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Interpolation (music)
Interpolation is a method of constructing new data points within the range of a discrete set of known data points in the mathematical field of numerical analysis. Interpolation may also refer to: Science and technology *Interpolation space, in mathematical analysis, the space "in between" two other Banach spaces *Craig interpolation, in mathematical logic, a result about the relationship between logical theories * Interpolation (computer graphics), the generation of intermediate frames ** Image scaling, the resizing of a digital image ** Inbetweening, the generation of intermediate video frames ** Motion interpolation, a form of video processing * Interpolation theory, an explanation of the alternation of generations in plants * String interpolation, in computing, the substitution of variables by their values Music * Interpolation (classical music), musical material inserted between two logically succeeding functions * Interpolation (popular music), the inclusion of a re-recorde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Space
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of derivatives. History The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem. In simple terms, if a linear function is continuous on a certain space and also on a certain space , then it is also continuous on the space , for any intermediate between and . In other words, is a space which is intermediate between and . In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and Jacques-Louis Lions discovered that indeed these trace spaces were constituted of functions that have a no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig Interpolation
In mathematical logic, Craig's interpolation theorem is a result about the relationship between different logical theories. Roughly stated, the theorem says that if a formula φ implies a formula ψ, and the two have at least one atomic variable symbol in common, then there is a formula ρ, called an interpolant, such that every non-logical symbol in ρ occurs both in φ and ψ, φ implies ρ, and ρ implies ψ. The theorem was first proved for first-order logic by William Craig in 1957. Variants of the theorem hold for other logics, such as propositional logic. A stronger form of Craig's interpolation theorem for first-order logic was proved by Roger Lyndon in 1959; the overall result is sometimes called the Craig–Lyndon theorem. Example In propositional logic, let ::: \varphi = \lnot(P \land Q) \to (\lnot R \land Q) ::: \psi = (S \to P) \lor (S \to \lnot R) . Then \varphi tautologically implies \psi. This can be verified by writing \varphi in conjunctive normal form: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation (computer Graphics)
In the context of live-action and computer animation, interpolation is inbetweening,{{Cite web, url=https://www.freecodecamp.org/news/understanding-linear-interpolation-in-ui-animations-74701eb9957c/, title=Understanding Linear Interpolation in UI Animation, date=2017-05-14, website=Developer News, language=en, access-date=2019-08-26 or filling in frames between the key frames. It typically calculates the in-between frames through use of (usually) piecewise polynomial interpolation to draw images semi-automatically. For all applications of this type, a set of "key points" is defined by the graphic artist. These are values that are rather widely separated in space or time, and represent the desired result, but only in very coarse steps. The computed interpolation process is then used to insert many new values in between these key points to give a "smoother" result. In its simplest form, this is the drawing of two-dimensional curves. The key points, placed by the artist, are u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Scaling
In computer graphics and digital imaging, image scaling refers to the resizing of a digital image. In video technology, the magnification of digital material is known as upscaling or resolution enhancement. When scaling a vector graphic image, the graphic primitives that make up the image can be scaled using geometric transformations, with no loss of image quality. When scaling a raster graphics image, a new image with a higher or lower number of pixels must be generated. In the case of decreasing the pixel number (scaling down) this usually results in a visible quality loss. From the standpoint of digital signal processing, the scaling of raster graphics is a two-dimensional example of sample-rate conversion, the conversion of a discrete signal from a sampling rate (in this case the local sampling rate) to another. Mathematical Image scaling can be interpreted as a form of image resampling or image reconstruction from the view of the Nyquist sampling theorem. According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inbetweening
Inbetweening, also known as tweening, is a process in animation that involves creating intermediate Film frame, frames, called inbetweens, between two key frame, keyframes. The intended result is to create the illusion of movement by smoothly transitioning one image into another. Traditional animation Traditional inbetweening involves the use of a light table to draw a set of pencil and paper drawings. The process of inbetweening in traditional animation starts with a primary artist, who draws key frames to define movement. After the testing and approval of a rough animation, the scene is passed down to assistants, who perform Cleanup (animation), clean-up and add necessary inbetweening. In large studios, assistants usually add breakdowns, which define the movement in more detail. The scene is then passed down to another assistant, the inbetweener who completes the animation. In small animation teams, animators will often carry out the full inbetweening process themselves. Dick H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motion Interpolation
Motion interpolation or motion-compensated frame interpolation (MCFI) is a form of video processing in which intermediate animation frames are generated between existing ones by means of interpolation, in an attempt to make animation more fluid, to compensate for display motion blur, and for fake slow motion effects. Hardware applications Displays Motion interpolation is a common, optional feature of various modern display devices such as HDTVs and video players, aimed at increasing perceived framerate or alleviating display motion blur, a common problem on LCD flat-panel displays. Difference from display framerate A display's framerate is not always equivalent to that of the content being displayed. In other words, a display capable of or operating at a high framerate does not necessarily mean that it can or must perform motion interpolation. For example, a TV running at 120 Hz and displaying 24 FPS content will simply display each content frame for five of the 120 dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Theory
{{about, Interpolation Theory in biology, other types of interpolation, interpolation (other) 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 organographically and anatomically. The Interpolation Theory was introduced by Čelakovský (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, sporoph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Interpolation
In computer programming, string interpolation (or variable interpolation, variable substitution, or variable expansion) is the process of evaluating a string literal containing one or more placeholders, yielding a result in which the placeholders are replaced with their corresponding values. It is a form of simple template processing or, in formal terms, a form of quasi-quotation (or logic substitution interpretation). The placeholder may be a variable name, or in some languages an arbitrary expression, in either case evaluated in the current context. String interpolation is an alternative to building string via concatenation, which requires repeated quoting and unquoting; or substituting into a printf format string, where the variable is far from where it is used. Compare: apples = 4 print("I have $ apples.") # string interpolation print("I have " + apples + " apples.") # string concatenation print("I have %s apples.", apples) # format string Two types of literal expression ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation (classical Music)
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". This device is commonly used to extend what would normally be a regular 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). A clear example exists in the second movement of Mozart's Piano Sonata No. 10, K.330. Formerly, in the sung portions of the Mass, such as the introit or kyrie, it was permissible, especially during the medieval period, to amplify a liturgical formula by interpolating a "farse" (from Medieval Latin ''farsa'', forcemeat), also called "trope". New Advent. This mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation (popular Music)
In popular music, interpolation (also called a replayed sample) refers to using a melody—or portions of a melody (often with modified lyrics)—from a previously recorded song but re-recording the melody instead of sampling it. Interpolation is often used when the artist or label who owns the piece of music declines to license the sample, or if licensing the piece of music is considered too costly. Interpolation examples Interpolation is prevalent in many genres of popular music; early examples are the Beatles interpolating "La Marseillaise" and "She Loves You", among three other interpolations in the 1967 song "All You Need Is Love", and Lyn Collins interpolating lyrics from the 5 Royales' "Think" in her similarly titled 1972 song "Think (About It)". One genre where interpolating (as well as sampling) is highly prevalent is hip hop music; prominent examples include Stevie Wonder's "Pastime Paradise" interpolated in Coolio's hit song "Gangsta's Paradise", and Sting's " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation (manuscripts)
An interpolation, in relation to literature and especially ancient manuscripts, 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. Overview Interpolations originally may be inserted as an authentic explanatory note (for example, ), 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'', i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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