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Horn–Schunck Method
The Horn–Schunck method of estimating optical flow is a global method which introduces a global constraint of ''smoothness'' to solve the ''aperture problem'' (see Optical Flow for further description). Mathematical details The Horn-Schunck algorithm assumes smoothness in the flow over the whole image. Thus, it tries to minimize distortions in flow and prefers solutions which show more smoothness. The flow is formulated as a global energy functional which is then sought to be minimized. This function is given for two-dimensional image streams as: : E=\iint \left I_xu + I_yv + I_t)^2 + \alpha^2(\lVert\nabla u\rVert^2+\lVert\nabla v\rVert^2)\right where I_x, I_y and I_t are the derivatives of the image intensity values along the x, y and time dimensions respectively, \vec = (x,y),v(x,y)\top is the optical flow vector (which is to be solved ''for''), and the parameter \alpha is a regularization constant. Larger values of \alpha lead to a smoother flow. This functional can be mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Flow
Optical flow or optic flow is the pattern of apparent motion of objects, surfaces, and edges in a visual scene caused by the relative motion between an observer and a scene. Optical flow can also be defined as the distribution of apparent velocities of movement of brightness pattern in an image. The concept of optical flow was introduced by the American psychologist James J. Gibson in the 1940s to describe the visual stimulus provided to animals moving through the world. Gibson stressed the importance of optic flow for affordance perception, the ability to discern possibilities for action within the environment. Followers of Gibson and his ecological approach to psychology have further demonstrated the role of the optical flow stimulus for the perception of movement by the observer in the world; perception of the shape, distance and movement of objects in the world; and the control of locomotion. The term optical flow is also used by roboticists, encompassing related techniq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperture Problem
Motion perception is the process of inferring the speed and direction of elements in a scene based on visual, vestibular and proprioceptive inputs. Although this process appears straightforward to most observers, it has proven to be a difficult problem from a computational perspective, and difficult to explain in terms of neural processing. Motion perception is studied by many disciplines, including psychology (i.e. visual perception), neurology, neurophysiology, engineering, and computer science. Neuropsychology The inability to perceive motion is called akinetopsia and it may be caused by a lesion to cortical area V5 in the extrastriate cortex. Neuropsychological studies of a patient who could not see motion, seeing the world in a series of static "frames" instead, suggested that visual area V5 in humans is homologous to motion processing area V5/MT in primates. First-order motion perception Two or more stimuli that are switched on and off in alternation can produce two di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Flow
Optical flow or optic flow is the pattern of apparent motion of objects, surfaces, and edges in a visual scene caused by the relative motion between an observer and a scene. Optical flow can also be defined as the distribution of apparent velocities of movement of brightness pattern in an image. The concept of optical flow was introduced by the American psychologist James J. Gibson in the 1940s to describe the visual stimulus provided to animals moving through the world. Gibson stressed the importance of optic flow for affordance perception, the ability to discern possibilities for action within the environment. Followers of Gibson and his ecological approach to psychology have further demonstrated the role of the optical flow stimulus for the perception of movement by the observer in the world; perception of the shape, distance and movement of objects in the world; and the control of locomotion. The term optical flow is also used by roboticists, encompassing related techniq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional (mathematics)
In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). * In linear algebra, it is synonymous with linear forms, which are linear mapping from a vector space V into its Field (mathematics), field of scalars (that is, an element of the dual space V^*) "Let ''E'' be a free module over a commutative ring ''A''. We view ''A'' as a free module of rank 1 over itself. By the dual module ''E''∨ of ''E'' we shall mean the module Hom(''E'', ''A''). Its elements will be called functionals. Thus a functional on ''E'' is an ''A''-linear map ''f'' : ''E'' → ''A''." * In functional analysis and related fields, it refers more generally to a mapping from a space X into the field of Real numbers, real or complex numbers. "A numerical function ''f''(''x'') defined on a normed linear space ''R'' will be called a ''functional''. A functional ''f''(''x'') is said to be ''linear'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Lagrange Equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that densi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramer's Rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729). Cramer's rule implemented in a naive way is computationally inefficient for systems of more than two or three equations. In the case of equations in unknowns, it requires computation of determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be nume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. Regular splittings We seek to solve the matrix equation where A is a given ''n'' × ''n'' non-singular matrix, and k is a given column vector with ''n'' components. We split the matrix A into where B and C are ''n'' × ''n'' matrices. If, for an arbitrary ''n'' × ''n'' matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Method
In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after Carl Gustav Jacob Jacobi. Description Let :A\mathbf x = \mathbf b be a square system of ''n'' linear equations, where: A = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_ \end, \qquad \mathbf = \begin x_ \\ x_2 \\ \vdots \\ x_n \end , \qquad \mathbf = \begin b_ \\ b_2 \\ \vdots \\ b_n \end. Then ''A'' can be decomposed into a diagonal component ''D'', a lower triangular part ''L'' and an upper triangular part ''U'': :A=D+L+U \qquad \text \qquad D = \begin a_ & 0 & \cdots & 0 \\ 0 & a_ & \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas–Kanade Method
In computer vision, the Lucas–Kanade method is a widely used differential method for optical flow estimation developed by Bruce D. Lucas and Takeo Kanade. It assumes that the flow is essentially constant in a local neighbourhood of the pixel under consideration, and solves the basic optical flow equations for all the pixels in that neighbourhood, by the least squares criterion.B. D. Lucas and T. Kanade (1981), An iterative image registration technique with an application to stereo vision.' Proceedings of Imaging Understanding Workshop, pages 121--130Bruce D. Lucas (1984) Generalized Image Matching by the Method of Differences' (doctoral dissertation) By combining information from several nearby pixels, the Lucas–Kanade method can often resolve the inherent ambiguity of the optical flow equation. It is also less sensitive to image noise than point-wise methods. On the other hand, since it is a purely local method, it cannot provide flow information in the interior of uniform reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
