Non-linear Minimization Process Research Articles

Scanning Electron Microscope Calibration Using a Multi-Image Non-Linear Minimization Process

A scanning electron microscope (SEM) calibrating approach based on non-linear minimization procedure is presented in this article11A part of this article has been published in IEEE International Conference on Robotics and Automation (ICRA), 2014.. Both the intrinsic parameters and the extrinsic parameters estimations are achieved simultaneously by minimizing the registration error. The proposed approach considers multi-images of a multi-scale calibration pattern view from different positions and orientations. Since the projection geometry of the scanning electron microscope is different from that of a classical optical sensor, the perspective projection model and the parallel projection model are considered and compared with distortion models. Experiments are realized by varying the position and the orientation of a multi-scale chessboard calibration pattern from 300× to 10,000×. The experimental results show the efficiency and the accuracy of this approach.

Accurate calibration with highly distorted images

Camera calibration is a two-step process where first a linear algebraic approximation is followed by a nonlinear minimization. The nonlinear minimization adjusts the pin-hole and lens distortion models to the calibrating data. Since both models are coupled, nonlinear minimization can converge to a local solution easily. Moreover, nonlinear minimization is poorly conditioned since parameters with different effects in the minimization function are calculated simultaneously (some are in pixels, some in world coordinates, and some are lens distortion parameters). A local solution is adapted to parameters, which minimize the function easily, and the remaining parameters are just adapted to this solution. We propose a calibration method where traditional calibration steps are inverted. First, a nonlinear minimization is done, and after, camera parameters are computed in a linear step. Using projective geometry constraints in a nonlinear minimization process, detected point locations in the images are corrected. The pin-hole and lens distortion models are computed separately with corrected point locations. The proposed method avoids the coupling between both models. Also, the condition of nonlinear minimization increases since points coordinates are computed alone.

Analytical potentiometric data analyses of metal–ligand equilibria by microcomputer. A general procedure for minimizing potential sources of errors and a fast linear-scaled least-squares algorithm

Analytical potentiometry is a powerful technique to investigate multiple species in solution and their formation constants. However, there have been some difficulties in this method resulting primarily from the theoretical approach. In this paper we critically examined them and improvements have been implemented successfully. Suitable experimental designs are adapted and preliminary computation of potentiometric data are performed to minimize potential sources of errors. Criteria are proposed to choose the best set of titration curves. A correction is introduced to partially minimize the errors originating from the volume variation. A linear-scaled minimization algorithm has been implemented to extract the value of the complex stability constants from the experimental free ion concentrations. This has two advantages over the earlier non-linear minimization process. The execution time is faster and it facilitates the selection of the best fit of species. A series of 14 programs are introduced to run on a low memory microcomputer.