Adaptive spline fitting with particle swarm optimization

In fitting data with a spline, finding the optimal placement of knots can significantly improve the quality of the fit. However, the challenging high-dimensional and non-convex optimization problem associated with completely free knot placement has been a major roadblock in using this approach. We present a method that uses particle swarm optimization (PSO) combined with model selection to address this challenge. The problem of overfitting due to knot clustering that accompanies free knot placement is mitigated in this method by explicit regularization, resulting in a significantly improved performance on highly noisy data. The principal design choices available in the method are delineated and a statistically rigorous study of their effect on performance is carried out using simulated data and a wide variety of benchmark functions. Our results demonstrate that PSO-based free knot placement leads to a viable and flexible adaptive spline fitting approach that allows the fitting of both smooth and non-smooth functions.Comment: Accepted version; Typo corrected in equation 3; Minor changes to tex

Adaptive spline fitting with particle swarm optimization

In fitting data with a spline, finding the optimal placement of knots can significantly improve the quality of the fit. However, the challenging high-dimensional and non-convex optimization problem associated with completely free knot placement has been a major roadblock in using this approach. We present a method that uses particle swarm optimization (PSO) combined with model selection to address this challenge. The problem of overfitting due to knot clustering that accompanies free knot placement is mitigated in this method by explicit regularization, resulting in a significantly improved performance on highly noisy data. The principal design choices available in the method are delineated and a statistically rigorous study of their effect on performance is carried out using simulated data and a wide variety of benchmark functions. Our results demonstrate that PSO-based free knot placement leads to a viable and flexible adaptive spline fitting approach that allows the fitting of both smooth and non-smooth functions

Multiple 2D self organising map network for surface reconstruction of 3D unstructured data

Surface reconstruction is a challenging task in reverse engineering because it must represent the surface which is similar to the original object based on the data obtained. The data obtained are mostly in unstructured type whereby there is not enough information and incorrect surface will be obtained. Therefore, the data should be reorganised by finding the correct topology with minimum surface error. Previous studies showed that Self Organising Map (SOM) model, the conventional surface approximation approach with Non Uniform Rational B-Splines (NURBS) surfaces, and optimisation methods such as Genetic Algorithm (GA), Differential Evolution (DE) and Particle Swarm Optimisation (PSO) methods are widely implemented in solving the surface reconstruction. However, the model, approach and optimisation methods are still suffer from the unstructured data and accuracy problems. Therefore, the aims of this research are to propose Cube SOM (CSOM) model with multiple 2D SOM network in organising the unstructured surface data, and to propose optimised surface approximation approach in generating the NURBS surfaces. GA, DE and PSO methods are implemented to minimise the surface error by adjusting the NURBS control points. In order to test and validate the proposed model and approach, four primitive objects data and one medical image data are used. As to evaluate the performance of the proposed model and approach, three performance measurements have been used: Average Quantisation Error (AQE) and Number Of Vertices (NOV) for the CSOM model while surface error for the proposed optimised surface approximation approach. The accuracy of AQE for CSOM model has been improved to 64% and 66% when compared to 2D and 3D SOM respectively. The NOV for CSOM model has been reduced from 8000 to 2168 as compared to 3D SOM. The accuracy of surface error for the optimised surface approximation approach has been improved to 7% compared to the conventional approach. The proposed CSOM model and optimised surface approximation approach have successfully reconstructed surface of all five data with better performance based on three performance measurements used in the evaluation

Bayesian Nonparametric Differential Equation Models for Functions

Bayesian nonparametric methods develop priors over a large class of functions that essentially allow any continuous function to be modeled. Though these methods are flexible, they are black box approaches that do not explicitly incorporate additional information on the shape of the curve. In many contexts, though the exact parametric form of the curve is unknown, additional scientific information is available in the form of differential operators. This dissertation develops nonparametric priors over function spaces that are specified by differential operators. Here two novel approaches to nonparametric function estimation are considered. In the first approach the prior is specified by a linear differential equation. The Mechanistic Hierarchical Gaussian process defines a prior over functions consistent with a differential operator. The method is applied to muscle force tracings in a functional ANOVA context, and is shown to adequately describe the between subject variability often seen in such tracings. In the second case a novel spline based approach is considered. Here prior information is specifies the maximum number of extrema (changepoints) for an arbitrary function located on an open set in R. The Local Extrema (LX) spline models the first derivative of the curve and puts a prior over the maximum number of changepoints. This method is applied to animal toxicology studies, human health surveys, and seasonal data; and it is shown to remove artifactual bumps common to other nonparametric methods. It is further shown to superior in terms of estimated squared error loss in simulation studies.Doctor of Philosoph

Reconstruction of moving surfaces of revolution from sparse 3-D measurements using a stereo camera and structured light

Das Ziel dieser Arbeit ist die Entwicklung und Analyse der algorithmischen Methodik zur Rekonstruktion eines parametrischen Oberflächenmodells für ein rotationssymmetrisches Objekt aus einer Sequenz von dünnen 3D-Punktwolken. Dabei kommt ein neuartiges Messsystem mit großem Sichtfeld zum Einsatz, das auch in schwierigen Bedingungen eingesetzt werden kann. Das zu vermessende Objekt kann während der Aufnahme der Sequenz einer als analytisches Modell formulierbaren Bewegung unterliegen. Das Verfahren wird anhand einer praktischen Anwendung zur Oberflächenrückgewinnung eines Rades analysiert und entwickelt. Es wird gezeigt, dass die durch Fit eines einfachen Models für jede Einzelmessung erzielbare Genauigkeit durch Anpassung eines globalen Modells unter gleichzeitiger Einbeziehung aller Einzelmessungen und unter Berücksichtigung eines geeigneten Bewegungsmodells erheblich verbessert werden kann. Die Gewinnung der dreidimensionalen Punktdaten erfolgt mit einem Stereokamerasystem in Verbindung mit aktiver Beleuchtung in Form eines Punktmusters. Eine relativ hohe Punktdichte im gesamten Sichtfeld des Stereokamerasystems wird durch Verbindung mehrerer Laserprojektoren zu einer Projektionseinheit erzielt. Durch exakte Kalibrierung des Kamerasystems und der Projektionseinheit wird trotz großer Streuung der Laserpunkte im Kamerabild unter Ausnutzung der trifokalen geometrischen Bedingungen eine hohe Genauigkeit in den dreidimensionalen Punktdaten erzielt

Optimal Surface Fitting of Point Clouds Using Local Refinement : Application to GIS Data

This open access book provides insights into the novel Locally Refined B-spline (LR B-spline) surface format, which is suited for representing terrain and seabed data in a compact way. It provides an alternative to the well know raster and triangulated surface representations. An LR B-spline surface has an overall smooth behavior and allows the modeling of local details with only a limited growth in data volume. In regions where many data points belong to the same smooth area, LR B-splines allow a very lean representation of the shape by locally adapting the resolution of the spline space to the size and local shape variations of the region. The iterative method can be modified to improve the accuracy in particular domains of a point cloud. The use of statistical information criterion can help determining the optimal threshold, the number of iterations to perform as well as some parameters of the underlying mathematical functions (degree of the splines, parameter representation). The resulting surfaces are well suited for analysis and computing secondary information such as contour curves and minimum and maximum points. Also deformation analysis are potential applications of fitting point clouds with LR B-splines

Optimal Surface Fitting of Point Clouds Using Local Refinement

This open access book provides insights into the novel Locally Refined B-spline (LR B-spline) surface format, which is suited for representing terrain and seabed data in a compact way. It provides an alternative to the well know raster and triangulated surface representations. An LR B-spline surface has an overall smooth behavior and allows the modeling of local details with only a limited growth in data volume. In regions where many data points belong to the same smooth area, LR B-splines allow a very lean representation of the shape by locally adapting the resolution of the spline space to the size and local shape variations of the region. The iterative method can be modified to improve the accuracy in particular domains of a point cloud. The use of statistical information criterion can help determining the optimal threshold, the number of iterations to perform as well as some parameters of the underlying mathematical functions (degree of the splines, parameter representation). The resulting surfaces are well suited for analysis and computing secondary information such as contour curves and minimum and maximum points. Also deformation analysis are potential applications of fitting point clouds with LR B-splines.publishedVersio