Linear Combination Of B-splines Research Articles

Software compensation of undesirable racking motion of H-frame 3D printers using filtered B-splines

The H-frame (also known as H-Bot) architecture is a simple and elegant two-axis parallel positioning system used to construct the XY stage of 3D printers. It holds potential for high speed and excellent dynamic performance due to the use of frame-mounted motors that reduce the moving mass of the printer while allowing for the use of (heavy) higher torque motors. However, the H-frame's dynamic accuracy is limited during high-acceleration and high-speed motion due to racking -- i.e., parasitic torsional motions of the printer's gantry due to a force couple. Mechanical solutions to the racking problem are either costly or detract from the simplicity of the H-frame. In this paper, we introduce a feedforward software compensation algorithm, based on the filtered B-splines (FBS) method, that rectifies errors due to racking. The FBS approach expresses the motion command to the machine as a linear combination of B-splines. The B-splines are filtered through an identified model of the machine dynamics and the control points of the B-spline based motion command are optimized such that the tracking error is minimized. To compensate racking using the FBS algorithm, an accurate frequency response function of the racking motion is obtained and coupled to the H-frame's x- and y-axis dynamics with a kinematic model. The result is a coupled linear parameter varying model of the H-frame that is utilized in the FBS framework to compensate racking. An approximation of the proposed racking compensation algorithm, that decouples the x- and y-axis compensation, is developed to significantly improve its computational efficiency with almost no loss of compensation accuracy. Experiments on an H-frame 3D printer demonstrate a 43 percent improvement in the shape accuracy of a printed part using the proposed algorithm compared to the standard FBS approach without racking compensation.

Bivariate B-splines from convex configurations

An order-k univariate spline is a function defined over a set S of at least k+2 real parameters, called knots. Such a spline can be obtained as a linear combination of B-splines, each of them being defined over a subset of k+2 consecutive knots of S, called a configuration of S. In the bivariate setting, knots are pairs of reals and B-splines are defined over configurations of k+3 knots. Using convex pseudo-circles, we define a family of configurations that gives rise to bivariate B-splines that retain the fundamental properties of univariate B-splines. We also give an algorithm to construct such configurations.

Numerical Study of the Inverse Problem of Generalized Burgers–Fisher and Generalized Burgers–Huxley Equations

In this paper, the boundary value inverse problem related to the generalized Burgers–Fisher and generalized Burgers–Huxley equations is solved numerically based on a spline approximation tool. B-splines with quasilinearization and Tikhonov regularization methods are used to obtain new numerical solutions to this problem. First, a quasilinearization method is used to linearize the equation in a specific time step. Then, a linear combination of B-splines is used to approximate the largest order of derivatives in the equation. By integrating from this linear combination, some approximations have been obtained for each of the functions and derivatives with respect to time and space. The boundary and additional conditions of the problem are also applied in these approximations. The Tikhonov regularization method is used to solve the system of linear equations using noisy data. Several numerical examples are provided to illustrate the accuracy and efficiency of the method.

Fast dynamic nonparametric distribution tracking in electron microscopic data

In situ transmission electron microscope (TEM) adds a promising instrument to the exploration of the nanoscale world, allowing motion pictures to be taken while nano objects are initiating, crystalizing and morphing into different sizes and shapes. To enable in-process control of nanocrystal production, this technology innovation hinges upon a solution addressing a statistical problem, which is the capability of online tracking a dynamic, time-varying probability distribution reflecting the nanocrystal growth. Because no known parametric density functions can adequately describe the evolving distribution, a nonparametric approach is inevitable. Towards this objective, we propose to incorporate the dynamic evolution of the normalized particle size distribution into a state space model, in which the density function is represented by a linear combination of B-splines and the spline coefficients are treated as states. The closed-form algorithm runs online updates faster than the frame rate of the in situ TEM video, making it suitable for in-process control purpose. Imposing the constraints of curve smoothness and temporal continuity improves the accuracy and robustness while tracking the probability distribution. We test our method on three published TEM videos. For all of them, the proposed method is able to outperform several alternative approaches.

Bayesian sample size determination for phase IIA clinical trials using historical data and semi-parametric prior's elicitation.

The Simon's two-stage design is the most commonly applied among multi-stage designs in phase IIA clinical trials. It combines the sample sizes at the two stages in order to minimize either the expected or the maximum sample size. When the uncertainty about pre-trial beliefs on the expected or desired response rate is high, a Bayesian alternative should be considered since it allows to deal with the entire distribution of the parameter of interest in a more natural way. In this setting, a crucial issue is how to construct a distribution from the available summaries to use as a clinical prior in a Bayesian design. In this work, we explore the Bayesian counterparts of the Simon's two-stage design based on the predictive version of the single threshold design. This design requires specifying two prior distributions: the analysis prior, which is used to compute the posterior probabilities, and the design prior, which is employed to obtain the prior predictive distribution. While the usual approach is to build beta priors for carrying out a conjugate analysis, we derived both the analysis and the design distributions through linear combinations of B-splines. The motivating example is the planning of the phase IIA two-stage trial on anti-HER2 DNA vaccine in breast cancer, where initial beliefs formed from elicited experts' opinions and historical data showed a high level of uncertainty. In a sample size determination problem, the impact of different priors is evaluated.

Penalized B-spline estimator for regression functions using total variation penalty

We carry out a study on a penalized regression spline estimator with total variation penalty. In order to provide a spatially adaptive method, we consider total variation penalty for the estimating regression function. This paper adopts B-splines for both numerical implementation and asymptotic analysis because they have small supports, so the information matrices are sparse and banded. Once we express the estimator with a linear combination of B-splines, the coefficients are estimated by minimizing a penalized residual sum of squares. A new coordinate descent algorithm is introduced to handle total variation penalty determined by the B-spline coefficients. For large-sample inference, a nonasymptotic oracle inequality for penalized B-spline estimators is obtained. The oracle inequality is then used to show that the estimator is an optimal adaptive for the estimation of the regression function up to a logarithm factor.

Discontinuous Galerkin method for hyperbolic equations involving $$\delta $$ -singularities: negative-order norm error estimates and applications

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $$\delta $$ -singularities. Negative-order norm error estimates for the accuracy of DG approximations to $$\delta $$ -singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $$k$$ th degree polynomials, at time $$t$$ , the error in the $$H^{-(k+2)}$$ norm over the whole domain is $$(k+1/2)$$ th order, and the error in the $$H^{-(k+1)}(\mathbb R \backslash \mathcal R _t)$$ norm is $$(2k+1)$$ th order, where $$\mathcal R _t$$ is the pollution region due to the initial singularity with the width of order $$\mathcal O (h^{1/2} \log (1/h))$$ and $$h$$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $$L^2$$ error estimate of $$(2k+1)$$ th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel's principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $$\delta $$ -singularities.

Particle Swarm Optimization of Multiple-Burn Rendezvous Trajectories

The particle swarm algorithm is a population-based heuristic method successfully applied in several fields of research, only recently to aerospace trajectories. It represents a very intuitivemethodology for optimization, inspired by the behavior of bird flocks while searching for food. In this work, the method is applied to (impulsive and finitethrust) multiple-burn rendezvous trajectories. First, the technique is employed to determine the globally optimal four-impulse rendezvous trajectories for two challenging test cases. Second, the same problems are solved under the assumption of using finite thrust. In this context, the control function is assumed to be a linear combination of B-splines. Themethod at hand is relatively straightforward to implement anddoes not require an initial guess, unlike gradient-based solvers. Despite its simplicity and intuitiveness, the particle swarm methodology proves to be quite effective in finding the optimal solution to orbital rendezvous optimization problems with considerable numerical accuracy.

A Parameter Optimization Method to Determine Ski Stiffness Properties From Ski Deformation Data

The deformation of skis and the contact pressure between skis and snow are crucial factors for carved turns in alpine skiing. The purpose of the current study was to develop and to evaluate an optimization method to determine the bending and torsional stiffness that lead to a given bending and torsional deflection of the ski. Euler-Bernoulli beam theory and classical torsion theory were applied to model the deformation of the ski. Bending and torsional stiffness were approximated as linear combinations of B-splines. To compute the unknown coefficients, a parameter optimization problem was formulated and successfully solved by multiple shooting and least squares data fitting. The proposed optimization method was evaluated based on ski stiffness data and ski deformation data taken from a recently published simulation study. The ski deformation data were used as input data to the optimization method. The optimization method was capable of successfully reproducing the shape of the original bending and torsional stiffness data of the ski with a root mean square error below 1 N m2. In conclusion, the proposed computational method offers the possibility to calculate ski stiffness properties with respect to a given ski deformation.

A Note on B-Splines for Semiparametric Elicitation

In this article we introduce a novel method for transferring expert statements about an uncertain bounded quantity into a probability distribution. This is a challenging problem, as these statements are usually sparse and imprecise. We will approach both issues by penalization, such that the final distribution will contain the stated information, but subject to that distributes its probability mass as uniformly as possible. As a functional basis for the modeled distribution we use linear combinations of B-splines. B-splines are highly flexible, have excellent shape-preserving properties, and are also well suited for evaluation, differentiation, and integration.An R-package implementing the methods described in this article is available as online supplemental material.

A Bayesian Model for Sparse Functional Data

We propose a method for analyzing data which consist of curves on multiple individuals, i.e., longitudinal or functional data. We use a Bayesian model where curves are expressed as linear combinations of B-splines with random coefficients. The curves are estimated as posterior means obtained via Markov chain Monte Carlo (MCMC) methods, which automatically select the local level of smoothing. The method is applicable to situations where curves are sampled sparsely and/or at irregular time points. We construct posterior credible intervals for the mean curve and for the individual curves. This methodology provides unified, efficient, and flexible means for smoothing functional data.

Semiparametric Estimation of Spectral Density With Irregular Observations

We propose a semiparametric method for estimating spectral densities of isotropic Gaussian processes with scattered data. The spectral density function (Fourier transform of the covariance function) is modeled as a linear combination of B-splines up to a cutoff frequency and, from this point, a truncated algebraic tail. We calculate an analytic expression for the covariance function and tackle several numerical issues that arise when calculating the likelihood. The parameters are estimated by maximizing the likelihood using the simulated annealing method. Our method directly estimates the tail behavior of the spectral density, which has the greatest impact on interpolation properties. The use of the likelihood in parameter estimation takes the correlations between observations fully into account. We compare our method with a kernel method proposed by Hall et al. and a parametric method using the Matérn model. Simulation results show that our method outperforms the other two by several criteria. Application to rainfall data shows that our method outperforms the kernel method.

New method for distinguishing planar rational cubic B-spline curve segments as monotone curvature variation

In order to fair and optimize rational cubic B-spline curves used frequently in engineering, and to improve design system function, some formulae on the degree and the knot vector, of the product of three B-spline functions, are presented; then Marsden’s identity is generalized, and by using discrete B-sline theory, the product of three B-spline functions is converted into a linear combination of B-splines. Consequently, a monotone curvature variation (MCV) discriminant for uniform planar rational cubic B-spline curves can be converted into a higher degree B-spline function. Applying the property of positive unit resolution of B-spline, an MCV sufficient condition for the curve segments is obtained. Theoretical reasoning and instance operation showed that the result is simple and applicable in curve design, especially in curve fair processing.

Least-squares image resizing using finite differences

We present an optimal spline-based algorithm for the enlargement or reduction of digital images with arbitrary (noninteger) scaling factors. This projection-based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are B-splines of any degree n. A noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a. For a given choice of basis functions, the results of our method are consistently better than those of the standard interpolation procedure; the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal-to-noise ratio. The method can be generalized to include other classes of piecewise polynomial functions, expressed as linear combinations of B-splines and their derivatives.

Modified B-splines for the sampling of bandlimited functions

It is shown that there is an optimal finite linear combination of B-splines (denominated modified B-splines) such that a pertinent low frequency condition called M-flatness is satisfied. A profound relationship of the modified B-splines with the beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling.

Estimation of complicated distributions using B-spline functions

The distributions of some of random variables are quite complicated and difficult to determine using ordinary statistical models. A method is presented in this paper which gives satisfactory estimation of the complicated distribution of a continuous random variable. There are two key steps in the method: one being that the probability density function (p.d.f.) of a random variable is approximated by a linear combination of B-splines and the other being that the best model is determined by entropy analysis. Extensive numerical experiments have made it clear that the proposed method is useful to determine the p.d.f. directly from a set of sample points without using any prior knowledge of the distribution form.

An Analysis of Paediatric CD4 Counts for Acquired Immune Deficiency Syndrome Using Flexible Random Curves

SUMMARY In this paper we analyse CD4 counts from infants born to mothers who are infected with the human immunodeficiency virus. A random effects model with linear or low order polynomials in time is unsatisfactory for these longitudinal data We develop an alternative approach based on a flexible family of models for which both the fixed and the random effects are linear combinations of B-splines. The fixed and random parts are smooth functions of time and the covariance structure is parsimonious. The procedure allows estimates of each individual's smooth trajectory over time to be exhibited. Model selection, estimation and computation are discussed. Centile curves are presented that take into account the longitudinal nature of the data. We emphasize a graphical approach to the presentation of results.

Least squares approximation by splines with free knots

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process. The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.

On convolutions of B-splines

A smooth approximation to a function ƒ is achieved by convolving ƒ with a smooth function φ. When φ is nonnegative, of unit mean value, compactly supported and has certain symmetry properties, convolving with φ respects the shape properties of the data ƒ such as local positivity, monotonicity and convexity. We study the convolution of ƒ and φ when φ is a univariate B-spline, tensor product B-spline, box spline or simplex spline, and ƒ is a linear combination of the same kind of splines as φ. In terms of divided differences and blossoms, we express the convolution of univariate splines over nonuniform knots as linear combinations of B-splines. This conversion can be carried out by a stable recurrence.

A fast algorithm to raise the degree of spline curves

An algorithm to perform degree elevation on a linear combination of B-splines is presented that is linear in both the number of knots and the degree. Explicit pseudo-code is provided for both the non-periodic and periodic case.