B-spline Approximation Research Articles

Trivariate B-spline Approximation of Spherical Solid Objects

Recently, novel application areas in digital geometry processing, such as simulation, dynamics, and medical surgery simulations, have necessitated the representation of not only the surface data but also the interior volume data of a given 3D object. In this paper, we present an efficient framework for the shape approximations of spherical solid objects based on trivariate B-splines. To do this, we first constructed a smooth correspondence between a given object and a unit solid cube by computing their harmonic mapping. We set the unit solid cube as a rectilinear parametric domain for trivariate B-splines and utilized the mapping to approximate the given object with B-splines in a coarse-to-fine manner. Specifically, our framework provides user-controllability of shape approximations, based on the control of the boundary condition of the harmonic parameterization and the level of B-spline fitting. Experimental results showed that our method is efficient enough to compute trivariate B-splines for several models, each of whose topology is identical to a solid sphere.

Operational method for solving fractional differential equations using cubic B-spline approximation

In the present paper we construct the cubic B-spline operational matrix of fractional derivative in the Caputo sense, and use it to solve fractional differential equation. The main characteristic of the approach is that it overcomes the computational difficulty induced by the memory effect. There is no need to save and call all historic information, which can save memory space and reduce computational complexity. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation. The results from this method are good in terms of accuracy.

Lookback option pricing using the Fourier transform B-spline method

We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes ( jump) diffusions, Lévy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications in which the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete lookback options under Brownian motion, jump diffusion models, and the variance gamma process.

Efficient estimation of varying coefficient seemly unrelated regression model

In this paper, we propose a class of varying coefficient seemingly unrelated regression models, in which the errors are correlated across the equations. By applying the series approximation and taking the contemporaneous correlations into account, we propose an efficient generalized least squares series estimation for the unknown coefficient functions. The consistency and asymptotic normality of the resulting estimators are established. In comparison with the ordinary least squares ones, the proposed estimators are more efficient with smaller asymptotical variances. Some simulation studies and a real application are presented to demonstrate the finite sample performance of the proposed methods. In addition, based on a B-spline approximation, we deduce the asymptotic bias and variance of the proposed estimators.

Automatic human body segmentation based on feature extraction

Purpose – Automatic segmentation of unorganized 3D human body scan data was developed without heuristic specified values. It was reliable in finding the upper body's primary landmarks. The paper aims to discuss these issues. Design/methodology/approach – Quasi boundary point sequence (QBPS) was defined to find the boundary of the human body. Body scan data were categorized by clustering the features extracted from the predefined QBPS. A non-uniform rational B-spline (NURBS) approximation was used to detect the landmarks of the segmented upper torso. Findings – The segmentation method based on feature extraction was reliable regardless of the scan data's fidelity. It was verified that the landmark detection method introduced in this work is more robust than a previous method that utilizes the position of point data. Originality/value – There are several studies of human body segmentation and body landmark detection. This work, however, aims to automate fully segmentation and develop more reliable searching methods. Unlike previous work that uses only 2D human body information, this work uses 3D body information. Furthermore, previous landmark searching methods were superseded by more robust methods applying NURBS approximations.

Meshless Local B-Spline Basis Functions-FD Method and its Application for Heat Conduction Problem with Spatially Varying Heat Generation

In this paper, a new meshless local B-spline basis functions-finite difference (FD) method is presented for two-dimensional heat conduction problem with spatially varying heat generation. In the method, governing equations are discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation. The key aspect of the method is that any derivative at a point or node is stated as neighbouring nodal values based on the B-spline interpolants. Compared with mesh-based method such as FEM the method is simple and efficient to program. In addition, as the method poses the Kronecker delta property, the imposition of boundary conditions is also easy and straightforward. Moreover, it poses no difficulties in dealing with arbitrary complex domains. Heat conduction problem in complex geometry is presented to demonstrate the accuracy and efficiency of the present method.

Simulation of spatiotemporal CT data sets using a 4D MRI-based lung motion model

Four-dimensional CT imaging is widely used to account for motion-related effects during radiotherapy planning of lung cancer patients. However, 4D CT often contains motion artifacts, cannot be used to measure motion variability, and leads to higher dose exposure. In this article, we propose using 4D MRI to acquire motion information for the radiotherapy planning process. From the 4D MRI images, we derive a time-continuous model of the average patient-specific respiratory motion, which is then applied to simulate 4D CT data based on a static 3D CT. The idea of the motion model is to represent the average lung motion over a respiratory cycle by cyclic B-spline curves. The model generation consists of motion field estimation in the 4D MRI data by nonlinear registration, assigning respiratory phases to the motion fields, and applying a B-spline approximation on a voxel-by-voxel basis to describe the average voxel motion over a breathing cycle. To simulate a patient-specific 4D CT based on a static CT of the patient, a multi-modal registration strategy is introduced to transfer the motion model from MRI to the static CT coordinates. Differences between model-based estimated and measured motion vectors are on average 1.39mm for amplitude-based binning of the 4D MRI data of three patients. In addition, the MRI-to-CT registration strategy is shown to be suitable for the model transformation. The application of our 4D MRI-based motion model for simulating 4D CT images provides advantages over standard 4D CT (less motion artifacts, radiation-free). This makes it interesting for radiotherapy planning.

Accuracy-Complexity Trade-Off for the Signal Reconstruction of Multifunctional Sensors

The on-chip signal reconstruction method based on B-spline approximation and Extended Kalman Filter (EKF) for multifunctional sensors has been studied previously. In this paper, we focus on the design for reducing the complexity of the reconstruction method without significant loss of reconstruction accuracy. The two-objective optimization design framework is proposed, where the reconstruction accuracy and complexity are considering as two conflicting costs to be decreased jointly. Genetic Algorithm (GA) is presented to achieve the accuracy-complexity trade-off by optimizing the B-spline structure parameters, i.e. the dimensions of knot vectors. The experimental results show that the proposed method provides a good improvement to the B-spline and EKF based on-chip signal reconstruction method.

Railroad inspection based on ACFM employing a non-uniform B-spline approach

The stresses sustained by rails have increased in recent years due to the use of higher train speeds and heavier axle loads. For this reason surface and near-surface defects generate by Rolling Contact Fatigue (RCF) have become particularly significant as they can cause unexpected structural failure of the rail, resulting in severe derailments. The accident that took place in Hatfield, UK (2000), is an example of a derailment caused by the structural failure of a rail section due to RCF. Early detection of RCF rail defects is therefore of paramount importance to the rail industry. The performance of existing ultrasonic and magnetic flux leakage techniques in detecting rail surface-breaking defects, such as head checks and gauge corner cracking, is inadequate during high-speed inspection, while eddy current sensors suffer from lift-off effects. The results obtained through rail inspection experiments under simulated conditions using Alternating Current Field Measurement (ACFM) probes, suggest that this technique can be applied for the accurate and reliable detection of surface-breaking defects at high inspection speeds. This paper presents the B-Spline approach used for the accurate filtering the noise of the raw ACFM signal obtained during high speed tests to improve the reliability of the measurements. A non-uniform B-spline approximation is employed to calculate the exact positions and the dimensions of the defects. This method generates a smooth approximation similar to the ACFM dataset points related to the rail surface-breaking defect.

VARIABLE SELECTION FOR PARTIALLY LINEAR VARYING COEFFICIENT QUANTILE REGRESSION MODEL

In this paper, we propose a variable selection procedure for partially linear varying coefficient model under quantile loss function with adaptive Lasso penalty. The functional coefficients are estimated by B-spline approximations. The proposed procedure simultaneously selects significant variables and estimates unknown parameters. The major advantage of the proposed procedures over the existing ones is easy to implement using existing software, and it requires no specification of the error distributions. Under the regularity conditions, we show that the proposed procedure can be as efficient as the Oracle estimator, and derive the optimal convergence rate of the functional coefficients. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed variable selection procedure.

B-spline Collocation with Domain Decomposition Method

A global B-spline collocation method has been previously developed and successfully implemented by the present authors for solving elliptic partial differential equations in arbitrary complex domains. However, the global B-spline approximation, which is simply reduced to Bezier approximation of any degree p with C0 continuity, has led to the use of B-spline basis of high order in order to achieve high accuracy. The need for B-spline bases of high order in the global method would be more prominent in domains of large dimension. For the increased collocation points, it may also lead to the ill-conditioning problem. In this study, overlapping domain decomposition of multiplicative Schwarz algorithm is combined with the global method. Our objective is two-fold that improving the accuracy with the combination technique, and also investigating influence of the combination technique to the employed B-spline basis orders with respect to the obtained accuracy. It was shown that the combination method produced higher accuracy with the B-spline basis of much lower order than that needed in implementation of the initial method. Hence, the approximation stability of the B-spline collocation method was also increased.

A semiparametric Bayesian approach to joint mean and variance models

We propose a fully Bayesian inference for semiparametric joint mean and variance models on the basis of B-spline approximations of nonparametric components. An efficient MCMC method which combines Gibbs sampler and Metropolis–Hastings algorithm is suggested for the inference, and the methodology is illustrated through a simulation study and a real example.

Explicit B-spline regularization in diffeomorphic image registration

Diffeomorphic mappings are central to image registration due largely to their topological properties and success in providing biologically plausible solutions to deformation and morphological estimation problems. Popular diffeomorphic image registration algorithms include those characterized by time-varying and constant velocity fields, and symmetrical considerations. Prior information in the form of regularization is used to enforce transform plausibility taking the form of physics-based constraints or through some approximation thereof, e.g., Gaussian smoothing of the vector fields [a la Thirion's Demons (Thirion, 1998)]. In the context of the original Demons' framework, the so-called directly manipulated free-form deformation (DMFFD) (Tustison et al., 2009) can be viewed as a smoothing alternative in which explicit regularization is achieved through fast B-spline approximation. This characterization can be used to provide B-spline “flavored” diffeomorphic image registration solutions with several advantages. Implementation is open source and available through the Insight Toolkit and our Advanced Normalization Tools (ANTs) repository. A thorough comparative evaluation with the well-known SyN algorithm (Avants et al., 2008), implemented within the same framework, and its B-spline analog is performed using open labeled brain data and open source evaluation tools.

Look-Back Option Pricing Using the Fourier Transform B-Spline Method

We derive a new, efficient closed-form formula approximating the price of discrete look-back options, whose underlying asset price is driven by an exponential semi-martingale process including (jump) diffusions, Levy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides look-back option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods.We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications where the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the variance gamma process.A new efficient and robust methodology is presented for pricing discrete look-back options whose underlying asset price is driven by an exponential semi-martingale process and no analytical pricing formula exists. Using B-spline interpolation we obtain an accurate closed-form representation of the look-back option price under an inverse generalized Fourier transform. This provides look-back option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods.We derive an explicit representation for the characteristic function of the maximum of a discretely observed stochastic process, which provides a significant improvement in terms of numerical efficiency over the Spitzer recurrence formula. This is of fundamental importance and could have a wide range of applications where the Spitzer formula is utilized. Examples considered include pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the Variance Gamma process.

Approximating uniform rational B-spline curves by polynomial B-spline curves

Approximation of rational B-spline curves by B-spline curves is an important issue in computer aided geometric design. This paper presents a method to approximate a uniform rational B-spline with B-spline curve sequence as follows. We first elevate the degree of the original rational B-spline curve and take the control points of the degree-elevated curve as new control points of the B-spline approximation curve. Next we take an extended knot vector of the original curve as a new knot vector of the approximation curve. This generates a B-spline approximation curve with the same degree as the degree-elevated curve. Based on the discrete B-spline and multiple products of B-spline functions, we finally prove that the derivatives of any given degree of the uniform B-spline approximation curve sequence converge uniformly to the corresponding derivatives of the original rational B-spline curve. This approximation method is very simple and guarantees the convergence of the approximation.

Modelling and control of Hammerstein system using B-spline approximation and the inverse of De Boor algorithm

In this article a simple and effective controller design is introduced for the Hammerstein systems that are identified based on observational input/output data. The nonlinear static function in the Hammerstein system is modelled using a B-spline neural network. The controller is composed by computing the inverse of the B-spline approximated nonlinear static function, and a linear pole assignment controller. The contribution of this article is the inverse of De Boor algorithm that computes the inverse efficiently. Mathematical analysis is provided to prove the convergence of the proposed algorithm. Numerical examples are utilised to demonstrate the efficacy of the proposed approach.

Mesh independent analysis of shell-like structures

Mesh independent analysis is motivated by the desire to use accurate geometric models represented as equations rather than approximated by a mesh. The trial and test functions are approximated or interpolated on a background mesh that is independent of the geometry. This background mesh is easy to generate because it does not have to conform to the geometry. Essential boundary conditions can be applied using the implicit boundary method where the trial and test functions are constructed utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. This approach has been demonstrated for two-dimensional (2D) and three-dimensional (3D) structural analysis and is extended in this paper to model shell-like structures. The background mesh consists of 3D elements that use uniform B-spline approximations, and the shell geometry is assumed to be defined as parametric surfaces to allow arbitrarily complex shell-like structures to be modeled. Several benchmark problems are used to study the validity of these 3D B-spline shell elements. Copyright © 2012 John Wiley & Sons, Ltd.

스플라인 곡선을 이용한 막구조물의 재단도 작성에 관한 연구

막 구조물은 일반적인 설계와는 달리 형상해석, 응력변형해석 및 재단도 과정을 수행하여야만 설계가 가능하다. 처음 두 과정과는 달리 재단은 3차원 곡면을 최소 오차를 가진 평면 스트립을 형성하는 과정이다. 경제적인 이유로 재단 선은 주로 측지선을 이용하여 작성된다. 그러나 일반적으로 측지선은 초기 형상탐색에서 구성된 삼각형 요소의 정보에서 추출됨으로 부드러운 곡선이 아니며, 불규칙한 직선이다. 그러므로 어떻게 불규칙한 직선을 곡선으로 표현할 것인가가 중요하다. 따라서 본 연구에서는 스플라인 함수를 이용한 보간 방법을 재단도 생성에 적용하였다. 이를 위해서 삼차 스플라인 함수, B-스플라인 함수 및 최소자승 스플라인 함수의 세 가지 경우에 대해서 고찰하고, 단순 모델 및 카테나리 모델을 대상으로 재단도 작성 결과를 검토하였다. 단순모델의 해석요소수와 추출된 불연속 절점 수에 따른 보간 곡선 비교결과는 요소수가 큰 경우 추출된 절점의 수가 적은 것이 효과적이며, 최소자승 보간이 다른 방법보다 더 부드러운 재단 경계선을 제공한다. For membrane structure, there are three main steps in design and construction, which are form finding, statistical load analysis, and cutting patterning. Unlike the first two stages, the step of cutting pattern involves the translation of a double-curved surface in 3D space into a 2D plane with minimal error. For economic reasons, the seam lines of generated cutting patterns rely greatly on the geodesic line. Generally, as searching regions of the seam line are plane elements in the step of shape analysis, the seam line is not a smooth curve, but an irregularly divided straight line. So, it is how we make an irregularly divided straight line a smooth curve that defines the quality of the pattern. Accordingly, in this paper, we analyzed interpolation schemes using spline, and apply these methods to cutting pattern generation on the curved surface. To generate the pattern, three types of spline functions were used, i.e., cubic spline function, B-spline, and least-square spline approximation, and simple model and the catenary-shaped membrane was adopted to examine the result of generation. The result of comparing the approximation curves by the number of elements and the number of extracted nodes of simple model revealed that the seam line for less number of extracted nodes with large number of elements were more efficient, and the least-square spline approximation provided smoother seam line than other methods.

Stochastic controller as an active queue management based on B-spline kernel observer via particle swarm optimization

Given the fact that the current Internet is getting more difficult in handling the traffic congestion control, the proposed method is compatible with the stochastic nature of network dynamics. Most conventional active queue management is based on the first stochastic moment. In stochastic theory, the first moment is not efficient for non-Gaussian systems that are the same as the network queue size. We propose a new stochastic active queue management technique, based on stochastic control and B-spline window observer, called intelligent probability density function AQM (IPDF-AQM). The IPDF-AQM is based on a PDF control and particle swarm optimization, which not only considers the average queue length at the current time slot, but also takes into consideration the PDF of queue lengths within a round-trip time. We provide a guideline for the selection of the probability of dropping as control input for TCP/AQM system to make the PDF of queue length converge at a certain PDF target based on B-spline approximation and improve the network performance. Simulation results show that the proposed stochastic AQM scheme does improve the end-to-end performance.

A quartic [formula omitted]-spline for second-order singular boundary value problems

Second-order singular boundary value problems are treated using quartic (fourth-degree) B-spline approximations. The values of the coefficients, Ci, are chosen via optimization. Error analysis is discussed. The efficiency of the method is illustrated with several problems from the literature.