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Box Spline
In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes. Definition A box spline is a multivariate function (\mathbb^d \to \mathbb) defined for a set of vectors, \xi \in \mathbb^d, usually gathered in a matrix \mathbf := \left xi_1 \dots \xi_N\right When the number of vectors is the same as the dimension of the domain (i.e., N = d ) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in \mathbf: : M_(\mathbf) := \frac\chi_(\mathbf) = \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Sum
In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowski decomposition'', or ''geometric difference'') is the corresponding inverse, where (A - B) produces a set that could be summed with ''B'' to recover ''A''. This is defined as the complement of the Minkowski sum of the complement of ''A'' with the reflection of ''B'' about the origin. \begin -B &= \\\ A - B &= (A^\complement + (-B))^\complement \end This definition allows a symmetrical relationship between the Minkowski sum and difference. Note that alternately taking the sum and difference with ''B'' is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing. \begin (A - B) + B &\subseteq A\\ (A + B) - B &\supseteq A\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
X-ray Transform
In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over line (geometry), lines rather than over hyperplanes as in the Radon transform. The X-ray transform derives its name from X-ray tomography (used in CT scans) because the X-ray transform of a function ''ƒ'' represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function ''ƒ''. Inverse problem, Inversion of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density ''ƒ'' from its known attenuation data. In detail, if ''ƒ'' is a compact support, compactly supported continuous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomographic Reconstruction
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projection (linear algebra), projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the operation of computed tomography#Tomographic reconstruction, reconstruction of CT scan, computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security. This article applies in general to reconstruction methods for all kinds of tomography, but some of the terms and physical descriptions refer directly to the reconstruction of X-ray computed tomography. Introducing formula The projection of an object, resulting from the tomograph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body Centered Cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc) Note: the term fcc is often used in synonym for the ''cubic close-packed'' or ccp structure occurring in metals. However, fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are fcc but not close-packed. Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais latices in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face-centered Cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc) Note: the term fcc is often used in synonym for the ''cubic close-packed'' or ccp structure occurring in metals. However, fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are fcc but not close-packed. Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive unit cells often are not. Bravais lattices The three Bravais latices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Body-centered Cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the Crystal structure#Unit cell, unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: *Primitive cubic (abbreviated ''cP'' and alternatively called simple cubic) *Body-centered cubic (abbreviated ''cI'' or bcc) *Face-centered cubic (abbreviated ''cF'' or fcc) Note: the term fcc is often used in synonym for the Close-packing of equal spheres, ''cubic close-packed'' or ccp structure occurring in metals. However, fcc stands for a face-centered cubic Bravais lattice, which is not necessarily close-packed when a motif is set onto the lattice points. E.g. the diamond and the zincblende lattices are fcc but not close-packed. Each is subdivided into other variants listed below. Although the ''unit cells'' in these crystals are conventionally taken to be cubes, the primitive cell, primitiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to Non-Euclidean geometry, non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the ''packing density'' of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Systems
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the surface of the soil, but roots can also be aerial or aerating, that is, growing up above the ground or especially above water. Function The major functions of roots are absorption of water, plant nutrition and anchoring of the plant body to the ground. Types of Roots (major rooting system) Plants exhibit two main root system types: ''taproot'' and ''fibrous'', with variations like adventitious, aerial, and buttress roots, each serving specific functions. Taproot System Characterized by a single, main root growing vertically downward, with smaller lateral roots branching off. Examples. Dandelions, carrots, and many dicot plants. Fibrous RootSystem Consists of a network of thin, branching roots that spread out from the base of the stem, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reconstruction Filter
In a mixed-signal system ( analog and digital), a reconstruction filter, sometimes called an anti-imaging filter, is used to construct a smooth analog signal from a digital input, as in the case of a digital to analog converter ( DAC) or other sampled data output device. Sampled data reconstruction filters The sampling theorem describes why the input of an ADC requires a low-pass analog electronic filter, called the anti-aliasing filter: the sampled ''input'' signal must be bandlimited to prevent aliasing (here meaning waves of higher frequency being ''recorded'' as a lower frequency). For the same reason, the output of a DAC requires a low-pass analog filter, called a reconstruction filter - because the ''output'' signal must be bandlimited, to prevent imaging (meaning Fourier coefficients being reconstructed as spurious high-frequency 'mirrors'). This is an implementation of the Whittaker–Shannon interpolation formula. Ideally, both filters should be brickwall filters, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multidimensional Sampling
In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and MiddletonD. P. Petersen and D. Middleton, "Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces", Information and Control, vol. 5, pp. 279–323, 1962. on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete Lattice (group), lattice of points. This result, also known as the Petersen–Middleton theorem, is a generalization of the Nyquist–Shannon sampling theorem for sampling one-dimensional band-limited functions to higher-dimensional Euclidean spaces. In essence, the Petersen–Middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
