In this work, we propose the general idea of using a path-finding algorithm to solve a variational problem. By interpreting a variational problem of finding the function that minimizes a functional integral as a shortest path finding, we can apply the shortest path-finding algorithm to numerically estimate the optimal function. This can be achieved by discretizing the continuous domain of the variational problem into a spatially weighted graph. The weight of each edge is defined according to the function of the original problem. We adopt the Moser lattice as the discretization scheme since it provides adjustable connections around a vertex. We find that this number of connections is crucial to the estimation of an accurate optimal path. Dijkstra’s shortest path-finding algorithm was chosen due to its simplicity and convenience in implementation. We validate our proposal by applying Dijkstra’s path-finding algorithm to numerically solve three famous variational problems, i.e., the optical ray tracing, the brachistochrone, and the catenary problems. The first two are examples of problems with no constraint. The standard Dijkstra’s algorithm can be directly applied. The third problem is an example of a problem with an isoperimetric constraint. We apply the Lagrangian relaxation technique to relax the optimization in the standard Dijkstra algorithm to incorporate the constraint. In all cases, when the number of sublattices is large enough, the results agree well with the analytic solutions. In all cases, the same path-finding code is used, regardless of the problem details. Our approaches provide more insight and promise to be more flexible than conventional numerical methods. We expect that our method can be useful in practice when an investigation of the optimal path in a complex problem is needed.

Linear least square is a common method to fit two-dimensional data set with linear relation. In non-linear case we need to do some linearization before fitting. Parameters can be extracted from the fitted line and used to predict new dependent variable values at new independent variable points. In this report we investigate the uncertainty introduced to a nonlinear model prediction from linear least square fitting. As an example, we fit a data set with the relation D = a/h, which is the physical model of measurement we are interested. The fitting goes well, and we get the curve with R-square very closed to one. However, when we use the fitted parameters to actual measurements, the accuracy is poor, especially at large y. At first, we expect that this due to the functional form of the model. When compare with high order polynomial fitting we realize that this is not the case, since, for example, the sixth-degree polynomial gives less than one percent error, about 10 times less than linear fitting prediction. It is because the linearization and the inverse transformation to the original space in the linear least square method that give rise to the uncertainty. Our analysis can be generalized to any non-linear model prediction. We expect our results to be a caution to anyone using linear fitting to their non-linear model prediction.

We demonstrate the analysis of numerical error introduced when a path-finding algorithm, such as Dijkstra’s algorithm, is applied to estimate the optimum path in a variational problem. The space discretization is crucial since it must provide as many paths as possible to approximate the exact solution. We use the concept of Moser’s lattice to pixelate the continuous domain into square pixels. Each pixel is surrounded by vertices, which are interpreted as the channels to enter or exit the pixel. Obviously, the accuracy of the estimated path depends on the pixel size and number of channels around a pixel. Since the path-finding algorithm ensures that the path obtained is the optimal path in the network, the convergency of the functional integral can be used to ensure the convergency of the numerical solution. To demonstrate the validity of the procedure, we apply the path-finding algorithm to solve the classic brachistochrone problem. The continuum limit of the time-of-flight with increasing numbers of channels is determined for various values of pixel sizes. We found the convergency of these limiting time-of-flight to the exact results with the decreasing pixel size. Analysis of the root-mean-square error of the pathfinding results confirm the convergencies of the numerical solutions. Since the effect of the discretization procedure is on its ability to provide a large variety of paths, the results here can be used as a guide for any variational problems where the shape of optimum paths are comparable.

We have developed a multi-wavelength light source and apply it to micro welding inspection. We project light rays from multiple white light super bright LEDs on a reflection grating. These LEDS are mounted on the specific position to give the diffracted ray with the wavelength range from 480 to 680 nm. The diffracted ray is pass to an optical fibre. The orientation of fibre tube is fixed so diffracted light from each LED will give a specific wavelength. The bandwidths of lights from all LEDS are quite narrow (< 50 nm), which is possible to be used as a light source for multi-wavelength imaging. The diffracted light is transported via optical fibre to a microscope used to inspect the micro welding points on a microelectronics circuit. Due to its very small size, the welding defects are hard to detected, even with high intensity white light. We find that, for typical welding points, the light with wavelength 500 to 600 nm can significantly enhance the contrast of welding image. This agree well with our preliminary result that, with our RGB light source, the maximum contrast can be obtained with the green - yellow to orange - red light. We expect our results to be useful to the inspection of micro welding, especially when integrated to some automation, e.g. machine vision system.

We investigate the uncertainty of Diameter at Breast Height, or DBH, of woods via a simple mathematical model. Usually, the DBH is not measured directly. The measurement is done on the Girth at Breast Height (GBH), which, by definition, is the circumference of wood at 130 cm above the ground. The DBH can then be calculated, assuming that the wood cross-section is a circle. This can introduce uncertainty to the DBH due to the irregularity of the shape of the wood cross-section. In this work, we approximate the wood cross-section by an ellipse, which can be fitted to the cross-section better than the circle. However, the eccentricity of the fitted ellipse is unknown, so we treat it as a random variable. The discrepancy between the diameter of the circle with the same area as the fitted ellipse, and the diameter of the circle with the same circumference as the GBH, is calculated. Monte Carlo simulations are performed to obtain the uncertainty of the DBH.