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Adjustment Of Observations
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 ''h(X)=Y'' relating observations ''Y'' explicitly in terms of parameters ''X'' (leading to the A-model below). * In conditional adjustment, there exists a condition equation which is ''g(Y)=0'' involving only observations ''Y'' (leading to the B-model below) — with no parameters ''X'' at all. * Finally, in a combined adjustment, both parameters ''X'' and observations ''Y'' are involved implicitly in a mixed-model equation ''f(X,Y)=0''. Clearly, parametric and conditional adjustments correspond to the more general combined case ... [...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 (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 unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a 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 system has been overconstrained — that is, when the equations outnumber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Methods
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called 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 finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations A\mathbf=\mathbf by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. Ho ... [...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, Christian August Nagel, hired him while still a student to work on the triangulation of the Erzgebirge and the drafting of the trigonometric network for Saxony. In 1863 Helmert became Nagel's assistant on the Central European Arc Measurement. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regression Analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'explanatory variables' or 'features'). The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a give ... [...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 fo ... [...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. 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,\, ..,\, n ), simultaneously.Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, 2nd, Ganga-Jamuna Press, 2006. # GNSS satellites broadcast the messages of satellites' ephemeris, \displaystyle \boldsymbol_i (t), and intrinsic clock bias (i.e., clock advance), \displaystyle\delta t_ (t) as the functions of ( atomic) standard time, e.g., GPST. # The transmitting t ... [...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."Multilateration (MLAT ... [...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 kno ... [...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 every feature-based 3D reconstruction algorithm. 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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