点模型数字几何处理若干技术研究
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摘要
传统的多边形网格模型除存储模型的几何信息之外,还需维护庞大的拓扑连接信息。而以离散点为表面表达方式的点模型,因具有数据获取方便、数据结构简单且无需维护其全局一致的拓扑结构等优点,故以点模型为研究对象的数字几何处理已成为计算机图形学中一个新兴的研究热点。三维点模型在医学辅助诊断、数字娱乐、工业设计、航天模拟、文物保护和修复等行业中具有广泛的应用前景。
本文以计算机图形学,计算机辅助几何设计,离散微分几何学和数字信号处理为坚实的理论基础,对点模型数字几何处理的若干核心技术进行了深入研究,提出了一些新的算法,初步形成了一个点模型数字几何处理框架。主要内容包括;
●提出了基于采样保真性和基于相似性的两种点模型去噪算法。在基于采样保真性的去噪算法中,通过MLS计算出采样点的保真性,基于法向张量投票测量采样点的特征性,并将两者相结合实现保持曲面几何特征的点模型去噪。在基于相似性的去噪算法中,通过本文设计的三边滤波算子计算采样点的几何强度,利用构建的均匀几何强度网格估算采样点间几何强度的相似性;通过Mean Shift聚类法对点模型进行几何特征相似性聚类;依据相似性实现点模型保持曲面特征的去噪。
●提出了一种生成细节的点模型几何修复算法。我们通过RBF曲面重建得到残缺区域的光滑修复曲面;利用三边滤波算子获取已知采样点的几何细节纹理;在孔洞光滑曲面上建立一个带约束的全局纹理合成能量方程,并通过求解出一个全局的能量最小值获得孔洞光滑曲面的几何细节。
●给出了两种曲率自适应的点模型简化方法。在基于几何图像的简化方法中,通过将点模型的球面极坐标映射到平面上,构造其几何图像;为了便于控制采样密度,将简化密度定义为简化时所允许的最大像素搜索半径;将曲率阈值和简化密度相结合实现点模型的曲率自适应简化。在基于相似性的简化方法中,通过法向张量投票方法将点模型分为强特征边性和非强特征边性两部分,利用Mean Shift聚类法对非强特征边性部分进行表面区域几何特征相似性聚类,然后分别进行简化。
●基于球面参数化,提出了一种鲁棒的点模型渐变算法。通过球面参数化将源和目标模型嵌入到单位球面上,在球面上自适应地对齐模型间的相应特征点之后,将球面映射到同一矩形参数域上,以建立模型间各采样点的对应关系。在渐变过程中,采用拉普拉斯算子求解中间点模型以保持模型的细节,利用MLS进行动态自适应的重采样以消除中间模型的裂缝。
●介绍了我们研发的点模型数字几何处理原型系统。该系统提供了一个包括数据获取、表示、处理以及绘制等技术在内的数字几何处理框架,并着重介绍和实现了点模型的绘制、点元预处理、点模型的去噪、几何修复、简化和重采样、参数化、渐变、点模型的纹理映射和纹理合成等功能模块。
点模型数字几何处理有着广阔的研究前景,基于我们研发的原型系统,本文最后给出了未来的研究方向。
The traditional geometry representation, such as polygonal meshes, needs to store and maintain the connectivity information between vertices as well as the geometry information. They are both memory consuming and computationally expensive. Due to convenient 3D point data acquirement, simple data structure and no need to maintain the globally consistent topology, however, point-sampled geometry is receiving a growing amount of attention as a new type of medium and surface presentation in Computer Graphics. Accordingly, considerable research has been devoted to the field of digital geometry processing of point-sampled models. Point-sampled geometry has thus been applied to many fields, such as computer-aided medical diagnosis, digital entertainment, industrial design, aviation simulation, protection and restoration of cultural relics, etc.
Based on the sound theoretical foundation on computer graphics, computer-aidedgeometric design, discrete differential geometry, digital signal processing, thisdissertation investigates some key research area for the digital geometry processing ofpoint-sampled models, and proposes a number of novel algorithms on them. At thesame time, we have presented a nearly complete geometry processing framework forpoint-sample geometry. Our main contributions focus on the following five aspects:
·Based on sampling likelihood, a robust denoising algorithm for point-sampledsurfaces is proposed. In terms of moving least squares surface, the samplinglikelihood for each point on point-sampled surfaces is computed. Based on thenormal tensor voting, the feature intensity of sample point is evaluated. Incombination with sampling likelihood and feature intensity, the point-sampledgeometry can be efficiently smoothed while preserving the surface features. Basedon the similarities including geometry intensity and features of sample points, anon-local denoising algorithm for point-sampled surfaces is presented. By usingthe trilateral filtering operator, the geometry intensity of sample point isdetermined. According to their regular grids of geometry intensity, the similarity of geometry intensity between points is measured. Base on Mean Shift clustering, the point-sampled surfaces are clustered into clusters according to the surface-features similarity. By the similarities, the filtered point-sampled geometry with the fine features is attainted.
·A method is presented for patching holes on point-sampled model and synthesizing surface with details. By the surface reconstruction based on radial basis functions, the hole is patched with a smooth surface. Using the trilateral filtering operator, the geometry textures of the existing sample points are produced. As a result, the geometric details on the smooth completed patch are generated by optimizing a constrained global texture energy function on the point-sampled surfaces.
·Based on geometry images, an adaptive curvature simplification method for point-sampled model is proposed. By projecting its spherical polar coordinates onto a plane, the point-sampled model is represented as geometry images. The simple density is defined as the maximum search radius on geometry images in order to conveniently control it. In combination with the surface variation and simple density, the point-sampled surfaces are simplified. Another simplification algorithm is based on similarity including strong feature-edge intensity and surface feature anisotropy. Using the normal tensor voting, the point-sampled surfaces are segmented into two parts, one for the strong feature-edge intensity and another for the nonstrong feature-edge intensity. Base on Mean Shift clustering, the second part is then clustered into clusters according to the surface-features similarity. The first part and all the clusters are respectively simplified.
·Based on spherical parameterization, a robust morphing algorithm for point-sampled geometry is presented. Source and target models represented by point-sampled geometry are first parameterized onto a sphere, respectively. After aligning the corresponding features of two models on their spheres, two spheres are projected onto a common rectangle-parameter domain and the correspondence between sample points on the two models is built using this rectangle domain. In order to preserve the geometric details of point set surfaces, the absolute geometry of the in-between model is computed by means of Laplacian operator and its surfaces are dynamically up-sampled using a moving least squares method so as to eliminate the cracks.
·A prototype system for digital geometry processing of point-sampled models is implemented and introduced. The system actually is a framework of 3D geometry acquisition, representation, processing and rendering. The main components of the system include various novel algorithms on rendering, estimation of local surface differentials, denoising, geometry completion, simplification and up-sampling, parameterization, morphing, texture mapping and synthesis.
As a new type of media data, the point-sampled geometry is an extreme promising presentation for 3D surface, and so we discuss some directions for future work in the last chapter.
本文以计算机图形学,计算机辅助几何设计,离散微分几何学和数字信号处理为坚实的理论基础,对点模型数字几何处理的若干核心技术进行了深入研究,提出了一些新的算法,初步形成了一个点模型数字几何处理框架。主要内容包括;
●提出了基于采样保真性和基于相似性的两种点模型去噪算法。在基于采样保真性的去噪算法中,通过MLS计算出采样点的保真性,基于法向张量投票测量采样点的特征性,并将两者相结合实现保持曲面几何特征的点模型去噪。在基于相似性的去噪算法中,通过本文设计的三边滤波算子计算采样点的几何强度,利用构建的均匀几何强度网格估算采样点间几何强度的相似性;通过Mean Shift聚类法对点模型进行几何特征相似性聚类;依据相似性实现点模型保持曲面特征的去噪。
●提出了一种生成细节的点模型几何修复算法。我们通过RBF曲面重建得到残缺区域的光滑修复曲面;利用三边滤波算子获取已知采样点的几何细节纹理;在孔洞光滑曲面上建立一个带约束的全局纹理合成能量方程,并通过求解出一个全局的能量最小值获得孔洞光滑曲面的几何细节。
●给出了两种曲率自适应的点模型简化方法。在基于几何图像的简化方法中,通过将点模型的球面极坐标映射到平面上,构造其几何图像;为了便于控制采样密度,将简化密度定义为简化时所允许的最大像素搜索半径;将曲率阈值和简化密度相结合实现点模型的曲率自适应简化。在基于相似性的简化方法中,通过法向张量投票方法将点模型分为强特征边性和非强特征边性两部分,利用Mean Shift聚类法对非强特征边性部分进行表面区域几何特征相似性聚类,然后分别进行简化。
●基于球面参数化,提出了一种鲁棒的点模型渐变算法。通过球面参数化将源和目标模型嵌入到单位球面上,在球面上自适应地对齐模型间的相应特征点之后,将球面映射到同一矩形参数域上,以建立模型间各采样点的对应关系。在渐变过程中,采用拉普拉斯算子求解中间点模型以保持模型的细节,利用MLS进行动态自适应的重采样以消除中间模型的裂缝。
●介绍了我们研发的点模型数字几何处理原型系统。该系统提供了一个包括数据获取、表示、处理以及绘制等技术在内的数字几何处理框架,并着重介绍和实现了点模型的绘制、点元预处理、点模型的去噪、几何修复、简化和重采样、参数化、渐变、点模型的纹理映射和纹理合成等功能模块。
点模型数字几何处理有着广阔的研究前景,基于我们研发的原型系统,本文最后给出了未来的研究方向。
The traditional geometry representation, such as polygonal meshes, needs to store and maintain the connectivity information between vertices as well as the geometry information. They are both memory consuming and computationally expensive. Due to convenient 3D point data acquirement, simple data structure and no need to maintain the globally consistent topology, however, point-sampled geometry is receiving a growing amount of attention as a new type of medium and surface presentation in Computer Graphics. Accordingly, considerable research has been devoted to the field of digital geometry processing of point-sampled models. Point-sampled geometry has thus been applied to many fields, such as computer-aided medical diagnosis, digital entertainment, industrial design, aviation simulation, protection and restoration of cultural relics, etc.
Based on the sound theoretical foundation on computer graphics, computer-aidedgeometric design, discrete differential geometry, digital signal processing, thisdissertation investigates some key research area for the digital geometry processing ofpoint-sampled models, and proposes a number of novel algorithms on them. At thesame time, we have presented a nearly complete geometry processing framework forpoint-sample geometry. Our main contributions focus on the following five aspects:
·Based on sampling likelihood, a robust denoising algorithm for point-sampledsurfaces is proposed. In terms of moving least squares surface, the samplinglikelihood for each point on point-sampled surfaces is computed. Based on thenormal tensor voting, the feature intensity of sample point is evaluated. Incombination with sampling likelihood and feature intensity, the point-sampledgeometry can be efficiently smoothed while preserving the surface features. Basedon the similarities including geometry intensity and features of sample points, anon-local denoising algorithm for point-sampled surfaces is presented. By usingthe trilateral filtering operator, the geometry intensity of sample point isdetermined. According to their regular grids of geometry intensity, the similarity of geometry intensity between points is measured. Base on Mean Shift clustering, the point-sampled surfaces are clustered into clusters according to the surface-features similarity. By the similarities, the filtered point-sampled geometry with the fine features is attainted.
·A method is presented for patching holes on point-sampled model and synthesizing surface with details. By the surface reconstruction based on radial basis functions, the hole is patched with a smooth surface. Using the trilateral filtering operator, the geometry textures of the existing sample points are produced. As a result, the geometric details on the smooth completed patch are generated by optimizing a constrained global texture energy function on the point-sampled surfaces.
·Based on geometry images, an adaptive curvature simplification method for point-sampled model is proposed. By projecting its spherical polar coordinates onto a plane, the point-sampled model is represented as geometry images. The simple density is defined as the maximum search radius on geometry images in order to conveniently control it. In combination with the surface variation and simple density, the point-sampled surfaces are simplified. Another simplification algorithm is based on similarity including strong feature-edge intensity and surface feature anisotropy. Using the normal tensor voting, the point-sampled surfaces are segmented into two parts, one for the strong feature-edge intensity and another for the nonstrong feature-edge intensity. Base on Mean Shift clustering, the second part is then clustered into clusters according to the surface-features similarity. The first part and all the clusters are respectively simplified.
·Based on spherical parameterization, a robust morphing algorithm for point-sampled geometry is presented. Source and target models represented by point-sampled geometry are first parameterized onto a sphere, respectively. After aligning the corresponding features of two models on their spheres, two spheres are projected onto a common rectangle-parameter domain and the correspondence between sample points on the two models is built using this rectangle domain. In order to preserve the geometric details of point set surfaces, the absolute geometry of the in-between model is computed by means of Laplacian operator and its surfaces are dynamically up-sampled using a moving least squares method so as to eliminate the cracks.
·A prototype system for digital geometry processing of point-sampled models is implemented and introduced. The system actually is a framework of 3D geometry acquisition, representation, processing and rendering. The main components of the system include various novel algorithms on rendering, estimation of local surface differentials, denoising, geometry completion, simplification and up-sampling, parameterization, morphing, texture mapping and synthesis.
As a new type of media data, the point-sampled geometry is an extreme promising presentation for 3D surface, and so we discuss some directions for future work in the last chapter.
引文
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