CN108961203A - A kind of three-dimensional rebuilding method of fusion ultrasound and the hollow plate type ceramic film defect of machine vision technique - Google Patents
A kind of three-dimensional rebuilding method of fusion ultrasound and the hollow plate type ceramic film defect of machine vision technique Download PDFInfo
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Abstract
一种融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法。本发明通过融合中空板式陶瓷膜内部缺陷三维重建数据和中空板式陶瓷膜表面缺陷三维重建数据实现中空板式陶瓷膜完整缺陷的三维重建，前者由超声检测技术采集，后者由机器视觉技术采集；采用证据理论分别计算由超声技术采集的中空板式陶瓷膜内部缺陷三维数据和基于机器视觉采集的中空板式陶瓷膜表面缺陷三维数据的权值，并对上述获得的权值设定闭值,以确定最终的缺陷边界并获取中空板式陶瓷膜完整缺陷的三维数据；对重新分配后的完整缺陷的三维数据进行基于约束的非刚性对齐，从而实现三维重构。本发明方法可以有效解决缺陷复杂各异、材料浪费的问题。
A method for 3D reconstruction of defects in hollow plate ceramic membranes that combines ultrasound and machine vision techniques. The present invention realizes the three-dimensional reconstruction of the complete defects of the hollow plate ceramic membrane by fusing the three-dimensional reconstruction data of internal defects of the hollow plate ceramic membrane and the three-dimensional reconstruction data of the surface defect of the hollow plate ceramic membrane. The former is collected by ultrasonic detection technology, and the latter is collected by machine vision technology; Evidence theory calculates the weights of the three-dimensional data of internal defects of hollow plate ceramic membranes collected by ultrasonic technology and the three-dimensional data of surface defects of hollow plate ceramic membranes collected based on machine vision, and sets closed values for the weights obtained above to determine the final The boundary of the defect and obtain the three-dimensional data of the complete defect of the hollow plate ceramic membrane; the non-rigid alignment based on the constraint is performed on the three-dimensional data of the complete defect after redistribution, so as to realize the three-dimensional reconstruction. The method of the invention can effectively solve the problems of complex and various defects and material waste.
Description
技术领域technical field
本发明涉及一种融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法，属于机械工程和计算机工程技术领域。The invention relates to a three-dimensional reconstruction method for hollow plate ceramic membrane defects which combines ultrasonic and machine vision technologies, and belongs to the technical fields of mechanical engineering and computer engineering.
背景技术Background technique
面对日益严峻的缺水形势, 向占地球水资源总储量96. 5%的大海要水是解决缺水问题的必由之路。在众多的海水淡化方法中, 反渗透海水淡化( )技术具有占地少、建造周期短、操作简单、比投资小、无相变、能耗低和起动运行快等特点, 在海水淡化领域发展迅速。目前预处理工艺可分为传统方法和膜法，膜法预处理工艺包括有机膜法和无机膜法。作为无机膜的一种, 中空板式陶瓷膜具有孔径分布窄、孔隙率高、分离层薄、过滤阻力小等优点，而且单位膜表面积处理量高、产水能力大，膜的化学性质稳定，可以在海水中长期稳定运行，更适用于海水淡化预处理。中空板式陶瓷膜是脆性材料，杨氏模量较高，即使微小的缺陷或轻度应变都会导致极大的机械应力。裂纹类缺、内部气孔、夹层等缺陷是陶瓷材料中经常出现的非完备现象。要准确地探知缺陷的形貌、位置、分布等性状特征，因此需要实现上述缺陷的快速检测与三维重建。针对中空板式陶瓷膜内部缺陷检测，传统的基于二维CT图像检测手段容易出现误判、漏判，超声技术用于三维空间的缺陷检测、识别和测量是今后的发展方向。针对中空板式陶瓷膜表面检测，在传统的管道内壁检测技术中，漏磁法和涡流法受于检测材料的限制不能使用，而超声波检测法检测精度太低。机器视觉检测技术作为发展前景最好的管道内壁检测方法，具有测量速度快，测量精度高，图像包含信息完整，容易实现自动连续检测，可满足生产线上的速度要求等诸多优点。由超声技术能够实现中空板式陶瓷膜深层缺陷的三维检测，基于机器视觉能够实现中空板式陶瓷膜表面缺陷的三维检测，基于非刚性对齐融合两者的检测数据，来准确地探知各种缺陷的形貌、位置、分布和大小等特征，并能构建相应的三维重建模型。根据缺陷的体积计算，估算中空板式陶瓷膜质量的分类，有效解决缺陷复杂各异、材料浪费的问题。Facing the increasingly severe water shortage situation, asking for water from the sea, which accounts for 96.5% of the earth's total water resources, is the only way to solve the water shortage problem. Among the many seawater desalination methods, reverse osmosis seawater desalination ( ) technology has the characteristics of less land occupation, short construction period, simple operation, small specific investment, no phase change, low energy consumption and fast start-up operation, etc., and has developed rapidly in the field of seawater desalination. Currently The pretreatment process can be divided into traditional method and membrane method, and the membrane method pretreatment process includes organic membrane method and inorganic membrane method. As a kind of inorganic membrane, the hollow plate ceramic membrane has the advantages of narrow pore size distribution, high porosity, thin separation layer, low filtration resistance, etc., and the treatment capacity per unit membrane surface area is high, the water production capacity is large, and the chemical properties of the membrane are stable. Long-term stable operation in seawater, more suitable for seawater desalination pretreatment. The hollow plate ceramic membrane is a brittle material with a high Young's modulus, and even small defects or slight strains will cause great mechanical stress. Defects such as cracks, internal pores, and interlayers are incomplete phenomena that often appear in ceramic materials. To accurately detect the shape, location, distribution and other characteristics of defects, it is necessary to realize the rapid detection and three-dimensional reconstruction of the above defects. For the detection of internal defects of hollow plate ceramic membranes, traditional detection methods based on two-dimensional CT images are prone to misjudgments and missed judgments. The use of ultrasonic technology in the detection, identification and measurement of defects in three-dimensional space is the future development direction. For the surface detection of the hollow plate ceramic membrane, in the traditional pipeline inner wall detection technology, the magnetic flux leakage method and the eddy current method cannot be used due to the limitation of the detection material, while the detection accuracy of the ultrasonic detection method is too low. As the most promising pipeline inner wall inspection method, machine vision inspection technology has many advantages such as fast measurement speed, high measurement accuracy, complete image information, easy automatic continuous inspection, and can meet the speed requirements of the production line. Ultrasonic technology can realize the three-dimensional detection of deep defects in the hollow plate ceramic membrane, based on machine vision can realize the three-dimensional detection of surface defects of the hollow plate ceramic membrane, based on non-rigid alignment and fusion of the detection data of the two, to accurately detect the shape of various defects features such as appearance, location, distribution, and size, and can construct corresponding 3D reconstruction models. According to the volume calculation of the defect, the classification of the quality of the hollow plate ceramic membrane is estimated, and the problems of complex and various defects and material waste are effectively solved.
已有技术中，挪威的Kumakiri等在“Membrane characterisation by a novel defectdetection technique”（Microporous and Mesoporous Materials, Vol115, No1-2(October),2008:33-39）一文中提出了一种新的薄膜表征技术，用于纳米缺陷的可视化和定位，这种新技术大大简化了膜的泄漏检测，使准确定位小泄漏也成为可能；加州大学的Francesco Lanza等在“Ultrasonic Tomography for Three-Dimensional Imaging ofInternal Rail Flaws Proof-of-Principle Numerical Simulations”（TransportationResearch Record.Vol2374，2013：162-168）一文中建立了缺陷轨道上超声层析成像阵列的有限元模型，提出了一种三维内轨缺陷层析成像算法；湖北工业大学汤亮等在“基于区域分级的陶瓷阀芯表面缺陷检测系统研究与实现”（组合机床与自动化加工技术，2017年第10卷第82-86页）一文中，依据陶瓷阀芯表面反射率的不同，提出了一种分区域、多级优化的陶瓷阀芯表面缺陷检测算法；北京航天航空大学的周正干、徐娜的“一种基于改进的动态深度聚焦的相控阵超声检测方法”（专利授权号：CN102809610B）获得国家发明专利。但这些研究均只采用一种检测技术手段，在融合不同检测技术所获得的数据方面仍存在一定的空白，离实际生产应用也还有相当长的距离。来自Gunadarma University的Dennis Christie等在“3D reconstruction of dynamic vehicles using sparse 3D-laser-scanner and 2Dimage fusion”一文中提出了一个三点ICP的RANSAC细化算法，从而进行刚性运动物体的三维重建；同济大学的王婷在“数据融合技术在混凝土结构检测中的应用研究”一文中建立了数据融合CT技术在混凝土结构检测领域的应用框架,将红外成像与超声技术相结合,实现了对缺陷大小的三维重构以及对缺陷类型的识别。但是这些研究均处于理论探索阶段，是否适用于融合中空板式陶瓷膜内部缺陷和表面缺陷的三维数据还有待验证。In the existing technology, Kumakiri et al. from Norway proposed a new thin film characterization in the article "Membrane characterization by a novel defect detection technique" (Microporous and Mesoporous Materials, Vol115, No1-2 (October), 2008:33-39) technology, for the visualization and localization of nano-defects, this new technology greatly simplifies the leak detection of membranes and makes it possible to accurately locate small leaks; Francesco Lanza of the University of California et al. Proof-of-Principle Numerical Simulations" (TransportationResearch Record.Vol2374, 2013: 162-168) established a finite element model of the ultrasonic tomography array on the defect track, and proposed a three-dimensional internal track defect tomography algorithm; In the article "Research and Implementation of Surface Defect Detection System of Ceramic Valve Core Based on Regional Classification" (Combined Machine Tool and Automatic Processing Technology, Vol. According to the difference in reflectivity, a sub-area, multi-level optimized ceramic valve core surface defect detection algorithm is proposed; Zhou Zhengqian and Xu Na of Beijing University of Aeronautics and Astronautics "A Phased Array Ultrasonic Detection Method Based on Improved Dynamic Depth Focusing "(patent authorization number: CN102809610B) obtained the national invention patent. However, these studies only use one detection technology, and there is still a certain gap in the integration of data obtained by different detection technologies, and there is still a long distance from the actual production application. Dennis Christie from Gunadarma University proposed a three-point ICP RANSAC refinement algorithm in the article "3D reconstruction of dynamic vehicles using sparse 3D-laser-scanner and 2Dimage fusion" to perform 3D reconstruction of rigid moving objects; Tongji University In the article "Application Research of Data Fusion Technology in Concrete Structure Inspection", Wang Ting established the application framework of data fusion CT technology in the field of concrete structure inspection, and combined infrared imaging with ultrasonic technology to realize the three-dimensional analysis of defect size. Refactoring and identification of defect types. However, these studies are still in the stage of theoretical exploration, and whether they are applicable to the fusion of three-dimensional data of internal defects and surface defects of hollow plate ceramic membranes remains to be verified.
发明内容Contents of the invention
为了克服现有技术的不足和缺陷，本发明采用先进的融合技术，提供一种融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法，该方法能够提高中空板式陶瓷膜缺陷三维重建的准确度。In order to overcome the deficiencies and defects of the existing technology, the present invention adopts advanced fusion technology to provide a three-dimensional reconstruction method of hollow plate ceramic membrane defects that integrates ultrasonic and machine vision technology, which can improve the accuracy of three-dimensional reconstruction of hollow plate ceramic membrane defects. Accuracy.
本发明的目的是通过以下技术方案实现的，融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法，包括以下步骤:The object of the present invention is achieved by the following technical solutions, the three-dimensional reconstruction method of the hollow plate ceramic membrane defect of fusion ultrasonic and machine vision technology, comprises the following steps:
1）基于超声数据的区域增长技术实现中空板式陶瓷膜内部缺陷三维重建；1) The region growth technology based on ultrasonic data realizes the three-dimensional reconstruction of internal defects of the hollow plate ceramic membrane;
2）基于机器视觉技术实现中空板式陶瓷膜表面缺陷的三维重建；2) Based on machine vision technology, the three-dimensional reconstruction of the surface defects of the hollow plate ceramic membrane is realized;
3）融合超声和机器视觉技术实现中空板式陶瓷膜完整缺陷的三维重建。3) Combining ultrasound and machine vision technology to realize the three-dimensional reconstruction of the complete defect of the hollow plate ceramic membrane.
优选的，所述步骤（1）中的超声数据是指由超声检测技术采集的数据进行体绘制后应用一种新的基于区域增长技术的混合绘制方法来实现三维重构的；Preferably, the ultrasonic data in the step (1) refers to volume rendering of data collected by ultrasonic detection technology and then applying a new hybrid rendering method based on region growing technology to realize three-dimensional reconstruction;
其中所述的区域增长技术的原理是通过选取一个种子像素作为生长点，然后与周围区域内像素的相似性(一般为平均灰度值)在阂值范围内做比较，若具有一致性则连接起来构成区域，将平面的生长延伸到三维空间场中即可以实现可视化分割，运用阈值选取技术，设置一阈值范围，以得到最为清晰的二维缺陷形态；The principle of the region growth technique described therein is to select a seed pixel as the growth point, and then compare it with the similarity (generally the average gray value) of the pixels in the surrounding area within the threshold range, and if there is consistency, connect To form a region, extend the growth of the plane to the three-dimensional space field to achieve visual segmentation, use the threshold selection technology, set a threshold range, in order to obtain the most clear two-dimensional defect shape;
所述的三维重建是将整个扫查区域置于笛卡尔坐标系中，每个位置的值对应当前位置的体像素，运用三维区域增长技术，即选取一个体像素作为种子点然后寻找相邻位置处阈值范围内的演算点，将阈值选取的图像相邻像素连接重组成实体三维图像。The three-dimensional reconstruction is to place the entire scanning area in the Cartesian coordinate system, and the value of each position corresponds to the voxel at the current position, using the three-dimensional area growth technology, that is, selecting a voxel as a seed point and then searching for adjacent positions At the calculation points within the threshold range, the adjacent pixels of the image selected by the threshold are connected and reconstructed into a solid three-dimensional image.
优选的，所述步骤（2）中的机器视觉技术是指采用双单目三维测量系统获取中空板式陶瓷膜表面缺陷的点云数据。Preferably, the machine vision technology in the step (2) refers to using a dual-monocular three-dimensional measurement system to obtain point cloud data of surface defects of the hollow plate ceramic membrane.
所述中空板式陶瓷膜表面缺陷的三维重建的方法为：The method for three-dimensional reconstruction of surface defects of the hollow plate ceramic membrane is:
（1）对获取的点云数据进行预处理：包括对数据进行中值滤波以提高数据的抗噪声能力、重采样、坐标归一化；(1) Preprocessing the obtained point cloud data: including median filtering of the data to improve the anti-noise ability of the data, resampling, and coordinate normalization;
（2）将拉普拉斯算子与点云数据卷积，除去无用的点云数据，拼合点云数据形成中空板式陶瓷膜表面缺陷的三维数据，并进行三维重建。(2) Convolve the Laplacian operator with the point cloud data, remove useless point cloud data, and merge the point cloud data to form the 3D data of the surface defects of the hollow plate ceramic membrane, and perform 3D reconstruction.
优选的，所述步骤（3）中数据融合是指融合由超声技术采集的数据三维重建后得到的数据和由机器视觉采集的数据三维重建后得到的数据, 采用证据理论计算两类数据在融合后各自所占的权值，接着进行基于约束的非刚性对齐，来获取中空板式陶瓷膜三维缺陷完整数据并进行三维重建。计算权值的过程如下：Preferably, the data fusion in the step (3) refers to the fusion of the data obtained by three-dimensional reconstruction of the data collected by ultrasonic technology And the data obtained after three-dimensional reconstruction of the data collected by machine vision , using the evidence theory to calculate the respective weights of the two types of data after fusion, and then perform non-rigid alignment based on constraints to obtain the complete data of the three-dimensional defects of the hollow plate ceramic membrane and perform three-dimensional reconstruction. The process of calculating weights is as follows:
（1）分别确定由超声技术采集的中空板式陶瓷膜内部缺陷三维数据和基于机器视觉采集的中空板式陶瓷膜表面缺陷三维数据的权值，即融合前的概率分配值；(1) Determine the weights of the three-dimensional data of internal defects of the hollow plate ceramic membrane collected by ultrasonic technology and the three-dimensional data of surface defects of the hollow plate ceramic membrane collected based on machine vision, that is, the probability distribution value before fusion;
（2）采用证据理论计算融合后两者的概率分配函数值，即确定不同三维数据采集方式的可信度；(2) Using evidence theory to calculate the probability distribution function value of the two after fusion, that is, to determine the credibility of different 3D data acquisition methods;
（3）对上述获得的融合后的概率分配函数值设定闭值，以确定最终的缺陷边界；(3) Set a closed value for the fused probability distribution function value obtained above to determine the final defect boundary;
（4）获取中空板式陶瓷膜完整缺陷的三维数据；(4) Obtain the three-dimensional data of the complete defect of the hollow plate ceramic membrane;
所述的证据理论是指理论或信任函数理论，通常简称为理论，其组合规则如下The stated theory of evidence refers to theory or belief function theory, often referred to simply as theory, the combination rules are as follows
设和是同一识别框架上的两个概率分配函数，则其正交和为Assume and is the same recognition frame Two probability assignment functions on , then their orthogonal sum for
当时， （1）when hour, (1)
当时， （2）when hour, (2)
其中， （3）in, (3)
如果，则正交和也是一个概率分配函数；如果，则不存在正交和，称与相矛盾。if , then the orthogonal sum is also a probability assignment function; if , then there is no orthogonal sum ,say and Contradictory.
与现有技术相比，本发明有益效果是:由于采用新型算法进行融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法，减少了数据计算、存储，简化成像算法，提高了陶瓷膜检测效率。Compared with the prior art, the beneficial effects of the present invention are: due to the adoption of a new algorithm for the three-dimensional reconstruction method of the hollow plate ceramic membrane defects of fusion ultrasound and machine vision technology, data calculation and storage are reduced, the imaging algorithm is simplified, and the ceramic membrane is improved. detection efficiency.
附图说明Description of drawings
图1 一种融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法的流程图。Fig. 1 Flowchart of a 3D reconstruction method for hollow plate ceramic membrane defects that combines ultrasound and machine vision technology.
具体实施方式Detailed ways
下面结合附图和一种融合超声和机器视觉技术的中空板式陶瓷膜缺陷的三维重建方法对本发明的具体实施作进一步描述。The specific implementation of the present invention will be further described below in conjunction with the accompanying drawings and a three-dimensional reconstruction method for hollow plate ceramic membrane defects that integrates ultrasonic and machine vision technology.
如图1所示，本发明融合超声和机器视觉技术实现中空板式陶瓷膜缺陷的三维重建方法，包括下述步骤:As shown in Figure 1, the present invention merges ultrasonic and machine vision technology to realize the three-dimensional reconstruction method of hollow plate type ceramic membrane defect, comprises the following steps:
1）基于超声数据的区域增长技术实现中空板式陶瓷膜内部缺陷三维重建1) The region growth technology based on ultrasonic data realizes the 3D reconstruction of internal defects of the hollow plate ceramic membrane
超声检测采集的数据进行体绘制后应用一种新的基于区域增长技术的混合绘制方法来实现三维重构。区域增长的原理是通过选取一个种子像素作为生长点，然后与周围区域内像素的相似性(一般为平均灰度值)在阂值范围内做比较，若具有一致性则连接起来构成区域，将平面的生长延伸到三维空间场中即可以实现可视化分割。运用阈值选取技术，设置一阈值范围，以得到最为清晰的二维缺陷形态。将整个扫查区域置于笛卡尔坐标系中，每个位置的值对应当前位置的体像素。运用三维区域增长技术，即选取一个体像素作为种子点然后寻找相邻位置处阈值范围内的演算点，将阈值选取的图像相邻像素连接重组成实体三维图像。After the volume rendering of the data collected by ultrasonic testing, a new hybrid rendering method based on region growing technology is applied to achieve 3D reconstruction. The principle of region growth is to select a seed pixel as the growth point, and then compare it with the similarity (generally the average gray value) of the pixels in the surrounding area within the threshold range, and if there is consistency, they will be connected to form a region. The growth of the plane is extended to the three-dimensional space field to realize the visual segmentation. Use threshold selection technology to set a threshold range to obtain the clearest two-dimensional defect shape. Place the entire scanning area in the Cartesian coordinate system, and the value of each position corresponds to the voxel at the current position. Using 3D region growth technology, that is, selecting a voxel as a seed point and then finding calculation points within the threshold range at adjacent positions, and connecting adjacent pixels of the image selected by the threshold to form a solid 3D image.
2）基于机器视觉实现中空板式陶瓷膜表面缺陷的三维重建2) 3D reconstruction of surface defects of hollow plate ceramic membrane based on machine vision
基于机器视觉技术是指采用双单目三维测量系统获取中空板式陶瓷膜表面缺陷的点云数据。首先对获取的点云数据进行预处理：包括对数据进行中值滤波以提高数据的抗噪声能力、重采样和坐标归一化等，接着将拉普拉斯算子与点云数据卷积，确定缺陷边缘并除去无用的点云数据，然后拼合点云数据形成中空板式陶瓷膜表面缺陷的三维数据，最后进行三维重建。Based on machine vision technology refers to the use of dual-monocular three-dimensional measurement system to obtain point cloud data of surface defects of hollow plate ceramic membrane. Firstly, the acquired point cloud data is preprocessed: including median filtering of the data to improve the anti-noise ability of the data, resampling and coordinate normalization, etc., and then the Laplacian operator is convolved with the point cloud data, Determine the edge of the defect and remove useless point cloud data, then merge the point cloud data to form the 3D data of the surface defect of the hollow plate ceramic membrane, and finally perform 3D reconstruction.
3）融合超声和机器视觉技术实现中空板式陶瓷膜完整缺陷的三维重建3) Combining ultrasound and machine vision technology to realize 3D reconstruction of complete defects in hollow plate ceramic membranes
设由超声技术采集的数据三维重建后得到数据，由机器视觉采集的数据三维重建后得到数据, 采用证据理论计算两类数据在融合后各自所占的权值，接着进行基于约束的非刚性对齐，来获取中空板式陶瓷膜三维缺陷完整数据。计算权值的过程如下：Assuming that the data collected by ultrasonic technology is reconstructed after three-dimensional reconstruction to obtain the data , after three-dimensional reconstruction of data collected by machine vision, the data obtained , using the evidence theory to calculate the respective weights of the two types of data after fusion, and then perform non-rigid alignment based on constraints to obtain the complete data of the three-dimensional defects of the hollow plate ceramic membrane. The process of calculating weights is as follows:
（1）分别确定由超声技术采集的中空板式陶瓷膜内部缺陷三维数据和基于机器视觉采集的中空板式陶瓷膜表面缺陷三维数据的权值，即融合前的概率分配值；(1) Determine the weights of the three-dimensional data of internal defects of the hollow plate ceramic membrane collected by ultrasonic technology and the three-dimensional data of surface defects of the hollow plate ceramic membrane collected based on machine vision, that is, the probability distribution value before fusion;
（2）采用证据理论计算融合后两者的概率分配函数值，即确定不同三维数据采集方式的可信度；(2) Using evidence theory to calculate the probability distribution function value of the two after fusion, that is, to determine the credibility of different 3D data acquisition methods;
（3）对上述获得的融合后的概率分配函数值设定闭值，以确定最终的缺陷边界；(3) Set a closed value for the fused probability distribution function value obtained above to determine the final defect boundary;
（4）获取中空板式陶瓷膜完整缺陷的三维数据；(4) Obtain the three-dimensional data of the complete defect of the hollow plate ceramic membrane;
所述的证据理论是指理论或信任函数理论，通常简称为理论，其组合规则如下The stated theory of evidence refers to theory or belief function theory, often referred to simply as theory, the combination rules are as follows
设和是同一识别框架上的两个概率分配函数，则其正交和为Assume and is the same recognition frame Two probability assignment functions on , then their orthogonal sum for
当时， （4）when hour, (4)
当时， （5）when hour, (5)
其中， （6）in, (6)
如果，则正交和也是一个概率分配函数；如果，则不存在正交和，称与相矛盾。if , then the orthogonal sum is also a probability assignment function; if , then there is no orthogonal sum ,say and Contradictory.
对重新分配后的完整缺陷的三维数据进行基于约束的非刚性对齐，从而实现中空板式陶瓷膜完整缺陷的三维重建。非刚性对齐选择的是薄板样条插值算法（），定义相应的误差函数，包括Constraint-based non-rigid alignment is performed on the 3D data of the reassigned complete defects, thereby realizing the 3D reconstruction of the complete defects of hollow plate ceramic membranes. For non-rigid alignment, the thin-plate spline interpolation algorithm ( ), define the corresponding error function, including
距离误差； （7）distance error ;(7)
平滑误差； （8）smoothing error ; (8)
优化公式定义如下：； （9）The optimization formula is defined as follows: ; (9)
采用算法进行求解。use algorithm to solve.
其中式（7）中代表数据，代表数据，为变换矩阵。采用算法（一种拟牛顿的方法）进行求解，数值试验表明算法是求解大规模边界问题的有效方法之一。where in formula (7) representative data , representative data , is the transformation matrix. use Algorithm (a quasi-Newton method) to solve, numerical experiments show that Algorithms are one of the effective methods for solving large-scale boundary problems.
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