This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.

Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g. computer graphics [1], coordinate metrology [2], petroleum engineering [11], statistics [7]. In the past, algorithms have been given which fit circles and ellipses in some least squares sense without minimizing the geometric distance to the given points [1], [6]. In this paper we present several algorithms which compute the ellipse for which the sum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods. Circles and ellipses may be represented algebraically i. e. by an equation of the form F (x) = 0. If a point is on the curve then its coordinates x are a zero of the function F . Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances.

An important investigation of time series involves searching for "movement" patterns, such as "going up" or "going down" or some combinations of them. Movement patterns can be in various scales: a large scale pattern may cover a long time period, while a small scale pattern usually covers a short time period. This paper considers such scale requirement. More specifically, a pattern is defined as a regular expression of letters, where each letter describes a movement direction and covers a specified length of time (called pattern unit length). To find if a time series (or a part of it) matches a pattern, the time series is first partitioned into consecutive sub-series of the unit length, and for each subseries, the direction of its best fitting line is taken as the movement direction of the sub-series if the distance between the best fitting line and the sub-series is within a specified tolerance (tolerance requirement). A direct implementation of pattern search will undoubtedly yield poor performance if the number of time series or the length of them is large. This paper introduces a pre-computation and indexing method to facilitate fast evaluation of pattern queries in user-specified scales. An efficient pre-computation algorithm is given to find the movement directions for all the sub-series that satisfy the tolerance requirement. Bounding triangles are used to represent clusters of sub-series. Relational database is then used to store these bounding triangles and relational operations are employed to facilitate the evaluation of pattern queries. The paper also reports some experiments performed on a real-life data set to show the efficiency and the scalability of the algorithms.

A unified formulation for the calibration of both serial-link robots and robotic mechanisms having kinematic closed-loops is presented and applied experimentally to two 6degree -of-freedom devices: the RSI 6-DOF Hand Controller and the MEL "Modified Stewart Platform." The unification is based on an equivalence between end-effector measurements and constraints imposed by the closure of kinematic loops. Errors are allocated to the joints such that the loop equations are satisfied exactly, which eliminates the issue of equation scaling and simplifies the treatment of multi-loop mechanisms. For the experiments reported here, no external measuring devices are used; instead we rely on measurements of displacements in some of the passive joints of the devices. Using a priori estimates of the statistics of the measurement errors and the parameter errors, the method estimates the parameters and their accuracy, and tests for unmodelled factors.

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This paper addresses two topics in robot calibration: the formulation of a calibration index and the presence of input noise. The calibration index is introduced to succinctly capture the essence of a kinematic calibration method and allow insightful comparison between that method and other methods. Input noise, e.g., joint angle errors, is a potential source of error that is usually ignored, but is known to lead to bias errors in the estimates (Norton 1986). The two topics also will be seen to have a relationship. Many kinematic calibration methods have been proposed by now, and more are continually introduced. These methods differ in terms of whether an external metrology system is required (openloop methods) or not (closed-loop methods), how many components of pose are measured and by what means, how many components of pose are passively constrained and how, whether there are unsensed joints, and whether a serial or parallel mechanism is being calibrated (Hollerbach 1993). It can be very difficult to understand how one method is related to another method, or what the tradeoffs are in selecting a calibration method. This statement is particularly true for parallel linkages, with different mixtures of sensed and unsensed joints, and passive and active joints. It is not always immediately obvious how easy a parallel linkage is to calibrate, especially by a closedloop method. The literature on camera calibration is usually distinct from that on manipulator calibration. By modeling cameras and light beams as prismatic legs, it is possible to fold camera calibration into manipulator calibration. An advantage is that mixed manipulator/camera calibration problems can be handled uniformly.

Parameter values for a kinetic model of the nuclear replication–division cycle in frog eggs are estimated by fitting solutions of the kinetic equations (nonlinear ordinary differential equations) to a suite of experimental observations. A set of optimal parameter values is found by minimizing an objective function defined as the orthogonal distance between the data and the model. The differential equations are solved by LSODAR and the objective function is minimized by ODRPACK. The optimal parameter values are close to the “guesstimates” of the modelers who first studied this problem. These tools are sufficiently general to attack more complicated problems, where guesstimation is impractical or unreliable. Key words: cyclin-dependent kinase, M-phase promoting factor, orthogonal distance regression, Levenberg-Marquardt method. 1.

This paper presents a practical numerical method for separating and estimating growth and mortality coefficients in stage- or size-structured populations using only observations of the relative or absolute abundance of each stage. The method involves writing a system of linear ordinary differential equations (ODE's) modelling the rate of change of abundance. The solution of the differential system can be numerically approximated using standard (e.g. sixth order Runge-Kutta-Felhberg) methods. An optimization problem whose solutions yield `optimal' coefficients for a given model is formulated. The ODE numerical integration technique can then be employed to furnish required function and gradient information to the optimization algorithm. Data fitting software package ORDPACK is then successfully employed to estimate optimal coefficients for the ODE population model. Simulation experiments with four- and eight-stage model populations illustrate that the method results in successful estimati...

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

Abstract A common method of fitting curves and surfaces to data is to minimize the sum of squares of the orthogonal distances from the data points to the curve or surface, a process known as orthogonal distance regression. Here we consider fitting geometrical objects to data when some orthogonal distances are not available. Methods based on the Gauss-Newton method are developed, analysed and illustrated by examples. Keywords: least squares, orthogonal distances, Gauss-Newton method. 1 Introduction In assessing the quality of manufactured parts, it is often appropriate to take measurements from the surface of the part, for example using a coordinate measuring machine, and fit a model to those data. In particular the part is often such that it can by modelled by a geometrical object, and so there is interest in the development of good numerical methods for fitting such a curve or surface to data. A commonly used fitting technique is orthogonal distance regression or ODR, where the sum of squares of the orthogonal distances from the data points to the model is minimized.