A robust surface fitting and reconstruction algorithm for form characterization of ultra-precision freeform surfaces

Ultra-precision freeform surfaces are non-rotational symmetric surfaces possessing sub-micrometer form accuracy and nanometric surface finish. Although they can be fabricated accurately by ultra-precision machining technology, their surface quality is difficult to be characterized. Surface reconstruction is a vital task in the form characterization of ultra-precision freeform surfaces. This paper presents a robust surface fitting algorithm to reconstruct a high fidelity surface from measured discrete points while the surface smoothness can be ensured as well. A fitting threshold named confidence interval of fitting error is proposed to strike the balance between fitting accuracy and surface smoothness in the fitting process. The fitting algorithm is in two steps. In the first step, bidirectional sampling method is developed to extract a curve network from measured points cloud to construct an initial surface. In the second step, the fitting error of the initial surface is minimized to meet the prescribed fitting error threshold. A series of experimental work has been conducted and the results show that the proposed algorithm is able to provide effective means for increasing the accuracy in the form characterization of ultra-precision freeform surfaces.     

Efficient analysis-suitable T-spline fitting for freeform surface reconstruction and intelligent sampling

Optical scanning instruments require sampling and reconstruction with high accuracy and low computational cost. T-splines have recently been developed that allow significant reductions in the number of control parameters by overcoming some of the topological constraints of B-splines and NURBS. As a subset of T-splines, analysis-suitable T-splines (ASTS) show promise due to the linear independence and partition of unity of their basis functions. In this paper, a computationally efficient ASTS fitting algorithm for freeform surface reconstruction is proposed. This algorithm starts with adaptive construction of an initial analysis-suitable T-mesh according to the distribution of high-curvature feature points. A local refinement and local optimisation algorithm of the analysis-suitable T-mesh is then iteratively performed until a preset accuracy condition is satisfied. Our experimental results show that the proposed ASTS fitting can produce over 50% root-mean-square reconstruction error reduction compared to NURBS fitting, with the same number of control parameters. The computing efficiency of the proposed algorithm is equivalent to or higher than that for simple T-spline fitting. Fast derivative analysis of the ASTS has also been carried out, where two automatic intelligent sampling design methods have been developed, namely element area sampling and element curvature sampling. Up to 50% reconstruction error reduction is observed when compared to uniform and statistically optimised sampling designs. With the novel reconstruction and compatible intelligent sampling design techniques, freeform surface measurement accuracy and efficiency could be effectively improved using coordinate measuring machines.     

Prediction of surface reconstruction uncertainties for freeform surface inspection

A method is introduced to predict uncertainties of the B-spline freeform surfaces that are reconstructed from the measurement points for inspection of freeform surfaces. The uncertainties of a reconstructed B-spline surface are modeled by variances of coordinates of points on this freeform surface. Prediction of the uncertainty at any location on the reconstructed B-spline surface is carried out in two steps: (1) estimation of variances of the B-spline surface’s control points introduced by the surface reconstruction process, and (2) propagation of the variances from the control points to the points on the B-spline freeform surface. In this research, the variances of the control points of the reconstructed B-spline surface in all three directions are considered to improve the accuracy of uncertainty prediction in the regions with significant changes of geometric shapes. Both the errors and uncertainties at different locations on the reconstructed surface are considered in freeform surface inspection to compare the manufactured surface against the design surface and its tolerance. The developed method has been applied to two case studies to demonstrate its effectiveness.     

Any-degrees-of-freedom (anyDOF) registration for the characterization of freeform surfaces

This paper presents an any-degrees-of-freedom (anyDOF) registration method for the characterization of freeform surfaces. The method attempts to fill the research gap regarding traditional surface registration methods which are normally dedicated to solving the global optimization problem with all DOF but they lack flexibility. The proposed anyDOF method is capable of registering surfaces with any specified combination of DOF. This is particularly useful when some of the DOF are known to be unchanged according to the a priori knowledge. The anyDOF surface registration method is regarded as a typical optimization problem of finding the minimum distance from target surface to the reference surface, with constraints of the unwanted DOF. The problem is solved by the Levenberg-Marquardt method. Simulated experiments for a two-dimensional (2D) profile and a three-dimensional (3D) surface were undertaken, together with three measurement experiments including a fluid-jet polished surface, a bonnet polished surface and a diamond machined freeform surface. Experimental results show that the anyDOF registration method is highly flexible in the characterization of freeform surfaces.     

<Emphasis Type="Bold">Determination of the Optimal Parameters for Freeform Surface Measurement and Data Processing in Reverse Engineering</Emphasis>

One objective of this work is to determine the optimal combination of the probe diameter and grid distance for freeform surface measurement, and another is to determine the optimal parameters for the local Shepard interpolation. The optimal combination of the probe diameter and grid distance for freeform surface measurement was determined through a Taguchi matrix experiment. The smaller the probe diameter and grid distance, the better the accuracy of the surface normal based on the configured matrix experimental result. The optimal parameters, namely the exponent and the radius R, for the local Shepard interpolation were determined by using the minimisation method of the root-mean-square normalised error (RMSNE) between the measured data points and the theoretical data points on a standard steel ball surface. The optimal parameters determined were actually applied to the measurement of a freeform surface (mouse surface) on a coordinate measuring machine (CMM). The local Shepard interpolation method was used to interpolate 16 control points from 1054 measured data points. Bi-cubic Bezier- and B-spline surface CAD models were constructed through these interpolated control points.     

散乱数据的曲面重构与深度图像多视匹配技术

针对激光扫描测量系统得到的大量散乱数据点 ,制定了基于曲率特征的自适应采样策略 ,研究了一种基于Delaunay三角剖分的三角Bezier曲面造型与NURBS曲面造型相结合的实用曲面重构技术 ,采用改进后的迭代最近点算法对深度图像数据多视匹配技术进行了研究。     

NON-CONTACT MEASUREMENT SYSTEM OF FREEFORM SURFACE AND NURBS RECONSTRUCTION OF MEASUREMENT POINTS

Based on the development of the non-contact measurement system of free-form surface, NURBS reconstruction of measurement points of freeform surface is effectively realized by modifying the objective function and recursiveprocedure and calculating the optimum number of control points. The reconstruction precision is evaluated through Ja-cobi's transformation method. The feasibility of the measurement system and effectiveness of the reconstruction algorithm above are proved by experiment.     

Calculation of the Normal Vector using the 3 × 3 Moving Mask Method for Freeform Surface Measurement and its Application

The concept of a 3 × 3 moving mask operation image-processing technique is proposed to calculate the normal vector of measuring points in this study. The method developed reduces greatly the calculation time of matrix operation and memory space in comparison with the traditional composite Ferguson-spline method. The methodology for calculating the normal vector is to select eight neighbouring points at equal distances in the vicinity of an arbitrary node on the surface, from which a small surface patch can then be constructed from the nine selected points. Different analytical methods are used to calculate the unit normal vector, namely the Bezier method with uniform parameters and the Bezier method with non-uniform parameters, and are discussed in this study. The accuracy of these two methods in calculating the unit normal vector was also verified by calculating different positions on a spherical surface. The Shepard interpolation method was adopted to interpolate a few control points from a massive number of measured data points to establish the CAD model of a freeform surface using a rectangular grid. The method developed was applied for the measurement of a freeform surface (mouse surface) using a coordinate measuring machine. The local Shepard interpolation method was used to interpolate 16 control points from 1054 measured data points. A bi-cubic Bezier- and B-spline surface CAD model were constructed through these interpolated control points.     

Grain mapping to freeform surface for machining using parameter-based incremental tracking method

This paper presents a novel method for mapping a texture for machining onto a freeform surface. When a grain texture designed on a 2D plane is projected onto a freeform surface simply, the texture is distorted along the curvature of the surface in the projection direction. Therefore, a method is required to suppress this distortion. The algorithm proposed in this paper rapidly generates a point cloud that represents a texture with less distortion on a simply connected parametric surface. Specifically, four equations that express local geometry constraints instead of conventional parameterization methods are presented to reduce the distortion between adjacent points on the point cloud. Solving these equations can generate grain textures on freeform surfaces. This paper examines two grain images mapped to two freeform surfaces that were prepared as case studies to demonstrate the effectiveness of the proposed method.     

自由曲面测量点云数据的建模方法研究

本文提出了一种散乱点云数据的建模新方法。通过对点云数据进行空间三维划分，实现了边界信息的高效提取。采用局部曲面拟合方式得到位于截平面上的有序数据，使得无序的散乱数据形成了有序的阵列数据，实现了NURBS曲面的精确拟合。实验证明，该方法非常适合卷曲类模型的自由曲面重构，在某零件的测量、建模和加工中已经得到了实际应用。     

多点成形件检测中三维数据配准方法的研究

针对多点成形件成形误差检测中的三维数据配准问题，提出了一种用两步法将CAD模型数据和三维测量数据点全局配准的方法。利用遗传算法具有全局搜索能力的特点，以四元数法中的3个参量作为优化解空间，采用自适应控制优化参数的方法，达到CAD模型数据和三维测量数据点的全局粗配准。以粗配准的结果作为初始值，用ICP算法修正误差以达到精确配准。采用两步法配准克服了标准ICP算法难以解决的局部最小问题，本算法也可广泛应用于精密测量时测量结果的比较分析。以马鞍形曲面制品为例，给出了配准结果。     

Design of a Universal Fixture for Laser Scanning

In this paper, a universal fixture for 3D laser scanning is presented. In order to reconstruct the surface or CAD model from the scanned data of an existing part, it is necessary to integrate the point data from several views of an object into a common reference frame. When the registration is implemented with a feature-based method using tooling balls, the reference coordinate system is established by several common tooling balls from different point clouds. The laser scanner is often used to acquire the surface information of the part, but is limited in its measuring direction, which is fixed only along the z-axis. A universal fixture using several joints and an adjustable tooling ball holder is designed considering the convenience of the part set-up and the accuracy of the registration. The location of the tooling balls can be arranged to avoid the occlusion of the part and to minimise the registration error. The universal fixture was applied to a wrap-around part having freeform surfaces to demonstrate the effectiveness of the proposed design.     

基于角点匹配的密封条截面图像分块配准算法

分析了汽车密封条传统检测方法的不足,针对现有方法无法测量的分块整体形变密封条,提出了一种基于角点匹配的轮廓分块配准算法。配准算法首先利用支持邻域求取轮廓角点,再利用分块角点匹配求取每块的仿射变换初值,最后利用基于最小二乘的搜索策略获得最佳配准效果。本算法在密封条存在较大分块形变时可实现有效测量,具有省去夹具装夹、节约测量时间和成本的优点。通过大量的实验证明,该方法的快速有效。     

Improved curvature-based registration methods for high-precision dimensional metrology

Multiple measurements using various data acquisition systems are generally required to substancially enhance measurement accuracy, reliability and holisticity of freeform shapes. The obtained multiple measurement data of the shape are transformed and fused into a common coordinate system within a registration technique involving coarse and fine alignments. Standardized methods have been established for fine registration such as Iterative Closest Points (ICP) and its variants. For coarse registration, no conventional method has been adopted yet despite a significant number of techniques which have been developed in the literature to supply an automatic rough matching between data sets.The work presented in this paper proposes an improvement of registration techniques by the consideration of new discrete curvature parameters. Two main issues are addressed in this paper: the coarse registration and the fine registration. For coarse registration, two novel automated methods based on the exploitation of discrete curvatures are presented: an enhanced Hough Transformation (HT) and an improved Ransac Transformation. The use of curvature features in both methods aims to reduce computational cost. For fine registration, a new variant of ICP method is proposed in order to reduce registration error using curvature parameters. A specific distance considering the curvature similarity is combined with Euclidean distance to define the distance criterion used for correspondences searching. Additionally, the objective function is improved by combining the point-to-point (P-P) minimization and the point-to-plane (P-Pl) minimization with automatic weights. The algorithms are applied on simulated and real data performed by a computed tomography (CT) system. The obtained results reveal the benefit of the proposed improved curvature-based registration methods.     

利用寻位信息的自由曲面在线测量路径规划方法

自由曲面的高速、高精度、扫描测量是曲面在线检测需要解决的首要问题.为此,提出一种基于寻位信息的测量路径规划方法,使得在满足精度要求的前提下,实现测点优化分布;在此基础上,将自适应控制理论应用在测头自动跟踪控制中,以便测头能够跟随曲面的变化自动调整其位置,完成自由曲面的高速、高精度、扫描测量,理论分析与仿真实验结果表明,所提出的测量方法可行,为曲面的精密测量开辟了一条新的有效途径.     

ALGORITHM OF PRETREATMENT ON AUTOMOBILE BODY POINT CLOUD

As point cloud of one whole vehicle body has the traits of large geometric dimension,huge data and rigorous reverse precision,one pretreatment algorithm on automobile body point cloud is put forward.The basic idea of the registration algorithm based on the skeleton points is to construct the skeleton points of the whole vehicle model and the mark points of the separate point cloud,to search the mapped relationship between skeleton points and mark points using congruence triangle method and to match the whole vehicle point cloud using the improved iterative closed point(ICP)algorithm. The data reduction algorithm,based on average square root of distance,condenses data by three steps, computing datasets' average square root of distance in sampling cube grid,sorting order according to the value computed from the first step,choosing sampling percentage.The accuracy of the two algo- rithms above is proved by a registration and reduction example of whole vehicle point cloud of a certain light truck.     

Adaptive sampling method for inspection planning on CMM for free-form surfaces

Coordinate measurement machines (CMMs) have been widely used in inspecting mechanical parts with higher accuracy. Both the distribution and the number of the sampled points on measurand have an important effect on the efficiency and quality of the measurement of CMMs. In this paper, an adaptive sampling method is proposed for inspection planning on CMM for free-form surface. The points are iteratively sampled from a form error model, which is constructed by superimposing appropriate form errors on the nominal data. Moreover, a modified algorithm is introduced to determine the deviation of two point sets for the improvement of the inspecting accuracy, and the inspection uncertainty is also analyzed. A comparison is performed between the proposed method and the two well-known sampling methods, which are the equi-parametric method and the patch mean Gaussian curvature-based method, both simulated and experiment results show the effectiveness and robustness of this method.     

基于激光扫描和SFM的非同步点云三维重构方法

室外场景具有测量数据量大、扫描数据易重叠及建筑物表面信息复杂等特点,单靠激光扫描方法能够获得场景精确的深度信息,但缺乏颜色和纹理信息,利用从运动中恢复结构(SFM)方法可获得丰富的彩色信息,但重构精度不高,若将两种设备固定进行在线实时同步测量,易受到测量环境和系统制约不易实现。针对此问题,提出了一种基于激光扫描和SFM结合的非同步点云数据融合的三维重构方法。首先,提出利用手动选择控制点进行7自由度初始配准,再利用迭代最近点(ICP)算法对初始配准结果进行精确配准,最后利用最近点搜索算法将分布在经基于面片的多视图立体视觉(PMVS)算法优化后的SFM数据中的颜色信息与激光扫描的点云坐标进行融合。实验结果和数据分析显示,本文的方法能有效地将激光扫描与SFM点云数据进行融合,实现了室外大场景的三维彩色重构。     

Calculation of the unit normal vector using the cross-curve moving mask method for probe radius compensation of a freeform surface measurement

This paper proposes a cross-curve moving mask method to calculate the unit normal vector based on 5 or 9 data points of a freeform surface measurement for probe radius compensation. This is done in order to simplify the calculation and thus, produce a more efficient and time-saving process. Passing through the middle point and 4 or 8 neighboring points, two crossed curves – longitude and latitude can be constructed. The unit normal vector at the middle point can be determined by calculating the cross product of two tangent vectors along these two crossed curves. Different curve fitting methods for the curves passing through 5 or 9 data points, such as Bézier and B-spline methodologies, have been investigated. Three kinds of surfaces, namely, a spherical surface, a cosine-like surface and a shoe-shaped surface (hereby termed “shoe last”) are selected for evaluating the accuracy of the calculated unit normal vectors.     

复杂型面工件超声自动检测中的匹配定位方法

为提高复杂型面工件超声自动检测的效率和通用性,研究了CAD模型已知时工件的匹配定位方法。首先使用人工缺陷贴片的超声C扫描数据获取待匹配特征点的位姿。然后以工件3个角点为特征点进行初始定位,由2个局部坐标系匹配得到初始匹配矩阵。最后根据点到曲面的最小距离原则进行精匹配定位,使用循环迭代及Menq算法进行寻优求解。实例表明,选用6个采样点时,匹配误差可控制在0.4mm以内,能很好地满足复杂型面工件超声自动检测的精度要求。