Christoph J. Witzgall

Ladars for construction assessment and update

Work at the National Institute of Standards and Technology (NIST) on laser radar imaging of a construction site is described. The objective of the NIST research is to make measurements required in a construction project quicker and cheaper than current practice and to do so without impacting existing operations. This can be done by developing techniques for real-time assessment and documentation in terms of 3-D as-built models of the construction process. Once developed, this technology may be used for other applications such as condition assessment of a hazardous environment where human intervention would be impossible.

Arc tolerances in shortest path and network flow problems

This paper studies one aspect of the “robustness” of optimal solutions to shortest path and, more generally, network flow problems. Specifically, we characterize the maximum increase and the maximum decrease in an arcs cost that can be tolerated without changing optimality of the current solution. Calculation of these quantities is quite simple for nonbasic arcs, and somewhat more involved for basic arcs. When such tolerances are to be determined simultaneously for all arcs in the network, considerable duplication of effort can be avoided through the use of specialized algorithms. Several algorithms for calculating all arc tolerances are presented, one of which is shown to have complexity order n2 for general networks with n nodes.

Fitting circles and spheres to coordinate measuring machine data

This work addresses the problem of enclosing given data points between two concentric circles (spheres) of minimum distance whose associated annulus measures the out-of-roundness (OOR) tolerance. The problem arises in analyzing coordinate measuring machine (CMM) data taken against circular (spherical) features of manufactured parts. It also can be interpreted as the “geometric” Chebychev problem of fitting a circle (sphere) to data so as to minimize the maximum distance deviation. A related formulation, the “algebraic” Chebychev formula, determines the equation of a circle (sphere) to minimize the maximum violation of the equation by the data points. In this paper, we describe a linear-programming approach for the algebraic Chebychev formula that determines reference circles (spheres) and related annuluses whose widths are very close to the widths of the true geometric Chebychev annuluses. We also compare the algebraic Chebychev formula against the popular algebraic least-squares solutions for various data sets. In most of these examples, the algebraic and geometric Chebychev solutions coincide, which appears to be the case for most real applications. Such solutions yield concentric circles whose separation is less than that of the corresponding least-squares solution. It is suggested that the linear-programming approach be considered as an alternate solution method for determining OOR annuluses for CMM data sets.

Construction object identification from LADAR scans :: an experimental study using I-Beams

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Algorithmic Enhancements to the Method of Centers for Linear Programming Problems

Interior point algorithms for solving linear programming problems are considered. The techniques are derived from a continuous version of Huards method of centers that yields a family of trajectories in the feasible region that all converge to an optimal solution. The tangential direction of these trajectories is the dual affine direction. Deficiencies in some of these trajectories are discussed, and the need to recenter is argued. Several new algorithms that use the dual affine direction and a recentering direction in a multidirection approach are then derived. The most promising of these algorithms is based on minimizing the cost function on a sequence of two-dimensional cross sections of the feasible region. Numerical results are presented. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Fitting Spheres to Range Data From 3-D Imaging Systems

Two error functions used for nonlinear least squares (LS) fitting of spheres to range data from 3-D imaging systems are discussed: the orthogonal error function and the directional error function. Both functions allow unrestricted gradient-based minimization and were tested on more than 40 data sets collected under different experimental conditions (e.g., different sphere diameters, instruments, data density, and data noise). It was found that the orthogonal error function results in two local minima and that the outcome of the optimization depends on the choice of starting point. The centroid of the data points is commonly used as the starting point for the nonlinear LS solution, but the choice of starting point is sensitive to data segmentation and, for some sparse and noisy data sets, can lead to a spurious minimum that does not correspond to the center of a real sphere. The directional error function has only one minimum; therefore, it is not sensitive to the starting point and is more suitable for applications that require fully automated sphere fitting.

Optimizing Over Three-Dimensional Subspaces in an Interior-Point Method for Linear Programming*

Abstract Interior-point algorithms for solving linear programming problems are considered. A three-dimensional method is developed that, at each iteration, solves a subproblem based on minimizing the cost function on low-dimensional cross sections of the feasible region. The generators for the three-dimensional subproblem include the dual affine search direction and two higher-order search directions. One of the higher-order directions is a third-order correction to the Newton recentering direction, and the other is a correction to the dual affine direction that is motivated by the use of rank-one updates of the second-derivative information. Numerical results are presented for this method that indicate a nearly 20% reduction in CPU time compared to our best dual affine implementation.

Evaluation of Aerodynamic Drag and Torque for External Tanks in Low Earth Orbit.

A numerical procedure is described in which the aerodynamic drag and torque in low Earth orbit are calculated for a prototype Space Shuttle external tank and its components, the “LO2” and “LH2” tanks, carrying liquid oxygen and hydrogen, respectively, for any given angle of attack. Calculations assume the hypersonic limit of free molecular flow theory. Each shell of revolution is assumed to be described by a series of parametric equations for their respective contours. It is discretized into circular cross sections perpendicular to the axis of revolution, which yield a series of ellipses when projected according to the given angle of attack. The drag profile, that is, the projection of the entire shell is approximated by the convex envelope of those ellipses. The area of the drag profile, that is, the drag area, and its center of area moment, that is, the drag center, are then calculated and permit determination of the drag vector and the eccentricity vector from the center of gravity of the shell to the drag center. The aerodynamic torque is obtained as the cross product of those vectors. The tanks are assumed to be either evacuated or pressurized with a uniform internal gas distribution: dynamic shifting of the tank center of mass due to residual propellant sloshing is not considered.

On an approximate minimax circle closest to a set of points

We show how the Chebychev minimax criterion for finding a circle closest to a set of points can be approximated well by standard linear programming procedures.

Logarithmic SUMT limits in convex programming

Abstract.The limits of a class of primal and dual solution trajectories associated with the Sequential Unconstrained Minimization Technique (SUMT) are investigated for convex programming problems with non-unique optima. Logarithmic barrier terms are assumed. For linear programming problems, such limits – of both primal and dual trajectories – are strongly optimal, strictly complementary, and can be characterized as analytic centers of, loosely speaking, optimality regions. Examples are given, which show that those results do not hold in general for convex programming problems. If the latter are weakly analytic (Bank et al. [3]), primal trajectory limits can be characterized in analogy to the linear programming case and without assuming differentiability. That class of programming problems contains faithfully convex, linear, and convex quadratic programming problems as strict subsets. In the differential case, dual trajectory limits can be characterized similarly, albeit under different conditions, one of which suffices for strict complementarity.