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Spline Interpolation
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. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-nine polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline. Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials. Introduction Originally, '' spline'' was a term for elastic rulers that were bent to pass through a number of predefined points, or ''knots''. These we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Splines Three Points
Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system where the unit cell is in the shape of a cube * Cubic function, a polynomial function of degree three * Cubic equation, a polynomial equation (reducible to ''ax''3 + ''bx''2 + ''cx'' + ''d'' = 0) * Cubic form, a homogeneous polynomial of degree 3 * Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3 * Cubic plane curve (mathematics), a plane algebraic curve ''C'' defined by a cubic equation * Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity * Cubic surface, an algebraic surface in three-dimensional space * Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra * CUBIC, a histology method Computing * Cubic IDE, a modular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thin Plate Spline
Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm. Physical analogy The name ''thin plate spline'' refers to a physical analogy involving the bending of a plate or thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. In 2D cases, given a set of K corresponding control points (knots), the TPS warp is described by 2(K+3) parameters which include 6 global affine motion parameters a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spline Wavelet
In the mathematics, mathematical theory of wavelets, a spline wavelet is a wavelet constructed using a spline function. There are different types of spline wavelets. The interpolatory spline wavelets introduced by C.K. Chui and J.Z. Wang are based on a certain Spline (mathematics), spline interpolation formula. Though these wavelets are Orthogonality, orthogonal, they do not have Compact space, compact Support (mathematics), supports. There is a certain class of wavelets, unique in some sense, constructed using B-splines and having compact supports. Even though these wavelets are not orthogonal they have some special properties that have made them quite popular. The terminology ''spline wavelet'' is sometimes used to refer to the wavelets in this class of spline wavelets. These special wavelets are also called B-spline wavelets and cardinal B-spline wavelets. The Battle-Lemarie wavelets are also wavelets constructed using spline functions. Cardinal B-splines Let ''n'' be a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing Spline
Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of \hat f(x). They provide a means for smoothing noisy x_i, y_i data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where x is a vector quantity. Cubic spline definition Let \ be a set of observations, modeled by the relation Y_i = f(x_i) + \epsilon_i where the \epsilon_i are independent, zero mean random variables. The cubic smoothing spline estimate \hat f of the function f is defined to be the unique minimizer, in the Sobolev space W^2_2 on a compact interval, of : \sum_^n \^2 + \lambda \int \hat^(x)^2 \,dx. Remarks: * \lambda \ge 0 is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate. This is oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no two x_j the same, a polynomial function p(x)=a_0+a_1x+\cdots+a_nx^n is said to interpolate the data if p(x_j)=y_j for each j\in\. There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. Applications The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in numerical quadrature ( Simpson's rule) and numerical ordinary differential equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Interpolation
In numerical analysis, multivariate interpolation or multidimensional interpolation is interpolation on ''multivariate functions'', having more than one variable or defined over a multi-dimensional domain. A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions. When the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points (x_i, y_i, z_i, \dots) and the interpolation problem consists of yielding values at arbitrary points (x,y,z,\dots). Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey). Regular grid For function values known on a regular grid (having predetermined, not necessarily uniform, spacing), the following methods are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-uniform Rational B-spline
Non-uniform rational basis spline (NURBS) is a mathematical model using B-spline, basis splines (B-splines) that is commonly used in computer graphics for representing curves and Surface (mathematics), surfaces. It offers great flexibility and precision for handling both analytic (defined by common mathematical formulae) and 3D modeling, modeled shapes. It is a type of 3D modeling#Process, curve modeling, as opposed to polygonal modeling or digital sculpting. NURBS curves are commonly used in computer-aided design (CAD), Computer-aided manufacturing, manufacturing (CAM), and Computer-aided engineering, engineering (CAE). They are part of numerous industry-wide standards, such as IGES, ISO 10303, STEP, ACIS, and PHIGS. Tools for creating and editing NURBS surfaces are found in various 3D computer graphics software, 3D graphics, Rendering (computer graphics), rendering, and 3D Animation, animation software packages. They can be efficiently handled by computer programs yet allow for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Cubic Interpolation
In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation. Monotone cubic Hermite interpolation Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m_i modified to ensure the monotonicity of the resulting Hermite spline. An algorithm is also available for monotone quintic Hermite interpolation. Interpolant selection There are several ways of selecting interpolating tangents for each data point. This section will outline the use of the Fritsch–Carlson method. Note that only one pass of the algorithm is required. Let the data points be (x_k,y_k) indexed in sorted order for k=1,\,\dots\,n. # Compute the slopes of the secant lines between successive points:\delta_k =\frac for k=1,\,\dots\,n-1. # These assignments are provisional, and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Spline Interpolation
In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...s are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous. Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences. Discrete cubic splines Let ''x''1, ''x''2, . . ., ''x''''n''-1 be an incre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centripetal Catmull–Rom Spline
In computer graphics, the centripetal Catmull–Rom spline is a variant form of the Catmull–Rom spline, originally formulated by Edwin Catmull and Raphael Rom, which can be evaluated using a recursive algorithm proposed by Barry and Goldman. It is a type of interpolating spline (a curve that goes through its control points) defined by four control points \mathbf_0, \mathbf_1, \mathbf_2, \mathbf_3, with the curve drawn only from \mathbf_1 to \mathbf_2. Definition Let \mathbf_i = [x_i \quad y_i]^T denote a point. For a curve segment \mathbf defined by points \mathbf_0, \mathbf_1, \mathbf_2, \mathbf_3 and knot sequence t_0, t_1, t_2, t_3, the centripetal Catmull–Rom spline can be produced by: : \mathbf = \frac\mathbf_1+\frac\mathbf_2 where : \mathbf_1 = \frac\mathbf_1+\frac\mathbf_2 : \mathbf_2 = \frac\mathbf_2+\frac\mathbf_3 : \mathbf_1 = \frac\mathbf_0+\frac\mathbf_1 : \mathbf_2 = \frac\mathbf_1+\frac\mathbf_2 : \mathbf_3 = \frac\mathbf_2+\frac\mathbf_3 and :t_ = \left[ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Hermite Spline
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the corresponding domain interval. Cubic Hermite splines are typically used for interpolation of numeric data specified at given argument values x_1,x_2,\ldots,x_n, to obtain a continuous function. The data should consist of the desired function value and derivative at each x_k. (If only the values are provided, the derivatives must be estimated from them.) The Hermite formula is applied to each interval (x_k, x_) separately. The resulting spline will be continuous and will have continuous first derivative. Cubic polynomial splines can be specified in other ways, the Bezier cubic being the most common. However, these two methods provide the same set of splines, and data can be easily converted between the Bézier and Hermite forms; so the names are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
