|
Scale Space Representation
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theory for handling image structures at different scales, by representing an image as a one-parameter family of smoothed images, the scale-space representation, parametrized by the size of the smoothing kernel used for suppressing fine-scale structures.Ijima, T. "Basic theory on normalization of pattern (in case of typical one-dimensional pattern)". Bull. Electrotech. Lab. 26, 368– 388, 1962. (in Japanese) The parameter t in this family is referred to as the ''scale parameter'', with the interpretation that image structures of spatial size smaller than about \sqrt have largely been smoothed away in the scale-space level at scale t. The main type of scale space is the ''linear (Gaussian) scale space'', which has wide applicability as well ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiscale Mathematics
Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion). An example of such problems involve the Navier-Stokes equations for incompressible fluid flow. \begin \rho_0(\partial_t\mathbf+(\mathbf\cdot\nabla)\mathbf)=\nabla\cdot\tau, \\ \nabla\cdot\mathbf=0. \end In a wide-variety of applications, the stress tensor \tau is given as a linear function of the gradient \nabla u. Such a choice for \tau has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, it’s use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impulse Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anisotropic Diffusion
In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details that are important for the interpretation of the image. Anisotropic diffusion resembles the process that creates a scale space, where an image generates a parameterized family of successively more and more blurred images based on a diffusion process. Each of the resulting images in this family are given as a convolution between the image and a 2D isotropic Gaussian filter, where the width of the filter increases with the parameter. This diffusion process is a ''linear'' and ''space-invariant'' transformation of the original image. Anisotropic diffusion is a generalization of this diffusion process: it produces a family of parameterized images, but each resulting image is a combination between the original image and a filter that d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f(x, y, \dots) with respect to the variable x is variously denoted by It can be thought of as the rate of change of the function in the x-direction. Sometimes, for z=f(x, y, \ldots), the partial derivative of z with respect to x is denoted as \tfrac. Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: :f'_x(x, y, \ldots), \frac (x, y, \ldots). The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition. The concept of diffusion is widely used in many fields, including physics ( particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas, and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. A gradient is the change in the value of a quantity, for example, concentration, pressure, or temperature with the change in another variable, usually distance. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion Equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation, when bulk velocity is zero. It is equivalent to the heat equation under some circumstances. Statement The equation is usually written as: where is the density of the diffusing material at location and time and is the collective diffusion coefficient for density at location ; and represents the vector differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear. The equation above applies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = x^2 + y^2 is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle ''θ'' :x' = x \cos \theta - y \sin \theta :y' = x \sin \theta + y \cos \theta the function, after some cancellation of terms, takes exactly the same form :f(x',y') = ^2 + ^2 The rotation of coordinates can be expressed using matrix form using the rotation matrix, :\begin x' \\ y' \\ \end = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end\begin x \\ y \\ \end. or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is :f(\mathbf') = f(\mathbf) = f(\mathbf) In words, the function of the rotated coordinates takes e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry. *In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity. *In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale. *In quantum field t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Extrema
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
