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Signed Distance
In mathematics and its applications, the signed distance function or signed distance field (SDF) is the orthogonal distance of a given point ''x'' to the boundary of a set Ω in a metric space (such as the surface of a geometric shape), with the sign determined by whether or not ''x'' is in the interior of Ω. The function has positive values at points ''x'' inside Ω, it decreases in value as ''x'' approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside). The concept also sometimes goes by the name oriented distance function/field. Definition Let be a subset of a metric space with metric , and \partial\Omega be its boundary. The distance between a point of and the subset \partial\Omega of is defined as usual as : d(x, \partial \Omega) = \inf_d(x, y), where \inf denotes the infimum. The ''sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Distance1
Signing or Signed may refer to: * Using sign language * Signature, placing one's name on a document * Signature (other) * Manual communication, signing as a form of communication using the hands in place of the voice * Digital signature, signing as a method of authenticating digital information * Traffic sign Traffic signs or road signs are signs erected at the side of or above roads to give instructions or provide information to road users. The earliest signs were simple wooden or stone milestones. Later, signs with directional arms were introduc ..., a road with a sign identifying is considered ''signed'' See also * Wikipedia:Sign your posts on talk pages, the Wikipedia policy of signing Talk pages {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zhao Hongkai
Hongkai Zhao is a Chinese mathematician and Ruth F. DeVarney Distinguished Professor of Mathematics at Duke University. He was formerly the Chancellor's Professor in the Department of Mathematics at the University of California, Irvine. He is known for his work in scientific computing, imaging and numerical analysis, such as the fast sweeping method for Hamilton-Jacobi equation and numerical methods for moving interface problems. Zhao had obtained his Bachelor of Science degree in the applied mathematics from the Peking University in 1990 and two years later got his Master's in the same field from the University of Southern California. From 1992 to 1996 he attended University of California, Los Angeles where he got his Ph.D. in mathematics. From 1996 to 1998 Zhao was a Gábor Szegő Assistant Professor at the Department of Mathematics of Stanford University and then got promoted to Research Associate which he kept till 1999. He has been at the University of California, Irvine since. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Sweeping Method
In applied mathematics, the fast sweeping method is a numerical method for solving boundary value problems of the Eikonal equation. : , \nabla u(\mathbf), = \frac 1 \text \mathbf \in \Omega : u(\mathbf) = 0 \text \mathbf \in \partial \Omega where \Omega is an open set in \mathbb^n, f(\mathbf) is a function with positive values, \partial \Omega is a well-behaved boundary of the open set and , \cdot, is the Euclidean norm. The fast sweeping method is an iterative method which uses upwind difference for discretization and uses Gauss–Seidel iterations with alternating sweeping ordering to solve the discretized Eikonal equation on a rectangular grid. The origins of this approach lie in the paper by Boue and Dupuis. Although fast sweeping methods have existed in control theory, it was first proposed for Eikonal equations by Hongkai Zhao, an applied mathematician at the University of California, Irvine The University of California, Irvine (UCI or UC Irvine) is a Public ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Marching Method
The fast marching methodJ.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591--1595, 1996/ref> is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation: : , \nabla u(x), =1/f(x) \text x \in \Omega : u(x) = 0 \text x \in \partial \Omega Typically, such a problem describes the evolution of a closed surface as a function of time u with speed f in the normal direction at a point x on the propagating surface. The speed function is specified, and the time at which the contour crosses a point x is obtained by solving the equation. Alternatively, u(x) can be thought of as the minimum amount of time it would take to reach \partial\Omega starting from the point x. The fast marching method takes advantage of this optimal control interpretation of the problem in order to build a solution outwards starting from the "known information", i.e. the boundary values. The algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a ''surface'' as shown in the illustration. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics. Surface integrals of scalar fields Assume that ''f'' is a scalar, vector, or tensor field defined on a surface ''S''. To find an explicit formula for the surface integral of ''f'' over ''S'', we need to parameterize ''S'' by defining a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely Integrable Function
In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since \int , f(x), \, dx = \int f^+(x) \, dx + \int f^-(x) \, dx where f^+(x) = \max (f(x),0), \ \ \ f^-(x) = \max(-f(x),0), both \int f^+(x) \, dx and \int f^-(x) \, dx must be finite. In Lebesgue integration, this is exactly the requirement for any measurable function ''f'' to be considered integrable, with the integral then equaling \int f^+(x) \, dx - \int f^-(x) \, dx, so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions. The same thing goes for a complex-valued function. Let us define f^+(x) = \max(\Re f(x),0) f^-(x) = \max(-\Re f(x),0) f^(x) = \max(\Im f(x),0) f^(x) = \max(-\Im f(x),0) where \Re f(x) and \Im f(x) are the real and imaginary parts In mathematics, a complex number is an element of a number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tubular Neighbourhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a tubular neighborhood. In general, let ''S'' be a submanifold of a manifold ''M'', and let ''N'' be the normal bundle of ''S'' in ''M''. Here ''S'' plays the role of the curve and ''M'' the role of the plane containing the curve. Consider the natural map :i : N_0 \to S which establishes a bijective correspondence between the zero section N_0 of ''N'' and the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometry, isometric embedding in Euclidean space. Surfaces naturally arise as Graph of a function, graphs of Function (mathematics), functions of a pair of Variable (mathematics), variables, and sometimes appear in parametric form or as Locus (mathematics), loci associated to Curve#Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
