B-spline Surface Approximation Research Articles

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

We introduce a progressive and iterative method for B-spline curve and surface approximation, incorporating parameter correction based on the Newton iterative method. While parameter corrections have been used in existing Geometric Approximation (GA) methods to enhance approximation quality, they suffer from low computational efficiency. Our approach unifies control point updates and parameter corrections in a progressive and iterative procedure, employing a one-step strategy for parameter correction. We provide a theoretical proof of convergence for the algorithm, demonstrating its superior computational efficiency compared to current GA methods. Furthermore, the provided convergence proof offers a methodology for proving the convergence of existing GA methods with location parameter correction.

B-spline surface approximation method for achieving optimum dwell time in deterministic polishing

In the computer-controlled optical surfacing (CCOS) process, the accurate calculation of the dwell time map is essential for achieving the desired amount of material removal distribution. However, it remains challenging for current methods to generate a smooth dwell time map that reduces the dynamic requirements of machine tools while maintaining the high finishing capability on freeform optics. This paper proposes a B-spline surface approximation method for obtaining the optimum dwell time in deterministic polishing. The B-spline surface representation allows the dwell time map to be expressed by a small number of control points instead of a large number of discrete dwell points. The dwell time deconvolution then becomes a linear least-squares problem with only a few control points as variables, which significantly reduces the scale of the problem. To achieve a more accurate and smoother dwell time map, a knot position optimization algorithm is proposed to adapt the knots to the geometric features of the target removal surface. Simulation results demonstrate that the proposed B-spline-based dwell time algorithm significantly outperforms common dwell time algorithms in terms of the computational accuracy, efficiency, as well as smoothness of the dwell time map. To validate the effectiveness and practicability of the proposed method, experimental polishing of a bi-sinusoidal optical surface has been carried out. The results show that the developed algorithm has greatly reduced the tracking error of the machine tool caused by dwell time fluctuations, and consequently, the corresponding residual error of material removal in deterministic polishing.

Evaluation Method for Shape Similarities between Archaeological Unorganized Point Cloud

Researchers assume that the bodies of several Tara statues may have been molded from the same template. Therefore, researchers are eager to study the shape similarities between the Tara statues. Motivated by the archaeologist's requirements, this study presents an efficient approach to evaluate shape similarities between point clouds of two different Tara statues right-arm models. The evaluation approach consists of two main stages. The first is the reconstruction of individual body parts without decoration using a B-spline surface approximation technique. In previous reconstruction process, the surrounding surface selection parameter was manually specified. If the manually applied parameter is inadequate, it reduces the accuracy of the surface approximation and hole-filling, resulting in reduced accuracy of the evaluation result. Therefore, in this study, the surrounding surface selection parameter is automatically determined based on the Golden-section-search algorithm. The second stage evaluates the shape similarities between two different point clouds. The evaluation method considers a simple point-to-surface distance metric method with overlapping-surface detection and when compared with generic point-to-point distance metric method, the evaluation results proved that the proposed methodology was reliable and effective for this application. The approximation accuracy affects the evaluation result when the reconstruction method fills the gap by adding new points through the approximated surface. Therefore, in this study, the shape similarities were investigated between the two arm models with holes, immediately after separating the decoration parts.

A Local Approach for Computing Smooth B-Spline Surfaces for Arbitrary Quadrilateral Base Meshes

Abstract We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees-of-freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

On the BIC for determining the number of control points in B-spline surface approximation in case of 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.

An Examination of Compression and Restoration Method of Measured Point Cloud Based on B-spline Surface Approximation of Flake Surfaces in Stone Implements

石器や土器などの遺物の3次元計測によって得られた点群データに基づき，考古学的な調査や分析が行われている．遺物は考古学上貴重なものであり，移動にリスクを伴うものが存在する．また，一つの遺跡から大量に出土する遺物を計測のために移動させることは，運搬に関わる梱包などの作業の負担が大きい．そこで，出土した遺物を現地で計測し，計測データをネットワークを介した通信により転送できると，効率的にデータ処理を行うことが可能である．しかし，3次元点群データは物体の細かい形状変化の表現に優れる一方，高精度計測を行った場合などに構成要素である点の数が増加し，データ量が大きくなりやすく，データ転送に時間がかかるという問題点がある．データ転送を効率よく行うためには，データの圧縮手法が有効である．そこで，本論文では，石器剥離面をB-spline曲面で近似して，剥離面を表す点群よりも少ない制御点数のトリム曲面を生成することで，データを圧縮する手法を提案する．また，トリム曲面から計測点群を再生成する手法もあわせて提案する． 本手法を実装し，計測間隔0.1mmで計測した石器計測点群に適用し，有効性を確認した．

Bioinspired Hydrodynamic Force Feedforward for Autonomous Underwater Vehicle Control

It is believed that the lateral line found in most fish and some other aquatic organisms plays an important role in many behaviors by providing hydrodynamic information about the surrounding fluid. This paper presents a lateral line inspired feedforward control design for the autonomous underwater vehicles. The goal of this paper is to improve maneuvering accuracy for underwater exploration and environmental monitoring. The proposed controller takes pressure measurements at multiple locations over the entire vehicle hull and forms a B-spline surface approximation of the pressure distribution. Hydrodynamic forces acting on the vehicle are then estimated and passed to the controller for the improved trajectory tracking performance. The algorithm is formulated in such a way that the force estimation is a linear, fixed weight combination of the pressure signals, which enables rapid online computation. The performance of the feedforward controller is simulated in conjunction with the “robust integral of the sign of the error” control design. Trajectory tracking is investigated for control accuracy in the presence of localization errors. Reduced tracking errors are observed with the feedforward design. Experimental tests with pressure sensors on a moving cylinder further vindicate the force estimation algorithm.

B-Spline Surface Approximation to Scanned Data Using Least Square Approximation

Lofting is widely used to approximate the scanned data in row-wise fashion, but this method is prone to result an astonishing number of control points in the process of making the rows curve compatible. A novel algorithm of B-spline surface approximation to the scanned data is presented in this paper to solve this problem. Firstly, the scanned data are interpolated by rows of curves; then these curves are approximated by other curves using least square approximation. In this process, all curves are approximated by a common knot vector, and it is different form the traditional method that each curve is approximated by a different knot vector, so we needn’t insert many knots in each curve to make curves compatible. We also can meet high accuracy without losing the shape of lofting surface because we firstly interpolate the data, the best least square approximation substitute insertion of knots in lofting. Numerical example shows that the proposed method is efficient in reducing control points of the lofting surface.

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.

A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function

In model selection, the most effective method requires much time.The analysis of the bivariate B-spline model with a penalized term has many difficulties.It has many factors and parameters such the number of the knots, the locations of those knots, number of B-spline functions and the value of the smoothing parameter of the penalized term.For the determination of the model we have to compare a large amount of the combinations of those parameters. Various information criteria are considered and the cross validation (CV) criterion is excellent but it requires a large amount of computational costs. The effect of the influence function and the techniques of the generalized cross validation (CV) are considered. The influence function is related to the first term of a Taylor expansion. Some alternative methods are tested and a new method is proposed. For the verification of this method theoretical proof and the computational results are shown.

Automatic knot adjustment for B-spline surface approximation

Automatic knot adjustment for B-spline surface approximation

3308 高速化された幾何アルゴリズムによるB-spline曲面近似(OS4-2 デジタルエンジニアリングII,OS4 デジタルエンジニアリング)

3308 高速化された幾何アルゴリズムによるB-spline曲面近似(OS4-2 デジタルエンジニアリングII,OS4 デジタルエンジニアリング)

Three-dimensional Technology for Apparel Mass Customization: Part II: Human Body Modeling from Unorganized Range Data

This paper presents a human body modeling method using the B-spline surface approximation algorithm and unorganized range data from 3D scanning systems. The scan data are firstly reorganized to have a uniform density through the Delaunay triangulation and linear interpolation. The scanned body is divided into six cylindrical segments, and the cubic B-spline surface approximation is performed on each segment separately. The individual B-spline surfaces are then merged to create a seamless model, which not only dramatically reduces the size of the data needed to reconstruct the surface but also smooths and completes the scan data. The fitted model is essential for e-tape measurement of the body and virtual try-on of clothes.

Ordering and parameterizing scattered 3D data for B-spline surface approximation

Surface representation is intrinsic to many applications in medical imaging, computer vision, and computer graphics. We present a method that is based on surface modeling by B-spline. The B-spline constructs a smooth surface that best fits a set of scattered unordered 3D range data points obtained from either a structured light system (a range finder), or from point coordinates on the external contours of a set of surface sections, as for example in histological coronal brain sections. B-spline stands as of one the most efficient surface representations. It possesses many properties such as boundedness, continuity, local shape controllability, and invariance to affine transformations that makes it very suitable and attractive for surface representation. Despite its attractive properties, however, B-spline has not been widely applied for representing a 3D scattered nonordered data set. This may be due to the problem in finding an ordering and a choice for the topological parameters of the B-spline that lead to a physically meaningful surface parameterization based on the scattered data set. The parameters needed for the B-spline surface construction, as well as finding the ordering of the data points, are calculated based on the geodesics of the surface extended Gaussian map. The set of control points is analytically calculated by solving a minimum mean square error problem for best surface fitting. For a noise immune modeling, we elect to use an approximating rather than an interpolating B-spline. We also examine ways of making the B-spline fitting technique robust to local deformation and noise.

Region coding using improved B-spline surface approximation

A new arbitrary region coding algorithm using a B-spline surface is presented. Pixels in the coded region are regarded as valid image data, and the parameters of the B-spline surface are calculated under certain optimisation criteria. Instead of the traditional least squares error (LSE) criterion, a least squares weighted error (LSWE) criterion is used as the approximation criterion. Simulation results show that the LSWE approach is much better for region coding than the LSE approach.

3D Recognition from Shading and B-Spline Surface Approximation. 3rd Report. Control Points Arrangements and Recognition Efficiency.

In the former report, we applied B-Spline surfaces to approximate the local surface of the shading and used the steepest descent method to minimize the difference between the shading of B-Spline surfaces and the image shading. Through the application of the method for the noises shading, we could reduce the effect of the noise and recognize the shape more correctly than the conventional method. However, being an interface for CAD, it is necessary to improve the recogniton efficiency. Therefore, in this report, we examine the control points arrangements and B-Spline degrees, and according to the result, we investigate a new process to set more control points around the extremun of the object. As a result, the proposed method can be available to recognize the shape form image shading.

3D Recognition from Shading and B-Spline Surface Approximation.

3D Recognition from Shading and B-Spline Surface Approximation.

3D Recognition from Shading and B-Spline Surface Approximation. 2nd Report. 3D Recognition from Shading and Stereo Vision.

In our previous study, we applied B-Spline surface method to estimate the local surface of the shading and used the steepest descent method to minimize the difference between the B-Spline surface and the image shading. These methods helped reduce the effects of the noise and also recognized the shape more accurately than by the conventional methods. However, we detected some problems in the applied methods, such as distortion in the result, presence of invisible occluded region, etc. Therefore, it is necessary to improve the accuracy in practical USe. In the present study, we observed the object from various directions by adopting a stereo vision for 3D recognition in the process. This newly proposed method was able to recognize the shape from image shading, and the method itself was also easy to achieve by selecting proper weights of two views.

A 3-D vision system model for automatic object surface sensing

The development of a noncontact “light striping” (structured light) based three-dimensional, six-degrees-of-freedom vision system for automatic object surface sensing is reported. The system modeling and world-point reconstruction methodology involve homogeneous-coordinate system transformations applied in two independent stages: the video imaging stage using three-dimensional perspective transformations and the mechanical scanning stage using three-dimensional affine transformations. Concatenation of the two independent matrix models leads to a robust four-by-four, six-degrees-of-freedom system model. The reconstructed sectional contours are registered automatically in real time with respect to a common world-coordinate system as a control net for Non-Uniform Rational B-spline (NURBS) surface approximation. The reconstruction process is demonstrated by measuring the surface of a 19.5×2 ft rowing shell. A detailed statistical accuracy and precision analysis shows an average error of 0.2% (0.002) of the camera's field-of-view. System sensitivity analysis reveals a nonlinear increase of sensitivity for angles higher than 45° between the normals of the image and laser planes.

Approximate conversion of rational splines

The exchange of data between different geometric modeling systems frequently frequires approximate conversions of rational Be´zier and B-spline curves and surfaces to lower degree integral B-spline representations. Similarly, processing of trimmed, free-form patches, such as their automated manufacture, may lead to high degree representations. Accurate conversion of such geometries to lower degree integral representations is required in many modeling environments. The objective of this paper is to present an algorithm to approximate non-uniform rational B-spline curves with non-uniform integral B-spline curves of lower degree. Global error bounds for the verification of the accuracy of the approximation are established. This algorithm forms the basis of a similar rational B-spline surface approximation procedure.