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Least Squares Adjustment
Least-squares adjustment is a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals. It is used extensively in the disciplines of surveying, geodesy, and photogrammetry—the field of geomatics, collectively. Formulation There are three forms of least squares adjustment: ''parametric'', ''conditional'', and ''combined'': * In parametric adjustment, one can find an observation equation relating observations explicitly in terms of parameters (leading to the A-model below). * In conditional adjustment, there exists a condition equation which is involving only observations (leading to the B-model below) — with no parameters at all. * Finally, in a combined adjustment, both parameters and observations are involved implicitly in a mixed-model equation . Clearly, parametric and conditional adjustments correspond to the more general combined case when and , respectively. Yet the special cases warrant simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overdetermined System
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent equations, inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting. Each Variable (mathematics), unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint (mathematics), constraint that restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The ''overdetermined'' case occurs when the syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Methods
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Robert Helmert
Friedrich Robert Helmert (31 July 1843 – 15 June 1917) was a German geodesist and statistician with important contributions to the theory of errors. Career Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and Dresden, he entered the Polytechnische Schule, now Technische Universität, in Dresden to study engineering science in 1859. Finding him especially enthusiastic about geodesy, one of his teachers, , hired him while still a student to work on the of the Ore Mountains and the drafting of the trigonometric network for Saxony. In 1863 Helmert became Nagel's assistant on the . After a year's study of mathematics and astronomy Helmert obtained his doctor's degree from the University of Leipzig in 1867 for a thesis based on his work for Nagel. In 1870 Helmert became instructor and in 1872 professor at RWTH Aachen, the new Technical University in Aachen. At Aachen he wrote ''Die mathematischen und physikalischen Theorieen der höheren Geodäsie' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces '' Disquisitiones Arithmeticae'' and ''Theoria motus corporum coelestium''. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Markov Model in mathematical statistics (in this theorem, one does ''not'' assume the probability distributions are Gaussian.)
{{mathematical disambiguation ...
The phrase Gauss–Markov is used in two different ways: * Gauss–Markov processes in probability theory *The Gauss–Markov theorem In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmert Transformation
The Helmert transformation (named after Friedrich Robert Helmert, 1843–1917) is a geometric transformation method within a three-dimensional space. It is frequently used in geodesy to produce datum transformations between datums. The Helmert transformation is also called a seven-parameter transformation and is a similarity transformation. Definition It can be expressed as: : X_T = C + \mu R X \, where * is the transformed vector * is the initial vector The parameters are: * – translation vector. Contains the three translations along the coordinate axes * – scale factor, which is unitless; if it is given in ppm, it must be divided by 1,000,000 and added to 1. * – rotation matrix. Consists of three axes (small rotations around each of the three coordinate axes) , , . The rotation matrix is an orthogonal matrix. The angles are given in either degrees or radians. Variations A special case is the two-dimensional Helmert transformation. Here, only f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNSS Positioning
Satellite navigation solution for the receiver's position (geopositioning) involves an algorithm. In essence, a GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites (giving the pseudorange) and these measurements are used to obtain its position (i.e., spatial coordinates) and reception time. The following are expressed in inertial-frame coordinates. The solution illustrated Image:Light cones.svg, Essentially, the solution shown in orange, \scriptstyle (\hat_,\, \hat_), is the intersection of light cones. Image:Evolution light cones 0.gif, The posterior distribution of the solution is derived from the product of the distribution of propagating spherical surfaces. (Seanimation) Calculation steps # A global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, \displaystyle \tilde_i, or "phase", of GNSS signals emitted from four or more GNSS satellites (\displaystyle i \;=\; 1,\, 2,\, 3,\, 4,\, .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trilateration
Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth ( geopositioning). When more than three distances are involved, it may be called multilateration, for emphasis. The distances or ranges might be ordinary Euclidean distances ( slant ranges) or spherical distances (scaled central angles), as in '' true-range multilateration''; or biased distances ( pseudo-ranges), as in ''pseudo-range multilateration''. Trilateration or multilateration should not be confused with ''triangulation'', which uses angles for positioning; and '' direction finding'', which determines the line of sight direction to a target without determining the radial distance. Terminology Multiple, sometimes overlapping and conflicting terms are employed for similar concepts – e.g., ''multilateration'' without modification has been used for aviation systems employing both true-ranges and pseudo-ranges."Multilateratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle measurements at known points, rather than measuring distances to the point directly as in trilateration; the use of both angles and distance measurements is referred to as triangulateration. In computer vision Computer stereo vision and optical 3D measuring systems use this principle to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base ''b'' and must be kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bundle Adjustment
In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from different viewpoints. Its name refers to the '' geometrical bundles'' of light rays originating from each 3D feature and converging on each camera's optical center, which are adjusted optimally according to an optimality criterion involving the corresponding image projections of all points. Uses Bundle adjustment is almost always used as the last step of feature-based 3D reconstruction algorithms. It amounts to an optimization problem on the 3D structure and viewing parameters (i.e., camera pose and possibly intrinsic calibration and radial distortion), to obtain a reconstruction which is optimal under certain assumptions regarding the noise pertaining to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
