Bierman, G. J.: Factorization Methods for Discrete Sequential Estimation.Aca- demic Press, New York, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....used in each iterative estimation step. These estimates of covariance matrices can easily loose their theoretical positive de niteness. Then, the subsequent estimation produces unbounded values. The aim of this paper is to utilize numerical procedures adopted from factorized on line estimation [4, 5] and to combine them with EM algorithm to improve its numerical properties. Indeed, both the single tasks are known and successfully used for a long time. It is astonishing that they were not coupled, yet. FD CVUT, Konviktsk a 20, Prague 1, Czech Republic email: nagy utia.cas.cz 2 UTIA, AV ....

.... notations, the recursion for R becomes L D L (L DL ) z t z t 1 = L Dff 1 f Df ; with f = L z t : The de nition of LD decomposition implies that the expression in brackets [ can be decomposed analytically into the LD form, say H DH , see [4]. The following algorithm provides it for i = n; n 1; 1 = 1 n k=i D k f k D i;i = D i;i i H i;i = 1 for j = i 1; i 2; n H j;i = f i f j D j;j end for j end for i We attach to this algorithm the multiplications f = z t and L = LH and organize ....

G.J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977.

....to estimate and predict T c values. A linear model is maintained for the time to contact in each of the three windows: 13) where t = 0, 1, 2, For each measurement of time to contact, model parameters and are updated by a weighted recursive least squares computation with exponential decay [5][9] 13] 19] This involves determining and such that the residual is minimized: 14) where ; is the present; is the forgetting factor; and is the confidence of the measurement (the number of flow data points in the window) In order to solve for a , a , the square root information filter ....

.... involves determining and such that the residual is minimized: 14) where ; is the present; is the forgetting factor; and is the confidence of the measurement (the number of flow data points in the window) In order to solve for a , a , the square root information filter (SRIF) algorithm [5] is used. The SRIF provides an efficient, numerically stable, closed form solution to the least squares problem. Z T 2 O , Figure 5. Flow Divergence Templates T c a 1 a 2 t = J J l w t [T c a 1 a 2 t ( n t 0 1 2 . n ....

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G. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press. New York, 1977.

....de Voluceau, Rocquencourt, France. 1 1 Introduction In spite of the fact that Kalman filtering relies on a simple Gram Schmidt orthogonalization principle, over the years, the literature devoted to Kalman filtering and smoothing [1] 2] and its square root or fast algorithms implementations [3], 4] has become relatively complex. To deal with numerical conditioning problems, such the possible singularity of the measurement noise, or large uncertainties in the initial state variance, a number of variants of the basic filtering and smoothing algorithms have been developed. Although most ....

....is that it relies exclusively on numerically stable operations, such as QR or singular value decompositions. In some sense, it can be viewed as just a formalization of procedures which occur repeatedly, but under different disguises, in the implementation of square root Kalman filters [3]. In the context of a discussion of generalized linear regression models in statistics, a decomposition of this type appears in [20] but no detailed numerical procedure was provided for its construction. 2) Given an observation o of the form (2.1) the generation of the last two equations of ....

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic Press, 1977.

....very small 0; 1: These are basic facts we need for applying the general algorithm of Section 5. When applying it, we have to evaluate many times the value of I(V; for various V; The ecient, numerically stable solution is obtained by a direct updating of a factorised version of V , [15]. It works on so called LD factorisation of V that always exists for any positive de nite V . Proposition 6.1 Let V = L 0 DL. L is a lower triangular matrix with a unit diagonal and D is diagonal matrix with positive entries. Then I(V; 2 ) 0:5 2 =2 ( 2)D =2 11 dim( Y i=2 ....

G.J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977.

....the matrix given by (5.4.5) has poor numerical properties. For this reason, other algorithms with better numerical properties have been derived which are based on a square root factorization of as , where is a lower triangular matrix. Such algorithms are called square root RLS algorithms [463] [443]. These algorithms update the matrix directly without computing explicitly, and have a computational complexity proportional to Other types of RLS algorithms appropriate for transversal FIR equalizers have been devised with a computational complexity proportional to [448] 508] 471] These ....

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic, 1977.

....Falle handelt es sich um die Rotationsmatrix R (dargestellt durch ein Einheitsquaternion 1 der Form e = e 0 ,e 1 ,e 2 ,e 3 )mit P 3 i=0 e 2 i = 1) den Translationsvektor t und den Ebenen Normalenvektor b. Fur den hier behandelten Fall planarer Welten ist allgemein bekannt (siehe z.B. [1]) da nur funf von den sechs Parametern des Translationsvektors und des Ebenen Normalenvektors bestimmt werden konnen. Um dieser Unbestimmtheit Rechnung zu tragen, legen wir eine der sechs Komponenten dieser beiden Vektoren auf einen Wert fest (b 2 : 1) Der Zusammenhang zwischen einem ....

....den aktuellen visuellen Messungen stammen, bezeichnen wir mit b x und b C x . Das Gau Markov Theorem der Schatztheorie (siehe z.B. 8] liefert in dieser Situation die eindeutige Vorschrift fur eine optimale Kombination mehrerer unabh angiger Schatzwerte mit Hilfe der folgenden Formel (siehe [1], S.16) x opt = b C 1 x C 1 x 1 b C 1 x b x C 1 x x (3) 6 Dirk Feiden et al. C opt = b C 1 x C 1 x 1 . 4) Die Verwendung dieser Formeln fuhrt zu einer wesentlichen Stabilisierung der Schatzwerte, weil immer dann, wenn aus der aktuellen visuellen ....

Bierman, G. J.: Factorization Methods for Discrete Sequential Estimation.Aca- demic Press, New York, 1977.

.... out to be insufficiently excited for identification purposes, which is a very common situation when working with servo positioning systems [5] 8] Therefore, parameter estimation is carried out by means of a well known version of RLS algorithm, where covariance matrix U D factorisation is used [3], 7] This technique for the covariance matrix update has proved its efficiency when sudden parameter variations occur during low excitation periods. In fact, the covariance matrix shows an outstanding recovering capability when, after having tracked parameters a 1 and b 1 , its main diagonal ....

Bierman G.J. "Factorization Methods for Discrete Sequential Estimation". Mathematics in Science and Engineering.

....the SRIF technique. 1. Introduction 1.1 Expanding the Capabilities of Square Root Information Filters and Smoothers. Square Root Information Filters and Smoothers (SRIF S) are special solution techniques for, respectively, the discrete time Kalman filter problem and the related smoothing problem [1]. Square root formulations increase numerical computation accuracy by guaranteeing positive definiteness of the associated covariances and by decreasing the condition numbers of the manipulated matrices. The present paper generalizes the SRIF S techniques to deal with singularities that can occur ....

....for the entire interval: x (j) E x (j) z a(0) z b(0) z a(N) z b(N) for j = 0, 1, 2, N. 3. Forward Pass Filter Algorithm 3.1 Overview of Filtering Procedure. The filtering problem can be solved by a least squares technique that determines the maximum likelihood solution [1]. This technique finds n b(0) and the x (j) w (j) n (j) and n w(j) sequences that satisfy constraint eqs. 1a) 1c) 2a) 3a) and (3b) and minimize J b(0) T b(0) j) T (j) j=0 N w(j) T w(j) j=0 N 1 = 1 2 n n n n n n (4) The algorithm works in an recursive ....

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, (New York, 1977).

....in each iterative estimation step. These estimates of covariance matrices can easily loose their theoretical positive de niteness. Then, the subsequent estimation produces unbounded values. The aim of this paper is to utilize numerical procedures adopted from fac16 torized on line estimation [5, 1] and to combine them with EM algorithm to improve its numerical properties. 3.2 EM algorithm Let us consider a system generating at time t data d t = y t ; c t ] The part y t of the data is measured and thus known, the rest c t cannot be measured and thus it is unknown to us. We suppose to have ....

.... for R becomes L D L 0 = h (L DL 0 ) 1 z t z 0 t i 1 = L D Dff 0 D 1 f 0 Df # L 0 ; with f = L 0 z t : The de nition of LD decomposition implies that the expression in brackets [ can be decomposed analytically into the LD form, say H DH 0 , see [5]. The following algorithm provides it for i = n; n 1; 1 2 = 1 n X k=i D k f 2 k D i;i = D i;i 2 i 1 2 i 21 H i;i = 1 for j = i 1; i 2; n H j;i = f i f j 2 i 1 D j;j end for j end for i We attach to this algorithm the multiplications f = L 0 z t ....

G. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic Press, 1977.

....estimation algorithms. In addition, in [10] Chisci and Zappa independently developed a square root Kalman filter filter for essentially the same problem studied in [9] The main feature of the ML approach, which was itself motivated by earlier work of Whittle [11] Chapter 11, and of Bierman [12] in the context of square root Kalman filtering, is that no distinction is made between system dynamics and observations. Specifically, all dynamic relations and initial or boundary conditions are viewed as observations, i.e. as noisy constraints on the state variables. Given a stream of ....

....two sets of observations are equivalent if they provide the same information about x. The idea of replacing a set of measurements by another containing the same information is not new and has been used informally in much of the recursive ML estimation and square root Kalman filtering literature [12]. A notion of equivalence similar to the one introduced here was proposed recently in [10] The definition we consider is slightly more general, since we require that equivalence should be contextfree. Specifically, given two sets of measurements for a vector x 1 , for these two measurements to ....

G.J. Bierman. Factorization Methods for Discrete Sequential Estimation. New York: Academic Press, 1977.

.... High level Primitives for Estimation 3 1 Introduction In spite of the fact that Kalman filtering relies on a simple Gram Schmidt orthogonalization principle, over the years, the literature devoted to Kalman filtering and smoothing [1] 2] and its square root or fast algorithms implementations [3], 4] has become relatively complex. To deal with numerical conditioning problems, such the possible singularity of the measurement noise, or large uncertainties in the initial state variance, a number of variants of the basic filtering and smoothing algorithms have been developed. Although most ....

....is that it relies exclusively on numerically stable operations, such as QR or singular value decompositions. In some sense, it can be viewed as just a formalization of procedures which occur repeatedly, but under different disguises, in the implementation of square root Kalman filters [3]. 2) Given an observation o of the form (2.1) the generation of the last two equations of (2.4) requires only the first step of the procedure employed to construct (2.4) Specifically, as indicated by (2.7a) 2.7b) if L is a matrix whose rows form a basis of the left null space of A, so that ....

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic Press, 1977.

....the gain matrix, and therefore it is numerically more stable. Consequently, in many engineering implementations of the Kalman filter Joseph s formula is preferably used. 93 5.3. 3 Serial Processing of Observations Serial processing of observations was introduced in the literature by Bierman [12], and discussed in Parrish Cohn [113] in the context of atmospheric data assimilation. In this section, we assume for simplicity that all the available observations are uncorrelated at all times t k . We have in mind the uncorrelatedness not only in time, but also among variables at a fixed ....

Bierman, G.J., 1977: Factorization Methods for Discrete Sequential Estimation. Academic Press, 241 pp.

....Gaussian transformations. The search, however, is combined with a conditional least squares adjustment and is based on the sequential least squares estimation, see [10] 11] and [12] Whitening is a factorization process used to project a sequence of data from one space to another, see [2]. It is used to obtain a decorrelated version of a data set by projecting it onto another space that is simpler to analyze. In the context of adaptive Kalman filtering, for instance, the whitening filter in its float version is used to decorrelate the innovation sequence used to fine tune (adapt) ....

....is not possible because of the integer constraint on the ATM entries. To overcome this constraint and get a V C matrix of the whitened observations which is close to a diagonal, an iterative procedure is needed which is shown in Fig. 4) below. The details of this procedure can be found in [2] and [5] 1. Starting with C e , do an upper triangular factorization (Eq. 7) and get a first realvalue approximation, say U 1 . 2. Round all elements of U 1 to their nearest integers, then Invert the result , get 1 1 U . 3. Whiten (transform) the V C matrix C e after rounding, and ....

Bierman, G.J. (1977), Factorization Methods for Discrete Sequential Estimation, Academic Press, Inc.

.... High level Primitives for Estimation 1 1 Introduction In spite of the fact that Kalman filtering relies on a simple Gram Schmidt orthogonalization principle, over the years, the literature devoted to Kalman filtering and smoothing [1] 2] and its square root or fast algorithms implementations [3], 4] has become relatively complex. To deal with numerical conditioning problems, such the possible singularity of the measurement noise, or large uncertainties in the initial state variance, a number of variants of the basic filtering and smoothing algorithms have been developed. Although most ....

....is that it relies exclusively on numerically stable operations, such as QR or singular value decompositions. In some sense, it can be viewed as just a formalization of procedures which occur repeatedly, but under different disguises, in the implementation of square root Kalman filters [3]. High level Primitives for Estimation 7 2) Given an observation o of the form (2.1) the generation of the last two equations of (2.4) requires only the first step of the procedure employed to construct (2.4) Specifically, as indicated by (2.7a) 2.7b) if L is a matrix whose rows form a ....

G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic Press, 1977.

....adoption of the weight function defined in Eq. 6, is that, simply by updating the parameter fi 1 (k) of the model identified using the k nearest neighbors, it is straightforward and inexpensive to obtain fi 1 (k 1) In fact, performing a step of the standard recursive least squares algorithm (Bierman, 1977), we have: 8 : P(k 1) P(k) Gamma P(k)x 1 (k 1)x 0 1 (k 1)P(k) 1 x 0 1 (k 1)P(k)x 1 (k 1) fl(k 1) P(k 1)x 1 (k 1) e(k 1) y(k 1) Gamma x 0 1 (k 1) fi 1 (k) fi 1 (k 1) fi 1 (k) fl(k 1)e(k 1) 7) where P(k) Z 0 ....

....function in the query point, and the vector of leave one out errors from which it is possible to extract an estimate of the variance of the prediction error. Notice that fi 1 is an a priori estimate of the parameter and P is the covariance matrix that reflects the reliability of fi 1 (Bierman, 1977). For non reliable initialization, the following is usually adopted: P = I, with large and where I is the identity matrix. The recursive algorithm described by Eq. 7 and Eq. 8 returns for a given query point x q , a set of predictions y 1;q (k) x 0 1;q fi 1 (k) together with a set of ....

Bierman G.J. 1977. Factorization Methods for Discrete Sequential Estimation.

....when the depth measurements are very sparse over the image frame. The so called square root form of an information Kalman filter, or a square root information filter (SRIF) improves numerical stability in such near singular estimation problems by reducing numerical dynamic ranges of the variables [9, 47]. Besides providing an increased margin for numerical roundoff errors, such reduction in the dy85 namic ranges can also relieve memory requirements (i.e. the number of bits required to represent each variable) in filter implementation a desirable feature as low level visual reconstruction ....

....section. 4.4.1 Data fusion in square root form The computational mechanisms for SRIF algorithms are considerably different from those for the information filter algorithms. Unitary transformation is a key operation in SRIF s. Here, we briefly review some rudiments of square root estimation theory [9]. Unitary transformation and unit white noise Let us use ffi to denote generically any zero mean white noise process whose covariance matrix is an identity matrix, i.e. ffi ( 0; I ) The dimension of ffi is context dependent. A unitary transformation on a (column) vector is a left ....

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G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 1977.

....(Kalman filtering and smoothing) to make optimal use of each measurement without computing differential quantities. First, we derive a linear set of equations between the unknown shape (surface point positions and radii of curvature) and the measurements. We then develop a robust linear smoother ([Gel74, Bie77]) to compute statistically optimal current and past estimates from the set of contours. Smoothing allows us to combine measurements on both sides of each surface point. Our technique produces a complete surface description, i.e. a network of linked 3D surface points, which provides us with a much ....

....of the osculating circle can be recovered given the projection of three non parallel tangent lines onto the epipolar plane, a much more reliable estimate can be obtained by using more views. Given the set of equations (2) how can we recover the best estimate for (x c ; y c ; r) Regression theory [Alb72, Bie77] tells us that the minimum least squared error estimate of the system of equations Ax = d can be found by minimizing e = jAx Gamma dj 2 = X i (a i Delta x Gamma d i ) 2 : 3) This minimum can be found by solving the set of normal equations 3 (A T A)x = A T d (4) or ( X i a i a ....

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G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, New York, 1977.

....are included which illustrate the properties of the extended Kalman filter in a nonlinear QG model. Conclusions are given in the final section. The Kalman Filter The Kalman filter has been derived in a number of books on control theory [e.g. Gelb, 1974; Jazwinski, 1970; Tzafestas, 1978; Bierman, 1977; Anderson, 1979; Stengel, 1986] In oceanography the Kalman filter has been used by Budgell [1986, 1987] to describe nonlinear and linear shallow water wave propagation in branched channels, using one dimensional cross sectionally integrated equations. Miller, 1986] used a one dimensional ....

Bierman, G. J., Factorization Methods for Discrete Sequential Estimation, Math. in Sci. and Eng., vol. 128, Academic, San Diego, Calif., 1977.

.... there are N of these matches and that each p k (that is the k th observation) is statistically independent of the previous k 1; otherwise, a procedure for obtaining a p 0 k independent from the past but containing all the information necessary to update the estimate is straightforward (e.g. [2]) The updating algorithm for the supplementary matched features is the following: for k=1 to N 8 : s = s s Q T k (Q k s Q T k pk ) Gamma1 (p k Gamma p I k ) s = s Gamma s Q T k (Q k s J T k pk ) Gamma1 Q k s s Gamma N Gamma s , s Delta By these ....

G. BIerman. Factorization Methods for Discrete Sequential Estimation. Academic Press, 1977.

.... methods which include the LU LDL T , QR, Cholesky decomposition and singular value decomposition forms of the least squares estimator (Golub Loan 1989) Potter s square root least squares method (Potter Stern 1963, Bennet 1963, Andrews 1968) and Bierman s UDU T factorization method (Bierman 1977, Thornton Bierman 1980) The Least Squares Lattice (LSL) method (Morf et al. 1977, Lee 1980, Lee et al. 1981, Makhoul 1977, Friedlander 1983, Aling 1993) which has been widely used in signal processing, is another example of improved least squares estimator. The second problem concerns the ....

....be regarded as an alternative proof of the MMLS method. 2. 3 Implementation The MMLS structure enables flexible and convenient implementation, where most of the well known decomposition methods such as the LU, LDL T , Cholesky, QR decompositions (Golub Loan 1989) and UDU T factorization (Bierman 1977) can all be used. For instance, the augmented data matrix defined in (14) can be decomposed with a QR decomposition Phi(t) Q(t)R(t) where Q(t) is an orthogonal matrix and R(t) is upper triangular. The parameter and loss function matrix U(t) and D(t) can be easily derived from R(t) with U(t) ....

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Bierman, G. J. (1977), Factorization Methods for Discrete Sequential Estimation, Academic Press, New York.

....the measurement X;Y and Theta are independent and accordingly can be split up in three measurements which can be processed one by one. This is advantageous regarding computational aspects and simplifies the implementation of a numerical stable algorithm which often builds on scalar measurements [3]. The vision system is limited in detection of guide marks due to a limited field of view for the camera and a constrained number of guide marks which implies interrupts in the measurements. Thus, the following algorithm has been applied. No vision measurements. Only the time update is executed: ....

G.J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, 1977.

....the adoption of the weight function defined in Eq. 6, is that, simply by updating the parameter fi(k) of the model identified using the k nearest neighbors, it is straightforward and inexpensive to obtain fi(k 1) In fact, performing a step of the standard recursive least squares algorithm (Bierman, 1977), we have: 8 : P(k 1) P(k) Gamma P(k)x(k 1)x 0 (k 1)P(k) 1 x 0 (k 1)P(k)x(k 1) fl(k 1) P(k 1)x(k 1) e(k 1) y(k 1) Gamma x 0 (k 1) fi(k) fi(k 1) fi(k) fl(k 1)e(k 1) 7) where P(k) Z 0 Z) Gamma1 when h = ....

....regression function in the query point, and the vector of leave one out errors from which it is possible to extract an estimate of the variance of the prediction error. Notice that fi is an a priori estimate of the parameter and P is the covariance matrix that reflects the reliability of fi (Bierman, 1977). For non reliable initialization, the following is usually adopted: P = I, with large and where I is the identity matrix. 4 Local Model Selection and Combination The recursive algorithm described by Eq. 7 and Eq. 8 returns for a given query point x q , a set of predictions y q (k) x 0 ....

Bierman G. J. 1977. Factorization Methods for Discrete Sequential Estimation. New York, NY: Academic Press.

....We need to refine the transformation estimate with each new pairing or group of pairings 1. LEARNING OBJECT RECOGNITION MODELS 17 adopted so that an improved estimate can then be used to identify additional pairings. Thus a recursive estimator is used. The square root information filter (SRIF) [3] is a recursive estimator well suited for this problem. Compared to the Kalman filter it is numerically more stable and faster for batched measurements; it also has the nice property of computing the total residual error as a side effect. As its name implies, the SRIF works by updating the square ....

G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 1977.

.... LU LDL T , QR, Cholesky, and singular value decomposition techniques (Golub Loan 1989) in the community of signal processing and system identification, Potter s square root least squares method (Potter Stern 1963, Bennet 1963, Andrews 1968) and later Bierman s UDU T factorization method (Bierman 1977, Thornton Bierman 1980) are examples of the successful endeavors. The Least Squares Lattice (LSL) method (Morf et al. 1977, Aling 1993) which has been widely used in signal processing, especially for recursive implementation, is another example of improved least squares estimator. In this ....

....the unknown and varying system dynamics (e.g. orders) 3. 4 Implementation The MMLS structure enables flexible and convenient implementation, where most of the well known decomposition methods such as the LU, LDL T , Cholesky, QR decompositions (Golub Loan 1989) and UDU T factorization (Bierman 1977) can all be used. For instance, the augmented data matrix defined in (3.14) can be decomposed with a QR decomposition Phi(t) Q(t)R(t) where Q(t) is an orthogonal matrix and R(t) is upper triangular. The parameter and loss function matrix U(t) and D(t) can be easily derived from R(t) with ....

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Bierman, G. J. (1977), Factorization Methods for Discrete Sequential Estimation, Academic Press, New York.

....matrix: maxfJg=min fJg = maxfJ 1 g=min fJ 1 g. The vector of parameter estimates is defined to 1 if the data functions are nearly orthogonal and of similar magnitude, and it becomes the greater the stronger linear dependencies are (fig. 7) 1 100000 1e 10 1e 15 1e 20 0 5 10 15 20 kappa [1] time [s] integral model Fig. 7: Condition number of the information matrix during identification by the integral model. It takes about two seconds to compute eigenvalues. Hence the steps. 200 150 100 50 0 50 100 150 200 0 1 2 acceleration [rad s 2] time [s] filtered accel. signal Fig. 8: ....

....scaled by the maximum eigenvalue maxfJ 1 g to produce a noise component of the parameter vector X . This scaling factor is related to the factor by which other (probably useful) components are scaled, namely the smallest one min fJ 1 g. 1 850 1e 05 1e 10 1e 15 1e 20 0 10 20 30 40 50 kappa [1] time [s] differential model Fig. 9: Condition number of the information matrix during identification by the differential model. number decreases as more data get available and finally it converges against a comparatively small value. 20 10 0 10 20 0 1 2 3 4 5 6 7 8 9 10 11 parameter X6 [Nms] ....

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G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 1977.

....and apply the generalised forgetting that counteracts both true parameter changes and approximation errors. Moreover, whole artillery is ready for solving other tasks when we concentrate on these statistics (e.g. structure estimation [14] estimation of the control period [20] factorised [21], fast and parallel implementations [22] etc. Analysis of the gained posterior pdf and or evaluation of its characteristics (like point estimates, confidence intervals, related predictors) is much harder task as a multivariate complex function of parameters has to be dealt with. It is solved, ....

G.J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977.

....and smoothing) to make optimal use of each measurement without computing differential quantities. First, we derive a linear set of equations between the unknown shape (surface point positions and radii of curvature) and the measurements. We then develop a robust linear smoother ( Gelb, 1974; Bierman, 1977] to compute statistically optimal current and past estimates from the set of contours. Smoothing allows us to combine measurements on both sides of each surface point. Our technique produces a complete surface description, i.e. a network of linked 3D surface points, which provides us with a ....

....circle can be recovered given the projection of three non parallel tangent lines onto the epipolar plane, a much more reliable estimate can be obtained by using more views. Given the set of equations (2) how can we recover the best estimate for (x c ; y c ; r) Regression theory [Albert, 1972; Bierman, 1977] tells us that the minimum least squared error estimate of the system of equations Ax = d can be found by minimizing e = jAx Gamma dj 2 = X i (a i Delta x Gamma d i ) 2 : 3) This minimum can be found by solving the set of normal equations 3 (A T A)x = A T d (4) or ( X i a i ....

[Article contains additional citation context not shown here]

G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, New York, 1977.

....(LS) problem is one of the oldest problems in applied mathematics, with numerous applications throughout modern engineering. Let us only mention linear estimation methods in adaptive signal processing and control, for adaptive antennas, beamforming, data communications [6, 7] space navigation [2], systems identification [1] etc. to name just a few. In general, without reference to any specific application, the problem can be stated as follows: Given a real m n matrix A (m n) with full column rank n, and a real m vector b, find x that solves min x kA x bk 2 The solution to the ....

G.J. Bierman, Factorization methods for discrete sequential estimation. Academic Press, New York, 1977.

....factor which controls the rate of tracking time varying parameters. Moreover, we want a multiplication division computational complexity of order N , where N is the filter memory length. A novel solution based on the Givens rotation and developed from an UDU T square root factorization [1] of the autocorrelation matrix is presented in this paper. The proposed algorithm belongs to the class of Fast QR and Lattice RLS algorithms. As in QR algorithms we do have a Q Givens rotation matrix and an R upper triangular matrix which is the Cholesky factor of the autocorrelation matrix. ....

.... the autocorrelation matrix Omega n = n X k=0 n Gammak X(k)X T (k) R T n PnRn (2) where X(k) is the input data vector at time k, Rn is an upper triangular with unit diagonal matrix and Pn is a positive diagonal matrix, such that the uniqueness of the factorization (2) will be assured [1]. As in many fast algorithms we exploit the relationship between forward and backward prediction filters. The following quantities are defined (see [2] for more details) ffl extended input vector, X(k) Theta v k . X T (k Gamma 1) T = Theta X T (k) r k Gamma1 T (3) ....

G.J.Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, 1977, pp.37-55

....[Bathe and Wilson, 1976] 4. 2 Statistical interpretation and robust estimation The weighted least squares formulation produces the minimum variance (and maximum likelihood) estimate for the unknown parameters under the assumption that each measurement is contaminated with additive Gaussian noise [Bierman, 1977]. In our structure from motion application, we can quantify the amount of error in the tracked feature locations by analyzing the response of the tracker (e.g. the shape of the correlation surface [Anandan, 1989] The statistical formulation enables us to make our technique more robust, by ....

G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, New York, 1977.

....of computations. Since the matrices P (k 1jk) and P (kjk) represent covariance matrices, they should be positive semidefinite. If due to approximations this is not true for the computed matrices, this may cause divergence of the solution. The square root algorithms as introduced by Potter (see [16, 2] for an introduction) avoid this problem by using the square root of the error covariance matrix, P (kjl) L(kjl)L(kjl) 0 , with L(kjl) a lower triangular matrix. Because the factors L(kjl) have a much smaller range of the eigenvalues these algorithms are also numerically better conditioned ....

G.J. Bierman. Factorization Methods for Discrete Sequential Estimation, volume 128 of Mathematics in Science and Engineering. Academic Press, Academic Press, New York, 1977.

....covariance (i.e. C j = U j U T j ) it weights both sides of the equation so that the residual error e has unit variance. A recursive estimator solves the system, efficiently updating the transformation estimate as pairings are adopted. We use the square root information filter (SRIF) [3] form of the Kalman filter for its numerical stability, and its efficiency with batched measurements. The SRIF works by updating the square root of the information matrix, which is the inverse of the estimate s covariance matrix. The initial square root, R 1 , and state vector, z 1 , are obtained ....

G.J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, 1977.

....square root information filter (SRIF) is a recursive estimator that is particularly well suited for this problem. Compared to the Kalman filter it is numerically more stable, it is faster for batched measurements, and it has the nice property of computing the total residual error as a side effect [3]. As its name implies, the SRIF works by updating the square root of the information matrix, which is the inverse of the estimate s covariance matrix. The initial square root, R 1 , and state vector, z 1 , are obtained from the first pairing hj; ki of model and image features: R 1 = U Gamma1 j ....

G. J. Bierman, Factorization methods for discrete sequential estimation, New York : Academic Press (1977).

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Bierman, G. J.: Factorization Methods for Discrete Sequential Estimation.Aca- demic Press, New York, 1977.

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G.J. Bierman, Factorization Method for Discrete Sequential Estimation, Academic Press, New York, 1977.

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977, pp. 69-76, 115-122.

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G. J. Bierman, Factorization methods for discrete sequential estimation, Academic Press, New York, 1977.

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Bierman, Gerald J., Factorization Methods for Discrete Sequential Estimation, Mathematics in Science and Engineering, Volume 128, Academic Press, New York, 1977.

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G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 1977.

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G. J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic Press, New York, 1977.

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G. J. Bierman, Factorization Methods for Discrete Sequential Estimation. New York: Academic, 1977.

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, (New York, 1977), pp. 69-76, 115-122, 214-217.

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, (New York, 1977), pp. 57-67, 69-76, 115-122, and 214-217.

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, (New York, 1977), pp. 69-76, 115-122.

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Bierman, G.J., Factorization Methods for Discrete Sequential Estimation, Academic Press, FL,

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G.J. Bierman. Factorization Methods for Discrete Sequential Estimation. Academic, New York, 1977.

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G. J. Bierman, Factorization Methods for Discrete Sequential Estimation, Academic Press, New York, 1977.

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G.J. Bierman, Factorization methods for discrete sequential estimation, Academic Press, New York, 1977.

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