Premultiplied alpha has some practical advantages over normal alpha blending because premultiplied alpha blending is associative and linear interpolation gives better results, although premultiplication can cause a loss of precision and, in extreme cases, a noticeable loss of quality.

and is equivalent to linear interpolation.

* The curve begins at P < sub > 0 </ sub > and ends at P < sub > n </ sub >; this is the so-called endpoint interpolation property.

While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.

This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies ( e. g. changing to ) without changing the interpolation property, but giving different values in between the points.

First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to ( instead of roughly to as above ), similar to the inverse DFT formula.

This interpolation does not minimize the slope, and is not generally real-valued for real ; its use is a common mistake.

" Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.

When that interval is 0. 5 / T, the applicable reconstruction formula is the Whittaker – Shannon interpolation formula.

In most cases, N is chosen equal to the length of non-zero portion of s. Increasing N, known as zero-padding or interpolation, results in more closely spaced samples of one cycle of S < sub > 1 / T </ sub >( ƒ ).

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

A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function.

Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present ; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in accuracy.

There is also another very different kind of interpolation in mathematics, namely the " interpolation of operators ".

Points through which curve is spline ( mathematics ) | splined are red ; the blue curve connecting them is interpolation.

The general scholarly view is that while the Testimonium Flavianum is most likely not authentic in its entirety, it is broadly agreed upon that it originally consisted of an authentic nucleus with a reference to the execution of Jesus by Pilate which was then subject to Christian interpolation.

Quarter-pixel motion is derived by linear interpolation of the halfpel values, to save processing power.

Shepard's interpolation for different power parameters p, from scattered points on the surface.

is a simple IDW weighting function, as defined by Shepard, x denotes an interpolated ( arbitrary ) point, x < sub > i </ sub > is an interpolating ( known ) point, is a given distance ( metric operator ) from the known point x < sub > i </ sub > to the unknown point x, N is the total number of known points used in interpolation and is a positive real number, called the power parameter.

it is easy to check two limits of this interpolation that follow laws similar to the Bloch law, for, and the critical behavior, for, respectively.

The most common solution is a technique called pixel interpolation that smoothly blends or interpolates the color of one pixel into the color of the next adjacent pixel at high levels of zoom.

*" On the analytical determination of relaxation modulus of viscoelastic materials by Prony's interpolation method " ( 1986 )

Cennétig was described as rígdamna Caisil, meaning that he was either heir or candidate (" king material ") to the kingship of Cashel or Munster, although this might be a later interpolation.

In film the term ad-lib usually refers to the interpolation of unscripted material in an otherwise scripted performance.

Differences often extend beyond minor variations and may involve, for instance, interpolation of material central to issues of historicity and doctrine, such as the ending of Mark 16.

Additionally, material may be displayed with synthetically created smoothness with the addition of motion interpolation abilities to the display, which has an even larger effect on filmed material.

For music of the Classical period, interpolation is defined in the context of a musical sentence or period as, " unrelated material inserted between two logically succeeding functions.

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form.

They are nevertheless useful in different frameworks: kriging is made for interpolation of a single realisation of a random field, while regression models are based on multiple observations of a multivariate dataset.

In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline.

Above all, the field interpolation scheme should conserve momentum.

As the final PAL DVD release shows, the film was shot 50i ( interlaced ), and only special effects shots were deinterlaced by means of smart field blending ( imitating a progressive-type amount of motion blur due to a different shutter speed of progressive modes ) during post-production, as the special effects crew obviously was in demand of progressive frames which are easier to process for special effects, before for the final release all normal shots were deinterlaced by means of simple line interpolation instead.

The breakthrough in the field of data assimilation was achieved by L. Gandin ( 1963 ) who introduced the " statistical interpolation " ( or " optimal interpolation ") method.

While Gouraud is a linear interpolation across the surface of a polygon, Nebulaud takes the luminosity of each pixel derived from Gouraud and shifts it within a finite field of predermined length.

Most of the physical properties of astatine have been estimated ( by interpolation or extrapolation ), using various theoretically-grounded or empirically-derived methods.

For 3D animation Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.

In automatic systems this can be done using a binary search algorithm or interpolation search ; manual searching may be performed using a roughly similar procedure, though this will often be done unconsciously.

where the coefficients X < sub > k </ sub > are given by the DFT of x < sub > n </ sub > above, satisfies the interpolation property for.

have often been regarded as a post-Pauline interpolation.

Other scholars, such as Schmithals, Eckhart, Demke and Munro, have developed complicated theories involving redaction and interpolation in 1 and 2 Thessalonians.