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Eight-point Algorithm
The eight-point algorithm is an algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding image points. It was introduced by Christopher Longuet-Higgins in 1981 for the case of the essential matrix. In theory, this algorithm can be used also for the fundamental matrix, but in practice the normalized eight-point algorithm, described by Richard Hartley in 1997, is better suited for this case. The algorithm's name derives from the fact that it estimates the essential matrix or the fundamental matrix from a set of eight (or more) corresponding image points. However, variations of the algorithm can be used for fewer than eight points. Coplanarity constraint One may express the epipolar geometry of two cameras and a point in space with an algebraic equation. Observe that, no matter where the point P is in space, the vectors \overline, \overline and \overline belong to the same plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human visual system can do. Computer vision tasks include methods for acquiring, processing, analyzing and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the forms of decisions. Understanding in this context means the transformation of visual images (the input of the retina) into descriptions of the world that make sense to thought processes and can elicit appropriate action. This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory. The scientific discipline of computer vision is concerned with the theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Matrix
In computer vision, the essential matrix is a 3 \times 3 matrix, \mathbf that relates corresponding points in stereo images assuming that the cameras satisfy the pinhole camera model. Function More specifically, if \mathbf and \mathbf' are homogeneous ''normalized'' image coordinates in image 1 and 2, respectively, then : (\mathbf')^\top \, \mathbf \, \mathbf = 0 if \mathbf and \mathbf' correspond to the same 3D point in the scene. The above relation which defines the essential matrix was published in 1981 by H. Christopher Longuet-Higgins, introducing the concept to the computer vision community. Richard Hartley and Andrew Zisserman's book reports that an analogous matrix appeared in photogrammetry long before that. Longuet-Higgins' paper includes an algorithm for estimating \mathbf from a set of corresponding normalized image coordinates as well as an algorithm for determining the relative position and orientation of the two cameras given that \mathbf is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Matrix (computer Vision)
In computer vision, the fundamental matrix \mathbf is a 3×3 matrix which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point x′ on the other image must lie. That means, for all pairs of corresponding points holds : \mathbf'^ \mathbf = 0. Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone. The term "fundamental matrix" was coined by QT Luong in his influential PhD thesis. It is sometimes also referred to as the "bifocal tensor". As a tensor it is a two-point tensor in that it is a bilinear form relating points in distinct coordinate systems. The above relation which defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Longuet-Higgins
Hugh Christopher Longuet-Higgins (April 11, 1923 – March 27, 2004) was a British scholar and teacher. He was the Professor of Theoretical Chemistry at the University of Cambridge for 13 years until 1967 when he moved to the University of Edinburgh to work in the developing field of cognitive science. He made many significant contributions to our understanding of molecular science. He was also a gifted amateur musician, both as performer and composer, and was keen to advance the scientific understanding of this art. He was the founding editor of the journal ''Molecular Physics''. Education and early life Longuet-Higgins was born on 11 April 1923 at The Vicarage, Lenham, Kent, England, the elder son and second of the three children of Henry Hugh Longuet Longuet-Higgins (1886-1966), vicar of Lenham, and his wife, Albinia Cecil Bazeley. He was educated at The Pilgrims' School, Winchester, and Winchester College. At Winchester College he was one of the "gang of four" consisting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normalized Algorithm
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to: * Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of social norms come to be regarded as "normal" Normalization or normalisation may also refer to: Science * Normalization process theory, a sociological theory of the implementation of new technologies or innovations * Normalization model, used in visual neuroscience * Normalization in quantum mechanics: see Mathematics and statistics * Normalization of an algebraic variety, the operation consisting in taking locally the integral closure of the ring of regular functions * Normalization (statistics), adjustments of values or distributions in statistics ** Quantile normalization, statistical technique for making two distributions identical in statistical properties * Normalizing (abstract rewriting), an abstract rewriting system in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Hartley (scientist)
Richard I. Hartley is an Australian computer scientist and a professor at the Australian National University, where he is head of the Computer Vision group in the Research School of Engineering. Biography In 1971, Hartley received a BSc degree from the Australian National University followed by MSc (1972) and PhD (1976) degrees in mathematics from the University of Toronto. He also obtained an MSc degree in computer science from Stanford University in 1983. Scientific work His work is primarily devoted to the fields of Artificial intelligence, Image processing, and Computer vision. He is best known for his 2000 book ''Multiple View Geometry in computer vision'', written with Andrew Zisserman, now in its second edition (2004). According to WorldCat, the book is held in 1428 libraries Publications Hartley has published a wide variety of articles in computer science on the topics of computer vision and optimization. The following are his most highly cited works * 2000 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epipolar Geometry1
Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. These relations are derived based on the assumption that the cameras can be approximated by the pinhole camera model. Definitions The figure below depicts two pinhole cameras looking at point X. In real cameras, the image plane is actually behind the focal center, and produces an image that is symmetric about the focal center of the lens. Here, however, the problem is simplified by placing a ''virtual image plane'' in front of the focal center i.e. optical center of each camera lens to produce an image not transformed by the symmetry. OL and OR represent the centers of symmetry of the two cameras lenses. X represents the point of interest in both cameras. Points xL and xR are the projections of point X onto t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epipolar Geometry
Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. These relations are derived based on the assumption that the cameras can be approximated by the pinhole camera model. Definitions The figure below depicts two pinhole cameras looking at point X. In real cameras, the image plane is actually behind the focal center, and produces an image that is symmetric about the focal center of the lens. Here, however, the problem is simplified by placing a ''virtual image plane'' in front of the focal center i.e. optical center of each camera lens to produce an image not transformed by the symmetry. OL and OR represent the centers of symmetry of the two cameras lenses. X represents the point of interest in both cameras. Points xL and xR are the projections of point X onto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Linear Equation
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Least Squares
In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models. The total least squares approximation of the data is generically equivalent to the best, in the Frobenius norm, low-rank approximation of the data matrix. Linear model Background In the least squares method of data modeling, the objective function, ''S'', :S=\mathbf, is minimized, where ''r'' is the vector of residuals and ''W'' is a weighting matrix. In linear least squares the model contains equations which are linear in the parameters appearing in the parameter vector \boldsymbol\beta, so the residuals are given by :\mathbf. There are ''m'' observations in y and ''n'' parameters in β with ''m''>''n''. X is a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows and n columns and entries in the field K. A matrix norm is a norm on K^. This article will always write such norms with double vertical bars (like so: \, A\, ). Thus, the matrix norm is a function \, \cdot\, : K^ \to \R that must satisfy the following properties: For all scalars \alpha \in K and matrices A, B \in K^, *\, A\, \ge 0 (''positive-valued'') *\, A\, = 0 \iff A=0_ (''definite'') *\left\, \alpha A\right\, =\left, \alpha\ \left\, A\right\, (''absolutely homogeneous'') *\, A+B\, \le \, A\, +\, B\, (''sub-additive'' or satisfying the ''triangle inequality'') The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative: *\le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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