literal is double quoted, but not when it is single quoted. The variables are recognized because variables start with a sigil computer programming sigil typically code code in these languages. For example, the following Perl code which would work identically in PHP source lang perl name Alice print name said Hello World to the crowd of people. source produces the output pre Alice said Hello World to the crowd of people. pre Ruby programming language Ruby uses the code number sign code symbol for interpolation, and allows one to interpolate any expression, not just variables. Other languages may support more advanced interpolation with a special formatting function, such as code printf code , in which the first argument, the format , specifies the pattern in which the remaining arguments are substituted. Category Variable computer programming Category Programming constructs ... more details
Image Enclosing points.svg right thumb 3D interpolation tries to assign a value at the red point math C math given values at the blue corner points. In the mathematical subfield numerical analysis , tricubic interpolation is a method for obtaining values at arbitrary points in Three dimensional space 3D space of a function defined on a regular grid . The approach involves approximating the function locally by an expression of the form math f x,y,z sum i 0 3 sum j 0 3 sum k 0 3 a ijk x i y j z k. math This form has 64 coefficients math a ijk math requiring the function to have a given value or given directional derivative at a point places one linear constraint on the 64 coefficients. The term tricubic interpolation is used in more than one context some experiments measure both the value of a function and its spatial derivatives, and it is desirable to interpolate preserving the values and the measured derivatives at the grid points. Those provide 32 constraints on the coefficients, and another 32 constraints can be provided by requiring smoothness of higher derivatives ref http citeseerx.ist.psu.edu viewdoc summary?doi 10.1.1.89.7835 Tricubic Interpolation in Three Dimensions 2005 , by F. Lekien, J. Marsden, Journal of Numerical Methods and Engineering ref . In other contexts, we .... The cubic interpolation article indicates that the method is equivalent to a sequential application ... ref http citeseerx.ist.psu.edu viewdoc summary?doi 10.1.1.89.7835 Tricubic Interpolation in Three ... more efficient. This aspect is even much more pronounced when interpolation is needed at several ... the interpolation coefficients for the entire cube. The coefficients are then stored and used for interpolation ... computational step must be repeated for each new location. See also Cubic interpolation Bicubic interpolation Trilinear interpolation References references External links http www.paulinternet.nl ?page bicubic Java C implementation of tricubic interpolation Category Multivariate interpolation ... more details
In mathematics , trigonometric interpolation is interpolation with trigonometric polynomial s. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of trigonometric function sines and cosines of given periods. This form is especially suited for interpolation of periodic function s. An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform . Formulation of the interpolation problem A trigonometric polynomial of degree n has the form math p x a 0 sum m 1 n a m cos mx sum m 1 n b m sin mx . , math This expression contains 2 n 1 coefficients, a sub 0 sub , a sub 1 sub , a sub n sub , b sub 1 sub , , b sub n sub , and we wish to compute those coefficients so that the function .... , math Note that we do not in general require these points to be equally spaced. The interpolation problem is now to find coefficients such that the trigonometric polynomial p satisfies the interpolation ... to the Lagrange polynomial Lagrange formula for polynomial interpolation math p x sum k 1 2n 1 y k prod ... the problem of trigonometric interpolation to that of polynomial interpolation on the unit circle . Existence and uniqueness for trigonometric interpolation now follows immediately from the corresponding results for polynomial interpolation. For more information on formulation of trigonometric interpolating ... , p135 Interpolation using Fourier Polynomials . Equidistant nodes and the discrete Fourier ... of the cosine only interpolation for equally spaced points, corresponding to a trigonometric interpolation when the points have Even and odd functions even symmetry , was treated by Alexis Clairaut ... since the orbits are periodic, a trigonometric interpolation was a natural choice. See also Heideman ... . Category Interpolation Category Trigonometry de Trigonometrische Interpolation es Interpolaci n ... more details
This article is about Interpolation Theory in Biology. See also interpolation mathematics , interpolation music , interpolation manuscripts and interpolation computer programming . The Interpolation Theory , also known as the Intercalation Theory or the Antithetic Theory , is a theory that attempts to explain the origin of the alternation of generations in plants . The Interpolation Theory suggests that the sporophyte generation progenated from a haploid , green algal thallus in which repeated mitotic cell divisions of a zygote produced an embryo retained on the thallus and gave rise to the diploid phase sporophyte . Ensuing evolution caused the sporophyte to become increasingly complex, both oranographically and anatomically. The Interpolation Theory was introduced by Ladislav Josef elakovsk elakovsk 1874 as the Antithetic Theory. Bower 1889 further developed this theory and renamed it the Interpolation Theory. The theory was later supported by Overton 1893 , Scott 1896 , Strasburger 1897 , Williams 1904 , and others. The gradual evolution of an independent, sporophyte phase was viewed by Bower as being closely related to the transition from aquatic to terrestrial plant life on Earth. The supporters of the Interpolation Theory believe that the progenitors of land plants were algal like, and evolved for terrestrial life under water stresses. The oldest known fossils of land plants were discovered in the cores of boreholes in Oman . The cores contained sets of four connected spore cells, divided from one mother cell. These sets of four are referred to as tetrads . The tetrads were enveloped by the remains of the spore sac in which they were developed. Study of the spore wall indicates a relationship with the liverworts . The fossils have been dated around the Middle Ordovician and are about 475 million years old. Evidence supporting this theory can be found in the life cycle of modern Bryophytes in which the sporophyte is physiologically dependent on the gametophyte ... more details
In the field of mathematical analysis , an interpolation space is a space which lies in between two other .... History The theory of interpolation of vector spaces began by an observation of J zef Marcinkiewicz ... transform , complex interpolation, real interpolation, as well as other tools see e.g. fractional ... spaces , between X and Y in the following sense Definition With X and Y as above, an interpolation ... A to B . If C 1 which is the smallest possible , we further say that W is an exact interpolation space . There are many ways of obtaining interpolation spaces and the Riesz Thorin theorem is an example of this for L sup p sup spaces . A method for arbitrary Banach spaces is the complex interpolation method. Complex interpolation If the Field mathematics field of scalars is the complex number s, then we may use properties of complex analytic function s to define an interpolation space. Definition For two Banach spaces X and Y , the complex interpolation method consists in looking at the space ... sub is an exact interpolation space of exponent &theta . Real interpolation the K method The K method of real interpolation can be used even when the field of scalars is the real number s. Definition ... method of real interpolation consists in taking math K theta,q X,Y math to be the set of all u in X Y such that math u theta,q K infty math . Then, math K theta,q X,Y math is an exact interpolation space of power &theta . Real interpolation the J method As with the K method, the J method can also ... of u is math u theta,q J inf v Phi v math . Again, math J theta,q X,Y math is an exact interpolation space of power &theta . Relations between the interpolation methods The two real interpolation ...,Y theta,q math for the real interpolation method. By contrast, the complex interpolation method is usually not equivalent to the real interpolation method. However, there is still a relationship. Theorem ... citation last Bergh first J ran last2 L fstr m first2 J rgen title Interpolation Spaces An Introduction ... more details
Refimprove date April 2009 Original research date April 2009 Interpolation search sometimes referred ... via a linear interpolation. The key value actually found at this estimated position is then compared ... any sorting. On average the interpolation search makes about log log n comparisons if the elements ... notation O n comparisons. In interpolation sequential search , interpolation is used to find an item ... big O notation, the performance of the interpolation algorithm on a data set of size N is O N however under the assumption of a uniform distribution of the data on the linear scale used for interpolation ... Wesley ref . However, Dynamic Interpolation Search is possible in o log log n time using a novel data structure. ref Andersson, Arne, and Christer Mattsson. Dynamic Interpolation Search in o log ... dx.doi.org 10.1007 3 540 56939 1 58. ref Practical performance of interpolation search depends on whether ... a disk seek and is much slower than the interpolation arithmetic. Index structures like B tree ... because they can index many types of data and can be updated Online algorithm online . Still, interpolation ..., linear interpolation is straightforward to implement and will find an index very near the sought value. On the other hand, for a phone book sorted by name, the straightforward approach to interpolation ... location. Some interpolation search implementations may not work as expected when a run of equal key values exists. The simplest implementation of interpolation search won t necessarily select the first or last element of such a run. The interpolation calculation also must be written to avoid ... that at a glance a segmented interpolation can be performed. Sample implementation The following code ... guarantees a halving of the interval s size with each stage, a misled interpolation may reduce increase ... s value will not be much different the interpolation calculation is not much improved by this one ... elements are stored to beat that the interpolation search as written above would be allowed no more ... more details
An interpolation , in relation to literature and especially ancient manuscript s, is an entry or passage in a text that was not written by the original author. As there are often several generations of copies between an extant copy of an ancient text and the original, each handwritten by different scribes, there is a natural tendency for extraneous material to be inserted into such documents over time. Interpolations may be inserted as an authentic explanatory note for example, bracket sic , but may also be included for fraudulent purposes. The forged passages and works attributed to the Pseudo Isidore are an example of the latter. Similarly, the letters of Ignatius of Antioch were interpolated by Apollinarian heretics, three centuries after the originals were written. Charters and legal texts are also subject to forgery of this kind. In the 13th Century, a medieval romance, the Prose Tristan , inserted another prose romance the Vulgate Queste del Saint Graal in its entirety in order to reinterpret the Quest for the Holy Grail through the optics of the Tristan story. ref On the Medieval technique of manuscript interpolation and the Prose Tristan , see Emmanu le Baumgartner, La pr paration la Queste del Saint Graal dans le Tristan en prose in Norris Lacy, ed. Conjunctures Amsterdam Rodopi, 1994 , pp. 1 14, Fanni Bogdanow, L Invention du texte, intertextualit et le probl me de ... and Abuse of the Grail Quest Tristania 15 1994 23 31. Gaston Paris, in 1897, also noted the interpolation ... interpolation, which are often apparent to modern observers, but would have been less so for medieval copyists. The Comma Johanneum is commonly regarded as interpolation. See also Western non interpolations Archive Media preservation References references DEFAULTSORT Interpolation Manuscripts Category Manuscripts Interpolation manuscripts cs Interpolace text de Interpolation Literatur ... ru sv Interpolation manuskript uk ... more details
Image BicubicInterpolationExample.png thumb right Bicubic interpolation on the square math 0,3 times 0,3 math consisting of 9 unit squares patched together. Bicubic interpolation as per MATLAB s implementation ... interpolation on the same dataset as above. Derivatives of the surface are not continuous over the square boundaries. Image Nearest2DInterpolExample.png thumb right Nearest neighbor interpolation .... In mathematics , bicubic interpolation is an extension of cubic interpolation for interpolation ... function smoother than corresponding surfaces obtained by bilinear interpolation or nearest neighbor interpolation . Bicubic interpolation can be accomplished using either Lagrange polynomials , cubic ... interpolation is often chosen over bilinear interpolation or nearest neighbor in resampling bitmap image resampling , when speed is not an issue. In contrast to bilinear interpolation, which only takes 4 pixels 2x2 into account, bicubic interpolation also considers the 16 pixels around it for a total of 4x4 pixels while computing an average. Images resampled with bicubic interpolation are smoother and have fewer interpolation anti aliasing artifacts . Bicubic interpolation Suppose the function ... The interpolation problem consists of determining the 16 coefficients math a ij math . Matching ... math 0,1 times 0,1 math which is continuous and with continuous derivatives. Bicubic interpolation ... right math . Bicubic convolution algorithm Bicubic spline interpolation requires the solution ... function. ref name Keys cite journal author R. Keys, year 1981 title Cubic convolution interpolation ... x,y p t y, b 1 , b 0 , b 1 , b 2 math Use in computer graphics Image Accutance.svg thumb Bicubic interpolation ... portal box Mathematics Computer graphics Anti aliasing B zier surface Bilinear interpolation Cubic ... interpolation Sinc filter Spline interpolation Tricubic interpolation References reflist External links http www.geovista.psu.edu sites geocomp99 Gc99 082 gc 082.htm Application of interpolation to elevation ... more details
In numerical analysis , polynomial interpolation is the interpolation of a given data set by a polynomial ... computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations . Polynomial interpolation is also essential to perform ... , where an interpolation through points on a polynomial which defines the product yields the product .... Interpolation based on those points will yield the terms of W x and subsequently the product ... matrix , as described below. The theorem states that for n 1 interpolation nodes x sub i sub , polynomial interpolation defines a linear bijection math L n mathbb K n 1 to Pi n math where math Pi n ... n . Constructing the interpolation polynomial Image Interpolation example polynomial.svg thumb ... the interpolation polynomial. Suppose that the interpolation polynomial is in the form math p x a n ... first a Newton polynomial Newton interpolation of the polynomial and then converting it to the monomial ... what method we use to do our interpolation direct, Spline mathematics spline , Lagrange polynomial .... Non Vandermonde solutions We are trying to construct our unique interpolation polynomial in the vector ... math we have to solve the Vandermonde matrix to construct the coefficients math a k math for the interpolation ... of the coefficients but then we have to do additional calculations when we want to express the interpolation polynomial in terms of a monomial basis . One method is to write the interpolation polynomial .... Another method is to use the Lagrange form of the interpolation polynomial. The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above ... of B zier curve s. Interpolation error clarify section date June 2011 When interpolating a given ... form of the Taylor s theorem Taylor theorem is a special case of interpolation error when all interpolation nodes x sub i sub are identical. In the case of equally spaced interpolation nodes ... more details
about music other uses Interpolation disambiguation Cleanup date September 2008 Interpolation in music refers to an abrupt change in musical elements from the main theme. In classical music In music and musical composition , especially twentieth century music 20th century and later, interpolation is an abrupt change of aspect of music elements , with almost immediate continuation of the first idea music idea . ref name Aspects Wittlich, Gary E. ed. 1975 . Aspects of Twentieth century Music , p.48 n.12 and p.49. Englewood Cliffs, N.J. Prentice Hall. ISBN 0 13 049 346 5. ref Pieces which are cited as featuring interpolation, among other musical technique techniques , are Music for Brass Quintet by Gunther Schuller and Threnody To the Victims of Hiroshima by Krzysztof Penderecki , both 1960 61 . ref name Aspects For music of the Classical period music Classical period , interpolation is defined in the context of a Sentence music musical sentence or period music period as, unrelated material inserted between two logically succeeding diatonic function functions . ref William E. Caplin, Classical Form A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven , p. 255. ISBN 019514399X. ref This device is commonly used to extend what would normally be a regular phrase music phrase into an irregular and extended phrase. Such expansion by interpolation is achieved by the addition of extra music in the middle of a phrase commonly through the use of sequence music sequence . A clear example exists in the second movement of Piano Sonata No. 10 Mozart Mozart s Piano Sonata No. 10, K.330 . Citation needed date July 2008 Formerly, in the sung portions of the Mass ..., minuet, musette or passepied. Citation needed date July 2008 In popular music Interpolation ... to be influential 1992 In rap music, interpolation refers to using a melody or portions of a melody ... DEFAULTSORT Interpolation Music Category Musical techniques de Interpolation Musik ... more details
In cryptography , an interpolation attack is a type of cryptanalysis cryptanalytic attack against block cipher s. In the attack, an algebraic function is used to represent an S box . This may be a simple quadratic function quadratic , or a polynomial or rational function over a Galois field . Its coefficients can be determined by standard Lagrange interpolation techniques, using known plaintext attack known plaintexts as data points. Alternatively, chosen plaintext attack chosen plaintexts can be used to simplify the equations and optimize the attack. Thomas Jakobsen introduced a randomized algorithm probabilistic version of the interpolation attack using Madhu Sudan s algorithm for improved decoding of Reed Solomon code s. This attack can work even when an algebraic relationship between plaintexts and ciphertexts holds for only a fraction of values. References cite conference author Thomas Jakobsen , Lars Knudsen title The Interpolation Attack on Block Ciphers conference 4th International Workshop on Fast Software Encryption FSE 97 , LNCS 1267 pages pp.28&ndash 40 publisher Springer Verlag month January year 1997 location Haifa url http citeseer.ist.psu.edu jakobsen97interpolation.html format PDF PostScript accessdate 2007 07 03 cite conference author Thomas Jakobsen title Cryptanalysis of Block Ciphers with Probabilistic Non linear Relations of Low Degree conference Advances in Cryptology &mdash CRYPTO 98 pages pp.212&ndash 222 publisher Springer Verlag date August 25, 1998 location Santa Barbara, California url http citeseer.ist.psu.edu jakobsen98cryptanalysis.html format PDF PostScript accessdate 2007 07 06 http video.google.com videoplay?docid 502705185794473481&hl en ..., Takeshi Shimoyama, Toshinobu Kaneko title Interpolation Attacks of the Block Cipher SNAKE conference ... conference author Amr M. Youssef, Guang Gong title On the Interpolation Attacks on Block Ciphers conference ... conference author Kaoru Kurosawa, Tetsu Iwata, Viet Duong Quang title Root Finding Interpolation ... more details
Motion interpolation is a form of video processing in which intermediate animation frames are generated between existing ones, in an attempt to make animation more fluid. Applications HDTV Motion interpolation is used in various display devices such as HDTV s and video players, aimed at alleviating the video artifact s introduced by framerate conversions in fixed framerate displays such as LCD TVs. Films are recorded at a frame rate of 24 frames per second frame s and television is typically filmed at 25, 50, 30 or 60 frames per second the first two being PAL, the other two from NTSC . Normally, when a fixed framerate display such as an LCD screen is used to display a video source whose framerate is less than that of the screen, frames are often simply duplicated as necessary until the timing ... , perceived as jumpiness in the picture. Motion interpolation intends to remedy this by generating ... interpolation feature. Also, anti judder technology is not the same as motion blur reducing technology ... given to motion interpolation technology varies across manufacturers, as does its implementation. Hitachi ... for frame interpolation. ref http www.corel.com servlet Satellite us en Product 1166553865668 Intervideo WinDVD 8 Platinum ref CrystalPlayer uses Motion Morphing MultiSampling for frame interpolation ... Motion for interpolation of DVD and video files to up to 72 frame s. ref cite web url http www.cyberlink.com ... HD video interpolation. ref http mirillis.com en products picture2.html Mirillis Motion technology ... avisynth.org.ru mvtools mvtools2.html MVTools plugin for AviSynth 2.5 ref libraries for frame interpolation. Possible Unwanted Effects Visual Artifacts Motion interpolation on certain brands of HDTVs ... Soap Opera Effect The video look is a byproduct of the perceived increase in framerate due to the interpolation ... CNET.com ref Others appreciate motion interpolation as it reduces motion blur produced by camera pans ... Display technology Category Video processing de Motion Interpolation ... more details
Barnes interpolation , named after Stanley L. Barnes, is the interpolation of unstructured data points from a set of measurements of an unknown function in two dimensions into an analytic function of two variables. An example of a situation where the Barnes scheme is important is in weather forecasting ref cite web url http www.bom.gov.au bmrc mdev expt rainanal rainanal.shtml title Objective Rainfall Analysis System accessdate 6th May 2009 ref ref cite journal title Thunderstorm distribution and frequency in Australia series Australian Meteorological Magazine year 2002 pages 145 154 author Y.Kuleshov, G. de Hoedt, W.Wright and A.Brewster ref where measurements are made wherever monitoring stations may be located, the positions of which are constrained by topography . Such interpolation is essential in data visualisation, e.g. in the construction of contour plot s or other representations of analytic surfaces. Introduction Barnes proposed an objective scheme for the interpolation of two dimensional data using a multi pass scheme. ref cite journal first S. L last Barnes title A technique for maximizing details in numerical weather map analysis journal Journal of Applied Meterology volume 3 issue 4 pages 396&ndash 409 year 1964 url http ams.allenpress.com perlserv ?request get abstract&doi 10.1175 2F1520 0450 1964 003 3C0396 ATFMDI 3E2.0.CO 3B2 doi 10.1175 1520 0450 1964 003 0396 ATFMDI 2.0.CO 2 bibcode 1964JApMe...3..396B ref ref cite journal last Barnes title Mesoscale objective analysis using weighted time series observations series NOAA Technical Memorandum publisher National ... the method of interpolation used in a multi pass Barnes interpolation. First pass For a given grid ... interpolation for the function from the measured values math f k x,y math then becomes math ... of interpolation integrity, x is argued to be constrained between 0.3 and 0.5. Notes references Category Interpolation Category Spatial data analysis ... more details
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