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Efficient Knot Optimization for Accurate B-spline-based Data Approximation
thesisposted on 14.12.2020, 21:20 by Yo-Sing YehYo-Sing Yeh
Many practical applications benefit from the reconstruction of a smooth multivariate function from discrete data for purposes such as reducing file size or improving analytic and visualization performance. Among the different reconstruction methods, tensor product B-spline has a number of advantageous properties over alternative data representation. However, the problem of constructing a best-fit B-spline approximation effectively contains many roadblocks. Within the many free parameters in the B-spline model, the choice of the knot vectors, which defines the separation of each piecewise polynomial patch in a B-spline construction, has a major influence on the resulting reconstruction quality. Yet existing knot placement methods are still ineffective, computationally expensive, or impose limitations on the dataset format or the B-spline order. Moving beyond the 1D cases (curves) and onto higher dimensional datasets (surfaces, volumes, hypervolumes) introduces additional computational challenges as well. Further complications also arise in the case of undersampled data points where the approximation problem can become ill-posed and existing regularization proves unsatisfactory.
This dissertation is concerned with improving the efficiency and accuracy of the construction of a B-spline approximation on discrete data. Specifically, we present a novel B-splines knot placement approach for accurate reconstruction of discretely sampled data, first in 1D, then extended to higher dimensions for both structured and unstructured formats. Our knot placement methods take into account the feature or complexity of the input data by estimating its high-order derivatives such that the resulting approximation is highly accurate with a low number of control points. We demonstrate our method on various 1D to 3D structured and unstructured datasets, including synthetic, simulation, and captured data. We compare our method with state-of-the-art knot placement methods and show that our approach achieves higher accuracy while requiring fewer B-spline control points. We discuss a regression approach to the selection of the number of knots for multivariate data given a target error threshold. In the case of the reconstruction of irregularly sampled data, where the linear system often becomes ill-posed, we propose a locally varying regularization scheme to address cases for which a straightforward regularization fails to produce a satisfactory reconstruction.