Moment-Preserving Spline Approximation and Turán Quadratures

1988 ◽  
pp. 357-365 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Milan A. Kovačević
Keyword(s):  
Spline Approximation

scholarly journals On the BIC for determining the number of control points in B-spline surface approximation in case of correlated observations

2020 ◽  
Vol 10 (1) ◽  
pp. 110-123
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib
Keyword(s):  
Shape Control ◽  
Simple Procedure ◽  
Spline Approximation ◽  
Autoregressive Process ◽  
Optimal Number ◽  
Information Criteria ◽  
General Effect ◽  
Control Points ◽  
B Spline ◽  
Correlated Observations

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.

scholarly journals On an n-dimensional quadratic spline approximation

1992 ◽  
Vol 68 (2) ◽  
pp. 113-135 ◽  
Author(s):  
Margit Lénárd
Keyword(s):  
Spline Approximation ◽  
Quadratic Spline

scholarly journals A stepwise-selected spline approximation to time-varying parameters, with application to occupancy modelling

2012 ◽  
Vol 4 (2) ◽  
pp. 123-132 ◽  
Author(s):  
James T. Thorson ◽  
Shijie Zhou ◽  
André E. Punt ◽  
Anthony D. M. Smith
Keyword(s):  
Spline Approximation ◽  
Time Varying ◽  
Occupancy Modelling ◽  
Varying Parameters ◽  
Time Varying Parameters

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

scholarly journals On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation

Sensors ◽  
2018 ◽  
Vol 18 (9) ◽  
pp. 2964 ◽  
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib ◽  
Ingo Neumann
Keyword(s):  
Stochastic Model ◽  
Good Description ◽  
Spline Approximation ◽  
Laser Scanner ◽  
Terrestrial Laser Scanner ◽  
Case Scenario ◽  
Constant Variance ◽  
Power Law Function ◽  
Stochastic Properties

For a trustworthy least-squares (LS) solution, a good description of the stochastic properties of the measurements is indispensable. For a terrestrial laser scanner (TLS), the range variance can be described by a power law function with respect to the intensity of the reflected signal. The power and scaling factors depend on the laser scanner under consideration, and could be accurately determined by means of calibrations in 1d mode or residual analysis of LS adjustment. However, such procedures complicate significantly the use of empirical intensity models (IM). The extent to which a point-wise weighting is suitable when the derived variance covariance matrix (VCM) is further used in a LS adjustment remains moreover questionable. Thanks to closed loop simulations, where both the true geometry and stochastic model are under control, we investigate how variations of the parameters of the IM affect the results of a LS adjustment. As a case study, we consider the determination of the Cartesian coordinates of the control points (CP) from a B-splines curve. We show that a constant variance can be assessed to all the points of an object having homogeneous properties, without affecting the a posteriori variance factor or the loss of efficiency of the LS solution. The results from a real case scenario highlight that the conclusions of the simulations stay valid even for more challenging geometries. A procedure to determine the range variance is proposed to simplify the computation of the VCM.

A small time solutions for the Korteweg–de Vries equation using spline approximation

2006 ◽  
Vol 173 (2) ◽  
pp. 834-846 ◽  
Author(s):  
Dursun Irk ◽  
İdris Dagˇ ◽  
Bülent Saka
Keyword(s):  
Vries Equation ◽  
Spline Approximation ◽  
Small Time ◽  
De Vries
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