Spline Surfaces Research Articles

Sampling inequalities for infinitely smooth radial basis functions and its application to error estimates

Recently, Rieger and Zwicknagl (2010) have introduced sampling inequalities for infinitely smooth functions to derive Sobolev-type error estimates. They introduced exponential convergence orders for functions within the native space associated with the given radial basis function (RBF). Our major concern of this paper is to extend the results made in Rieger and Zwicknagl (2010). We derive generalized sampling inequalities for the larger class of infinitely smooth RBFs, including multiquadrics, inverse multiquadrics, shifted surface splines and Gaussians.

An adaptive surface filter for airborne laser scanning point clouds by means of regularization and bending energy

The filtering of point clouds is a ubiquitous task in the processing of airborne laser scanning (ALS) data; however, such filtering processes are difficult because of the complex configuration of the terrain features. The classical filtering algorithms rely on the cautious tuning of parameters to handle various landforms. To address the challenge posed by the bundling of different terrain features into a single dataset and to surmount the sensitivity of the parameters, in this study, we propose an adaptive surface filter (ASF) for the classification of ALS point clouds. Based on the principle that the threshold should vary in accordance to the terrain smoothness, the ASF embeds bending energy, which quantitatively depicts the local terrain structure to self-adapt the filter threshold automatically. The ASF employs a step factor to control the data pyramid scheme in which the processing window sizes are reduced progressively, and the ASF gradually interpolates thin plate spline surfaces toward the ground with regularization to handle noise. Using the progressive densification strategy, regularization and self-adaption, both performance improvement and resilience to parameter tuning are achieved. When tested against the benchmark datasets provided by ISPRS, the ASF performs the best in comparison with all other filtering methods, yielding an average total error of 2.85% when optimized and 3.67% when using the same parameter set.

Polynomial spline interpolation of incompatible boundary conditions with a single degenerate surface

Coons’ construction generates a surface patch that interpolates four groups of specified boundary curves and the corresponding cross-boundary derivative curves. This constructive method is simple and widely used in computer aided design. However, at the corner points, it requires compatibility of the boundary conditions, which is usually difficult to satisfy in practice. In order to handle the incompatible case where the normal directions respectively indicated by two adjacent boundaries do not agree with each other at the common corner point, we utilize the property of degenerate parametric surfaces that the normal directions can converge to multiple values at degenerate points, and therefore the local degenerate geometry can satisfy conflicting conditions simultaneously. Following this idea, we use a single patch of (2(p+2),2(p+2))-degree polynomial spline surface with four degenerate corners to interpolate incompatible boundary conditions, which are represented by p-degree polynomial spline curves with G1 continuity. This method is based on symbolic operations and polynomial reparameterizations for polynomial splines, and without introducing any theoretical errors, it achieves G1 continuity on the boundary except for the four corner points.

A Novel Algorithm for Determining Contact Area Between a Respirator and a Headform

The contact area, as well as the contact pressure, is created when a respiratory protection device (a respirator or surgical mask) contacts a human face. A computer-based algorithm for determining the contact area between a headform and N95 filtering facepiece respirator (FFR) was proposed. Six N95 FFRs were applied to five sizes of standard headforms (large, medium, small, long/narrow, and short/wide) to simulate respirator donning. After the contact simulation between a headform and an N95 FFR was conducted, a contact area was determined by extracting the intersection surfaces of the headform and the N95 FFR. Using computer-aided design tools, a superimposed contact area and an average contact area, which are non-uniform rational basis spline (NURBS) surfaces, were developed for each headform. Experiments that directly measured dimensions of the contact areas between headform prototypes and N95 FFRs were used to validate the simulation results. Headform sizes influenced all contact area dimensions (P < 0.0001), and N95 FFR sizing systems influenced all contact area dimensions (P < 0.05) except the left and right chin regions. The medium headform produced the largest contact area, while the large and small headforms produced the smallest.

A novel gridding algorithm to create regional trace gas maps from satellite observations

Abstract. The recent increase in spatial resolution for satellite instruments has made it feasible to study distributions of trace gas column densities on a regional scale. For this application a new gridding algorithm was developed to map measurements from the instrument's frame of reference (level 2) onto a longitude–latitude grid (level 3). The algorithm is designed for the Ozone Monitoring Instrument (OMI) and can easily be employed for similar instruments – for example, the upcoming TROPOspheric Monitoring Instrument (TROPOMI). Trace gas distributions are reconstructed by a continuous parabolic spline surface. The algorithm explicitly considers the spatially varying sensitivity of the sensor resulting from the instrument function. At the swath edge, the inverse problem of computing the spline coefficients is very sensitive to measurement errors and is regularised by a second-order difference matrix. Since this regularisation corresponds to the penalty term for smoothing splines, it similarly attenuates the effect of measurement noise over the entire swath width. Monte Carlo simulations are conducted to study the performance of the algorithm for different distributions of trace gas column densities. The optimal weight of the penalty term is found to be proportional to the measurement uncertainty and the width of the instrument function. A comparison with an established gridding algorithm shows improved performance for small to moderate measurement errors due to better parametrisation of the distribution. The resulting maps are smoother and extreme values are more accurately reconstructed. The performance improvement is further illustrated with high-resolution distributions obtained from a regional chemistry model. The new algorithm is applied to tropospheric NO2 column densities measured by OMI. Examples of regional NO2 maps are shown for densely populated areas in China, Europe and the United States of America. This work demonstrates that the newly developed gridding algorithm improves regional trace gas maps; its application could be very helpful for the study of satellite-derived trace gas distributions.

L 1 C 1 polynomial spline approximation algorithms for large data sets

In this article, we address the problem of approximating data points by C 1-smooth polynomial spline curves or surfaces using L 1-norm. The use of this norm helps to preserve the data shape and it reduces extraneous oscillations. In our approach, we introduce a new functional which enables to control directly the distance between the data points and the resulting spline solution. The computational complexity of the minimization algorithm is nonlinear. A local minimization method using sliding windows allows to compute approximation splines within a linear complexity. This strategy seems to be more robust than a global method when applied on large data sets. When the data are noisy, we iteratively apply this method to globally smooth the solution while preserving the data shape. This method is applied to image denoising.

3207 操舵反力シミュレータにおけるステアリングギアの摩擦モデルの検討(OS9-1 シミュレータシステムの開発,OS9 ビークルシミュレータの開発と応用,オーガナイズド・セッション(OS))

The steering reactive torque of an automobile is one of the important information for a driver to judge the vehicle behavior and road conditions. This paper describes an empirical model of a friction characteristics of a steering gear for a steering torque simulator. In order to evaluate the friction characteristics, the authors conducted a test using the steering gear of the experimental vehicle. The friction characteristics were evaluated to the angular velocity and the rack force. Moreover, a curved surface approximation were performed to the experimental results using a spline surface, and the most suitable friction model of the steering gear was examined by changed a control point and an order of a spline surface.

Constrained smoothing and interpolating spline surfaces using normalized uniform B-splines

Constrained smoothing and interpolating spline surfaces using normalized uniform B-splines

21016 知識データベースを用いた欠損部を含んだ点群データの自由曲面への変換

Three-dimensional CAD becomes high-performance. And in late year's three-dimensional printer and scanners spread. Rendering can perform by using point group data which were taken in with three-dimensional scanners, but are not often suitable for various three dimensions processing such as a correction or the addition of the three-dimensional shape. Therefore convert point group data into masks such as irregular triangular mesh or non-same rational B spline surface form and often generally are handled it. When provided point group data include a loss area, there are many cases that an error occurs and cannot make a conversion to a polygon shape and a free-form surface shape. So we present way for the reduction of the conversion error to occur when I convert point group data having a loss department into the free curved surface. To solve the problem, knowledge data of the Delaunay triangulation to XML-based shape expression X3D.

Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

Monotonicity-preserving rational bi-cubic spline surface interpolation

We discuss the problem of monotonicity preservation of surfaces through 3D monotone data. This can be done using a rational bi-cubic blended function that is an extension of a rational cubic function in the form of a cubic numerator and quadratic denominator. The function involves twelve shape parameters in each rectangular patch. Data- dependent constraints are derived on four of these shape parameters to conserve the shape of the data while the other eight are left free to modify the monotone surface as desired. Several numerical examples are presented to show the effectiveness and capability of the scheme. The present scheme isC 1 , flexible, simple, local, and economical.

Positivity and Monotonicity Preserving Biquartic Rational Interpolation Spline Surface

A biquartic rational interpolation spline surface over rectangular domain is constructed in this paper, which includes the classical bicubic Coons surface as a special case. Sufficient conditions for generating shape preserving interpolation splines for positive or monotonic surface data are deduced. The given numeric experiments show our method can deal with surface construction from positive or monotonic data effectively.

Development of a novel adaptive nonuniform rational basis spline interpolator with limited kinematic error for hexapod machine tools

The kinematic nonlinearity of the Stewart platform–based machine tools produces a geometrical error which is shown to have significant value compared to the machining accuracy during the interpolation of non-uniform rational basis spline curves and surfaces, thus needs to be limited during the interpolation. In this study, a fast and sufficiently accurate algorithm is presented to estimate the maximum kinematic error based on the concept of the median osculating circle. Moreover, a weighted search interpolator and an adaptive non-uniform rational basis spline interpolator have been introduced, which are capable of controlling and limiting the feedrate fluctuations and limiting the maximum kinematic error within an interpolation segment. The experimental verification has shown the effectiveness of the proposed algorithm.

Surface spline interpolation method for thermal reconstruction with limited sensor data of non-uniform placements

With the characteristic size reducing as well as the power densities exponentially increasing, elevated chip temperatures are true limiters to the performance and reliability of integrated circuits. To address these thermal issues, it is essential to use a set of on-chip thermal sensors to monitor temperatures during operation. These temperature sampling results are then used by thermal management techniques to appropriately manage chip performance. In this paper, we propose a surface spline interpolation method to reconstruct the full thermal characterization of integrated circuits with non-uniform thermal sensor placements. We construct the thermal surface function using the mathematical tool of surface spline with the matrix calculation of the non-uniform sample data. Then, we take the coordinates of the points at grid locations into the surface function to get its temperature value so that we can reconstruct the full thermal signals. To evaluate the effectiveness of our method, we develop an experiment for reconstructing full thermal status of a 16-core processor. Experimental results show that our method outperforms the inverse distance weighting method based on dynamic Voronoi diagram and spectral analysis techniques both in the average absolute error metric and the hot spot absolute error metric with short enough runtime to meet the real-time process demand. Besides, our method still has the advantages such as its mathematical simplicity with no need of pre-process.

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

SUMMARYThe mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push‐forwards of B‐splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non‐uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push‐forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k‐refinement nor the h‐refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push‐forwards of NURBS via the design mapping are combined with push‐forwards of B‐splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X‐FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Application of Vortex Lattice Method and Surface Spline Function in Terrain-Induced Wind Field

A numerical simulation method for the wind field over complex terrain is developed in this paper. The complex terrain is represented by a surface, which is established by the surface spline method basing on the contour line of the map. Then it is divided into panels containing vortex ring singularities. The singularity distribution is solved by combining with the potential flow theory and the boundary condition of terrain surface. Finally, the wind field is simulated by linear superposition of the uniform stream and the induced velocity of vortex rings. A numerical example is studied, and the result indicates that the method proposed in this paper could describe the terrain much more accurately and provide a reasonable result of wind field distribution, but has a simple and efficient procedure, which is suitable for the engineering use in flight dynamics.

Application of Vortex Lattice Method and Surface Spline Function in Terrain-Induced Wind Field

Application of Vortex Lattice Method and Surface Spline Function in Terrain-Induced Wind Field

Hermite interpolation by triangular cubic patches with small Willmore energy

In this paper, a construction of a cubic Bézier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters.

A Robust Filtering Algorithm of LiDAR Data for DTM Extraction

For DTM extraction of road linear project from LiDAR data, this paper proposes a robust ground filtering algorithm. Firstly, a pyramidal hierarchical structure of ground candidate points is formed. Secondly, in each level, the Thin Plate Spline (TPS) surface interpolation is iteratively used to produces smooth surfaces, and based on the points residuals from the surface the ground candidate points are inspected. The experiments show that this method correctly determines DTM even in those cases of heavy vegetation, and achieve a better result for the road.

Two Dimensional Damage Localization Using the Interpolation Method

This paper presents a vibration based procedure for locating reductions of stiffness in two-dimensional structures that can be modeled as plates. This procedure is a generalization to the two-dimensional case of the previously published Interpolation Damage Detection Method (IDDM). The method is based on the definition of a damage sensitive feature in terms of the accuracy of a spline function in interpolating the operational displacement shapes of the structure. These latter are recovered from frequency response functions (FRFs) measured at different locations of the structure during vibrations. At the i-th location, the FRF is calculated through spline interpolation using the FRF’s recorded at the all the instrumented locations but the i-th. For two-dimensional structures a spline surface is defined to interpolate the operational shapes. The accuracy of the spline interpolation is measured by an error function defined as the difference between the measured and interpolated operational mode shapes. At a certain location an increase (statistically meaningful) of the interpolation error, with respect to a reference configuration, points out a localized variation of the operational shapes thus revealing the existence of damage. The two dimensional IDDM algorithm is checked herein through numerical simulations, using the FE model of a plate and modeling local reductions of stiffness through a reduction of the elastic modulus of the material of one or more elements of the model.