Shape reconstruction incorporating multiple non-linear geometric constraints

隐式曲面重建方法研究文章通过研究逆向工程中的关键技术三维散乱点云曲面重建技术，对现有的隐式曲面重建方法进行了总结分析，比较各方法的优缺点，以便在实际应用中能根据不同的需求进行相应的选择，也为曲面重建技术的进一步研究提供了方向。

标签：逆向工程；散乱点云；隐式曲面重建逆向工程（Reverse Engineering，RE）[1]，主要是对已有实物的原型或模型进行三维扫描以获取点云数据，然后对点云数据进行曲面重建，在曲面重建结果的基础上进行分析和修改，重建出新产品的模型，最后通过先进的制造技术对其新产品进行生产制造。

逆向工程具有快速研发新产品的特性，其技术已在众多领域得到应用，如机械制造、现实虚拟仿真、3D游戏、3D打印、人体器官仿真等。

在逆向工程中，根据三维扫描设备获取的点云数据信息重建出三维物体模型表面的技术，称之为三维曲面重建技术，见图1。

图1 点云模型曲面重建近年来，隐式曲面因其具备易于实现交、差、并等集合操作，能表示拓扑结构复杂的几何形体，对轻微的噪声不敏感等特点，使得隐式曲面造型技术受到了越来越多专家学者的重视和关注，并提出了一系列有效的隐式曲面重建算法。

1 RBF方法Carr[2]等人将RBF函数插值方法应用于点云数据的曲面重建中，该类算法以散乱数据点作为径向基函数插值中心，计算权值构造插值函数逼近模型曲面的表达函数。

其优点是不需要知道任何散乱数据点之间的拓扑结构信息，重构得到的曲面光顺，曲面细节特征明显，具备良好的孔洞修复能力。

但是由于求解径向基函数权重的方程组随输入点数目的增多而不断扩张，当点云数据的数目增多时，运算量将迅速增大，这样使得由大规模点云数据构成的隐式曲面在赋值计算时非常耗时，极大限制了算法的应用范围。

2 MPU方法在隐式曲面重建算法中，多层次单元划分（Multi-level Partition of Unity Implicits，MPU）曲面重构算法颇受国内外学者的关注。

基于时空注意力的空间关联三维形貌重建盖彦辛;闫涛;张江峰;郭小英;陈斌【期刊名称】《计算机应用》【年(卷),期】2024(44)5【摘要】聚焦形貌恢复通过对场景深度和散焦模糊之间的潜在关系进行建模实现三维形貌重建。

但现有的三维形貌重建网络无法有效利用图像序列的时序关联进行表征学习,因此,提出一种基于多景深图像序列空间关联特征的深度网络框架——三维空间相关水平分析模型(3D SCHAM)进行三维形貌重建。

该模型不仅可以精确捕获单帧图像中聚焦区域到离焦区域的边缘特征,而且可有效利用不同图像帧之间的空间依赖性特征。

首先,通过构建深度、宽度和感受野复合扩展的网络构造三维形貌重建的时域连续模型,进而确定单点深度结果;其次,引入基于空间关联的注意力模块,充分学习帧与帧间的“邻接性”与“距离性”空间依赖关系;另外,利用残差反转瓶颈进行重采样,以保持跨尺度的语义丰富性。

在DDFF 12-Scene真实场景数据集上的实验结果显示,相较于DfFintheWild模型,3D SCHAM在深度值准确度度量的3个阈值1.25,1.25^(2),1.25^(3)上的精确度分别提升了15.34%、3.62%、0.86%,验证了该模型在真实场景的鲁棒性。

【总页数】9页(P1570-1578)【作者】盖彦辛;闫涛;张江峰;郭小英;陈斌【作者单位】山西大学计算机与信息技术学院;山西大学大数据科学与产业研究院;山西大学自动化与软件学院;哈尔滨工业大学重庆研究院;哈尔滨工业大学(深圳)国际人工智能研究院【正文语种】中文【中图分类】TP391.41【相关文献】1.基于时空关联块匹配的动态变形表面三维重建2.基于多站点、多时间注意力机制的电磁强度时空关联分析与可视化3.通道与空间注意力结合的室内场景三维重建4.全局时空特征耦合的多景深三维形貌重建5.基于变分立体匹配算法的GMAW熔池形貌三维重建因版权原因，仅展示原文概要，查看原文内容请购买。

第42卷第22期 范劲松等：传统陶瓷艺术作品的三维数字化重建及应用的研究与实践 13模型，并开发了相关的虚拟现实应用。

随着科技的不断发展，这些数字内容将在未来有着越来越广阔的应用。

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Research and Design of the 3D Reconstruction System Based on Binocular Stereo VisionMajor: civil engineeringSchool of civil engineeringChongqing UniversityNovember, 2017Opening report1.The topic of this research:How to make artificial intelligence (automobile, robot) aware of our world is a complex problem. The 3D reconstruction based on stereo vision provides us a direction. Stereoscopic vision of three dimensional reconstruction means the way to restore the geometry of 3D visible surfaces from two or more two-dimensional images in computer vision. Stereo vision is a simple, reliable, flexible and widely used method to simulate human eyes' processing of scenery. In the computer stereo vision system, two images of the same scene can be obtained from different angles by using a camera. Then, the 3D shape of the scene is reconstructed by computer, and the spatial position information of the object is recovered.2.The purpose of this researchThe purpose of this research is to make computer have the ability of 3d environmental information, using digital camera as image sensor, using image processing, visual computing and other technologies for non-contact 3d measurement, using computer program to obtain 3d information of objects, and 3d model reconstruction, which will not only enable the machine to perceive the geometric information of objects in a three-dimensional environment, including its shape, position, posture, motion, etc., and can describe, store, identify and understand them.3. The research significance of the thesisHuman beings acquire information from the external environment, mainly through the eyes. There is a saying called "seeing is believing", is to illustrate the importance of the information obtained by human eyes. According to scientists statistics, most of the human perception of the world, about 60% from the visual, auditory information accounted for 20%, the other, such as taste, tactile information and so on add up to 20%. And human beings get visual information from the outside world, and use the brain to judge, processing is a very complex process. In the real world, any three-dimensional object has a very complex structure and color information, this information through the retina in order to convert the two-dimensional image information, the information transmitted in the brain pathways of photoreceptor cells, the brain for the final 3D reconstruction and color and position determination. The human brain is an extremely complex and developed processing system that can quickly process this information so that humans can quickly identify objects in the external environment.The purpose of computer vision system is to recognize the 3D world through the projection of the 3D scene on the camera plane. With the rapid development of computer hardware and software facilities and technology, computer vision theoryhas also been rapid developed.Allowing computers to recognize objects as fast as humans has been a relentless pursuit of human beings and decades of dreams. It is the main work of computer vision to use the computer to replace the human visual information in the environment of the outside world.4.Research methods (or experiment)To conduct this research, we will try to figure out some questions first.1.The argumentation of the subject, that is, the purpose of the subject and thebackground of project.2.The innovation point of the subject research. For example, is it new to most of us?Or is it possible that this idea will be widely used after most of us know about it?After solving these problems, the main research methods of this topic are as follows:(1)Document analysis(2)Combination of empirical analysis and logical analysisThe methods used in this study are:Monocular vision,Binocularvision,Trinocular vision5. Anticipated results3D reconstruction based on stereo matching and camera calibration, using SFM (Structure from Motion) algorithm to restore external camera parameters, and then calculate the 3D coordinates of discrete space points, triangulation and texture after Delaunay, finally through the OpenGL programming display 3D model.6. Details of the experimentOur research decides to adopt SFM which belongs to monocular vision method.SFM (Structure from motion) is a method to use numerical method to recover camera parameters and 3D information through detecting matching feature points in multiple images which is not calibrated. SFM requires very low image, and can be reconstructed by video or even random sequence of images. At the same time, the image sequence can be used to realize the camera self-calibration in the process of reconstruction, which eliminates the steps of camera calibration in advance. And because of the progress of feature point extraction and matching technology, the robustness of the SFM is also very strong. Another advantage of the SFM is that it can reconstruct large scale scenes, and the number of input images can reach millions. It is very suitable for 3D reconstruction of natural terrain and urban landscape. It is flexible and convenient to use. It is suitable for all kinds of complicated occasions with less cost. 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一种准线性光束平差方法刘侍刚;彭亚丽;韩崇昭【摘要】为了解决光束平差运行速度慢、复杂度高的缺点,提出了一种准线性光束平差方法(QBA),该方法利用深度因子等于投影矩阵的第3行与射影空间点相乘的特性,采用重投影点和已知图像点的代数距离建立目标函数,交替地将投影矩阵和射影空间点两个量中的一个保持不变,线性地求取另一个量,最后完成射影重建.实验结果表明,QBA方法具有较快的收敛速度,同时和传统的Leyenberg-Marquat方法及Mahamud方法比较,QBA方法运行时间约是L-M方法的1/8,是Mahamud方法的1/3.【期刊名称】《西安交通大学学报》【年(卷),期】2010(044)012【总页数】4页(P1-4)【关键词】光束平差;线性迭代;代数距离【作者】刘侍刚;彭亚丽;韩崇昭【作者单位】西安交通大学电子与信息工程学院,710049,西安;陕西师范大学计算机科学学院,710062,西安;西安交通大学电子与信息工程学院,710049,西安【正文语种】中文【中图分类】TP391.41;P232从图像序列中重建出三维场景结构是计算机视觉的主要目标之一[1-2].目前,它仍然是计算机视觉领域中的研究热点之一[3-4].如果没有任何先验知识,从图像测量中只能得到射影重建[5-6].现在所提出的射影重建算法大部分都是基于多线性约束[7-8].利用多线性约束关系进行射影重建的缺点是它并没有把所有的图像统一地看待,而是倚重某几幅图像,因此一旦获得了初始的射影重建,就要进一步求精,这一步叫做光束平差(bund le ad justment)[9].它是一种在计算机视觉领域有广泛应用的优化算法,其目的是以全局最优化来进一步求得所需要的精确解.在理想情况下,光束平差就是求已知图像点与重投影点之间的几何距离的最小值,可以采用Guass-New ton、Levenberg-M arquat(L-M)等求解非线性优化方法来求解[9-11],但这些求解方法计算量比较大,很难满足实时性的要求.有些文献将求解几何距离修改为求解代数距离,再利用奇异值分解(SVD)得到一个初始的射影重建,然后利用迭代的方法进一步求精,但这种方法需要对图像点数据矩阵的行和列进行归一化处理以避免全零解的出现,而这一步的出现,会导致算法不稳定.Mahamud等人对深度因子求导[12],令其等于0,再交替地求解投影矩阵和射影空间点.该方法最大的优点是能够线性迭代地求取在代数距离最小意义下的最优射影重建,而且可以保证算法的收敛,但该方法并没有考虑到深度因子实际上就是由投影矩阵的第3行行向量与射影空间点相乘而得到的,因此会影响该算法的收敛速度.本文针对上述缺点,提出了一种准线性光束平差方法(QBA),利用深度因子等于投影矩阵的第3行与射影空间点相乘的特性,采用重投影点和已知图像点的代数距离建立目标函数,线性迭代地求取投影矩阵和射影空间点,最后完成射影重建.1 基于代数距离的目标函数假定摄像机模型为经典的针孔模型,即成像过程可以用下列方程表示式中:λ为深度因子为3维空间点的齐次坐标为对应的图像平面点的齐次坐标;P为相机的投影矩阵,是一个3×4的矩阵.设有n个三维空间点,m幅图像,对于第i幅图像上的第j个图像平面点,由式(1)可得由式(2)可得基于代数距离的余差函数为式(3)可以利用线性的方法进行求解,但它是病态的.从式(3)可以看出,λi,j=0,Pi=0,Xj=0将是它的最优解,但是这种解并不是我们所期望的,因此应该增加一些附加的约束条件.2 准线性光束平差方法2.1 线性求解射影空间点首先,让投影矩阵Pi保持不变,线性地求解射影空间点Xj,使代数距离E最小.为了表示方便,令式中:pi,k表示投影矩阵Pi的第k列.由式(2)可得从式(9)可以看出,若不对Xj增加附加约束,将会出现全零解,这是本文不希望的.同时,若 Xj是它的一个解,则(α为常数)也是它的一个解,因此它有无穷多个解.为了求解方便,需要增加一个约束,通常可以令中的最后一个元素为1,或者增加一个约束条件进行SVD分解可得到射影空间点Xj,由此可对式(7)进行求解.2.2 线性求解投影矩阵现在让射影空间点 Xj保持不变,线性地求解投影矩阵Pi,使代数距离E最小.同样,为了表示方便,将投影矩阵Pi写成一个列向量,即同样,从式(15)可以看出,若不对qi增加附加约束,将会出现全零解.因此,本文增加一个约束条件|qi|=1.对Bi进行SVD分解可得到投影矩阵 Pi,由此可对式(15)进行求解.3 QBA方法本文提出的QBA方法步骤描述如下:(1)利用已知的初始射影重建},求到初始代数距离==,并令k=1及ε为任意小的一个正数;(2)令投影矩阵保持不变,利用式(9)求解每个射影空间点;(3)利用式(3)求重投影点到已知图像点的代数距离E(k)1,并判断时停止;否则,进行步骤(4);(4)令射影空间点保持不变,利用式(15)求解每个投影矩阵(5)利用式(3)求重投影点到已知图像点的代数距离并判断ε时停止;否则,k=k+1并转至第(2)步.收敛性分析:当投影矩阵保持不变时,是式(3)的极小值点,因此有.同样,当射影空间点保持不变时是式(3)的极小值点,因此也有.由以上分析可知,本文的方法能够保证收敛.4 实验为了检验本文提出的QBA方法的收敛性,用Matlab模拟产生8幅大小为640×480像素的图像,并在图像中分别加入均值为0、方差分别为1和2的高斯噪声,利用这些模拟图像点用文献[13]的方法完成初步的射影重建之后,分别用Levenberg-Marquat(L-M)非线性优化方法[9]、Mahamud方法[12]及本文提出的QBA方法进行光束平差,实验结果如图1所示.从图1可以看出,QBA方法具有良好的收敛性,在3～4步就可以达到收敛,而L-M方法和Mahamud方法都要5～6步才能够达到收敛.从图中还可以看出,QBA方法和Mahamud方法具有相同的收敛精度,而L-M方法的收敛精度要略高于本文方法和Mahamud方法,这是因为L-M方法求的是几何距离的最小值,而最后却用代数距离来衡量,通常情况下,几何距离的最小值点并不和代数距离的最小值重合.图1 代数距离随迭代次数变化情况同时,为了比较本文提出的QBA方法和L-M方法及Mahamud方法的运行速度,本文首先模拟产生8幅图像,每幅图像点由20个变化到400个.然后,固定每幅图像点数为100个,图像数由2幅变化到16幅.在所有的实验中,图像像素中都加入均值为0、方差为1的高斯噪声,当连续2个代数距离(对于L-M方法为几何距离)之差小于10-6时,就认为算法已经达到收敛.在每种情况下,实验重复100次,然后取其平均值,实验结果如图2和图3所示.图2 运行时间随空间点数变化图图3 运行时间随空间点数变化图从图2和图3中可以看,QBA方法运行时间约是L-M 方法的1/8,约是Mahamud 方法的1/3.由于L-M方法是采用非线性解法,因此它的运行速度最慢,而在Mahamud方法中,它并没有考虑深度因子实际上就是由投影矩阵和射影空间点相乘所组成,因此它的运行速度也较慢.5 结束语本文提出了一种准线性光束平差方法——QBA,利用重投影点和已知图像点的代数距离建立目标函数,通过线性迭代求取代数距离的极小值.由于本文方法考虑到深度因子实际上就是由投影矩阵的第3行行向量与射影空间点相乘而得到的,所以具有较快的运行速度,实验结果也表明了本文QBA方法具有收敛性好及运行速度快等优点.参考文献:【相关文献】[1]WANG Guanghui,WU J Q M.The quasi-perspective model:geometric properties and 3D reconstruction[J].Pattern Recognition,2010,43(5):1932-1942.[2]彭亚丽,刘芳,刘侍刚.一维子空间的三维重建方法[J].西安交通大学学报,2009,43(12):31-35.PENG Yali,LIU Fang,LIU Shigang.3D reconstruction method with 1D subspace[J].Journal of Xi′an Jiaotong University,2009,43(12):31-35.[3]彭亚丽,刘芳,焦李成,等.基于秩4约束的遮挡点恢复方法[J].机器人,2008,30(2):138-141.PENG Yali,LIU Fang,JIAO Licheng,et al.A method for occlusion recovery based on rank4[J].Robot,2008,30(2):138-141.[4]刘侍刚,彭亚丽,韩崇昭,等.3维子空间约束的遮挡点恢复方法[J].西安交通大学学报,2009,43(4):10-13.LIU Shigang,PENG Yali,HAN Chongzhao,et al.An occlusion recovery method based on 3D subspace[J].Journal of Xi′an Jiaoton g University,2009,43(4):10-13. 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Geometric ModelingGeometric modeling is a crucial aspect of computer graphics and design, playing a significant role in various fields such as engineering, architecture, animation, and gaming. It involves the creation and manipulation of geometric shapes and structures in a digital environment, allowing for the visualization and representation of complex objects and scenes. However, despite its importance, geometric modeling presents several challenges and limitations that need to be addressed in order to improve its efficiency and effectiveness. One of the primary issues in geometric modeling is the complexity of representing real-world objects and environments in a digital format. The process of converting physical objects into digital models involves capturing and processing a vast amount of data, which can be time-consuming and resource-intensive. This is particularly challenging when dealing with intricate and irregular shapes, as it requires advanced techniques such as surface reconstruction and mesh generation to accurately capture the details of the object. As a result, geometric modeling often requires a balance between precision and efficiency, as the level of detail in the model directly impacts its computational cost and performance. Another challenge in geometric modeling is the need for seamless integration with other design and simulation tools. In many applications, geometric models are used as a basis for further analysis and manipulation, such as finite element analysis in engineering or physics-based simulations in animation. Therefore, it is essential for geometric modeling software to be compatible with other software and data formats, allowing for the transfer and utilization of geometric models across different platforms. This interoperability is crucial for streamlining the design and production process, as it enables seamless collaboration and data exchange between different teams and disciplines. Furthermore, geometric modeling also faces challenges related to the representation and manipulation of geometric data. Traditional modeling techniques, such as boundary representation (B-rep) and constructive solid geometry (CSG), have limitations in representing complex and organic shapes, often leading to issues such as geometric inaccuracies and topological errors. To address this, advanced modeling techniques such as non-uniform rational B-splines (NURBS) and subdivision surfaces have been developed toprovide more flexible and accurate representations of geometric shapes. However, these techniques also come with their own set of challenges, such as increased computational complexity and difficulty in controlling the shape of the model. In addition to technical challenges, geometric modeling also raises ethical and societal considerations, particularly in the context of digital representation and manipulation. As the boundary between physical and digital reality becomes increasingly blurred, issues such as intellectual property rights, privacy, and authenticity of digital models have become more prominent. For example, the unauthorized use and reproduction of digital models can lead to copyright infringement and legal disputes, highlighting the need for robust mechanisms to protect the intellectual property of digital content creators. Similarly, the rise of deepfakes and digital forgeries has raised concerns about the potential misuse of geometric modeling technology for malicious purposes, such as misinformation and identity theft. It is crucial for the industry to address these ethical concerns and develop standards and regulations to ensure the responsible use of geometric modeling technology. Despite these challenges, the field of geometric modeling continues to evolve and advance, driven by the growing demand forrealistic and interactive digital experiences. Recent developments in machine learning and artificial intelligence have shown promise in addressing some of the technical limitations of geometric modeling, such as automated feature recognition and shape optimization. Furthermore, the increasing availability of powerful hardware and software tools has enabled more efficient and accessible geometric modeling workflows, empowering designers and artists to create intricate and immersive digital content. With ongoing research and innovation, it is likely that many of the current challenges in geometric modeling will be overcome, leading to more sophisticated and versatile tools for digital design and visualization. In conclusion, geometric modeling is a critical component of modern digital design and visualization, enabling the creation and manipulation of complex geometric shapes and structures. However, the field faces several challenges related to the representation, integration, and ethical implications of geometric models. By addressing these challenges through technological innovation and ethical considerations, the industry can continue to push the boundaries of what ispossible in digital design and create more immersive and impactful experiences for users.。

Shape from X从X恢复形状:x可以是，shading（单幅图像明暗）、stereo vision（立体视觉法）、photometric stereo(光度立体法)、texture(纹理)、motion(运动)、contour(轮廓)、shadow（阴影）。

从明暗恢复形状( shape f rom shading , 简称SFS)：是计算机视觉中三维形状恢复问题的关键技术之一，其任务是利用单幅图象中物体表面的明暗变化来恢复其表面各点的相对高度或表面法方向等参数值，为进一步对物体进行三维重构奠定基础。

由单幅图像灰度明暗变化恢复三维形状的过程可以看作成像过程的逆过程。

对实际图像而言，其表面点图像亮度受到了许多因素，如光源、物体表面材料性质和形状，以及摄像机(或观察者)位置和参数等的影响。

由单幅图像灰度明暗变化恢复三维形状是在一定的约束条件下从平滑变化的灰度图像恢复出表面法向信息，即根据物体表面反射模型建立物体表面三维形状与采集的图像灰度之间关系的反射图方程，以及由先验知识所建立的对物体表面形状参数的约束条件，对这些关系求解可得到物体表面三维形状。

传统SFS方法均进行了如下假设：( 1)光源为无限远处点光源；( 2)反射模型为朗伯体表面反射模型( Lambertian)；( 3)成象几何关系为正交投影。

立体视觉法(shape from Stereo vision)可以分为双目和多目立体视觉两种类型。

简要说明双目立体视觉的原理。

与人类双目视觉的感知过程类似，双目立体视觉从两个不同视点观察同一物体可以得到不同视角下的图像，通过分析不同图像中同一像点的不同视差来获取物体表面的三维空间信息。

立体视觉系统可以分为图像采集、摄像机标定、特征提取、立体匹配、深度恢复及三维表面插值等部分组成。

目前有MTI人工智能实验室、Yale 机器视觉机器人实验室、哈尔滨工业大学、中科院自动化所、西安交通大学、Sony 公司、Intel公司等国内外多家研究机构都在从事立体视觉方面的研究。