E. M. Mikhail and F. Ackermann. Observations and Least Squares. University Press of America, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the unknown entity. For example, in fig. 3, there are four cameras with two projecting lines and two projecting planes incident to the unknown space line. Gau Helmert model. To estimate the unknown entity, we use an iterative, linear estimation model, the so called Gau Helmert model (cf. 10] [16]) which is summarized as follows (cf. 6] the constraints between the true unknown entities construction expression X;Y L = X Y L = TT(X)Y = TT(Y)X A;B L = A B L = TT(A)B = TT(B)A X, L A = X L A = TT (X)L = I (L)X A,L X = A L X = TT (A)L = I (L)A , P A ; P) A = P ....

E. M. Mikhail and F. Ackermann. Observations and Least Squares. University Press of America, 1976.

....part of the response in the vicinity of the zero crossing, then taking mean square values. Thus, if the zero crossing occurs atx0 in the noise free signal andx0 iSx in the noisy signal, we have F(xo ax) F(Xo) F(xo) ax so that F, Xo ax) 0 [20] tSx = F (x ax) F(xo) [21] E, x0) The presence of a zero crossing implies that F(xo) 0 and the assumption of zero mean noise implies that E[F(x0) 0. Therefore, the variance of the edge position is Elbx 2] E[ F x) 2I [221 ( x0) 2 In a discrete implementation, E[ F, x0) 2] is the sum of the squares of the ....

E.M. Mikhail, Observations and Least Squares, University Press of America: Lanham, MD, 1976.

....of an entity, cf. table 1. 4 Estimation of an Unknown Entity We now have the relations between an unknown entity and a list of observation entities i expressed as an implicit form g i ( i ) 0. Taking this equation, we can use the Gauss Helmert model for estimating the unknown , cf. [8] or [9] Since the Gauss Helmert model is a linear estimation model, we need the Jacobians g i ( i ) i and g i ( i ) We already have given them in sec. 3 because all our expressions were bilinear: in the tables 2,3 and 4 the Jacobians g i ( i ) are given for each ....

E. M. Mikhail and F. Ackermann. Observations and Least Squares. University Press of America, 1976.

....parameters the procedure finds the parameters belonging to the grey value vector. For the 625 samples convergence was achieved, i.e. the model was invertible in all discretization points. The solution found by simulated annealing is improved by least squares adjustment using a Gauss Markov model (Mikhail 1976). This approach has the advantages that (1) convergence of simulated annealing can be considered as achieved after a relatively small number of it5 erations, and that (2) an analysis of error propagation is implicit to the computations. The least squares adjustment uses the solution found by ....

Mikhail, E. M. [1976]. Observations and Least Squares, IEP, New York.

....estimates for the tacheometry, assuming each survey observation to be independent. Data sets were simulated by adding a perturbation to the x , y, and z coordinates of each original data point, using the covariance matrix of that data point. The covariance matrices were estimated from error theory [Mikhail and Ackerman, 1976] by differentiation of the observation equations [Fryer et al. 1987] and evaluation using the raw observations. Given the large correlation between the x , y , and z coordinates of the individual data points, random perturbations to the data points were estimated by sampling from a multivariate ....

Mikhail, E. M., and F. Ackerman, Observations and Least Squares, IEP, New York, 497 pp., 1976.

....given in Table 2.3 all correspond to a single epoch in time. Assuming GPS observations are uncorrelated between epochs, the normal equations matrix, N, corresponding to a set of consecutive epochs of observations is block diagonal, meaning summation techniques may be used in solution computations (Mikhail, 1976). 2.4 PREANALYSIS Much can be learned about the relative accuracies achievable with the above models under a variety of observing conditions using dilution of precision (DOP) and statistical reliability. Both are suited for preanalysis since they require only a simulation of observing conditions ....

Mikhail, E.M. (1976): Observations and Least Squares. Harper & Row, New York.

.... the elimination of blunders is the use of redundant observations, accuracy being a second objective [Stefanovic, 1978] Systematic errors are theoretically eliminated by the use of an appropriate model, while random errors are taken into account by the use of a weight matrix in the cost function [Mikhail, 1976]. A good review on outliers analysis in statistics can be found in Beckman and Cook [Beckman and Cook, 1983] According to these authors, there exist two broad methods for dealing with the possibility of outliers. Identification of an outlier may lead to (a) its rejection, b) a revision of the ....

.... A Combinatorial and Local Optimization Strategy for 3D Model Based Image Registration10 January 1999 5 (EQ 4) An observation equation is consistent locally iff , where E is the ellipse of center and axes , where (EQ 5) EQ 6) Equations 5 and 6 are obtained by the rule of propagation of variances [Mikhail, 1976]. is assumed to be very well estimated, hence . This strong assumption permits to keep the rejection threshold strict to the presence of outliers. Anyhow, more work has to be done to obtain a reliable estimate of . Finally, criterion based on equation 5 and 6 permits to attribute the uncertainty ....

Mikhail, E. M. (1976). Observations and Least Squares, IEP - Harper and Row, New York.

.... and the Jacobian A = y i = fi j ) Deltay = E( Deltay) e = y Gamma f(fi (0) M (A Deltafi; Sigma yy ) 3) It is assumed to hold also for the estimated quantities, thus we have for the first moments: Deltay = A d Deltafi b e (4) The best linear unbiased estimate (BLUE, cf. e.g. [MA76]) for the unknown parameters is given by b fi = fi (0) d Deltafi with d Deltafi = A T Sigma Gamma1 yy A) Gamma1 A T Sigma Gamma1 yy Deltay with Sigma b fi b fi = N Gamma1 : A T Sigma Gamma1 yy A) Gamma1 (5) Observe that the covariance matrix of the ....

E. M. Mikhail and F. Ackermann. Observations and Least Squares. University Press of America, 1976.

.... T ( w) A w b that minimizes n X i=1 kT ( w i ) 0 z i k 2 : If A is the 2 2 2 matrix A = a 11 a 12 a 21 a 22 and b is the vector b = b 1 b 2 ; 27 the six unknown transformation parameters can be found by solving the following separable system of equations [6] 0 P w i x 2 P w i x w i y P w i x P w i x w i i P w i y 2 P w i y P w i x P w i y P 1 1 A 0 a 11 a 21 a 12 a 22 b 1 b 2 1 A 0 0 P w i x z i x P w i x z i y P w i y z i x P w i y z i y P z i x P z i y 1 A = 0; 45) where P means n P i=1 . If we first ....

Mikhail, E. M. and F. Ackermann, Observations and Least Squares, Dun Donnelley, New York, 1976.

....about the absolute orbit accuracy. This allows to control the magnitudes of the adjusted bias and drift parameters. The observations itself are assumed to be 0 for both bias and drift. The adjusted unknowns x are computed according to the well known equations of least squares adjustment (Mikhail, 1976): x = ffi x Deltax (7) where ffi x is the vector of approximations of the unknowns and Deltax is the vector of corrections to the approximations. The linearized observation equation system is v = A Deltax Gamma w (8) where v is the vector of residuals, A is the Jakobi matrix ....

Mikhail, E. M., 1976. Observations and Least Squares.

....all physical parameters simultaneously. The algorithm is mainly the same as earlier (Niini, 1994, 1995, 1996) only the optimization of the block is included in the algorithm. In the actual least squares adjustments, the parameters are always computed using the general least squares model Ad Bv=f (Mikhail, 1976) which uses only one residual per observation. The solution of the system is d= A T WA) 1 A T Wf where the weight matrix of the system W = BQB T ) 1 . A is the parameter coefficient matrix, B is the residual coefficient matrix, Q is the weight coefficient matrix of the original image ....

Mikhail, E., 1976. Observations and Least Squares. Thomas Y.

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