Optimal mass transport for geometric modeling based on variational principles in convex geometry

In geometric modeling, surface parameterization plays an important role for converting triangle meshes to spline surfaces. Parameterization will introduce distortions. Conventional parameterization methods emphasize on angle-preservation, which may induce huge area distortions and cause large spline fitting errors and trigger numerical instabilities.To overcome this difficulty, this work proposes a novel area-preserving parameterization method, which is based on an optimal mass transport theory and convex geometry. Optimal mass transport mapping is measure-preserving and minimizes the transportation cost. According to Brenier’s theorem, for quadratic distance transportation costs, the optimal mass transport map is the gradient of a convex function. The graph of the convex function is a convex polyhedron with prescribed normal and areas. The existence and the uniqueness of such a polyhedron have been proved by the Minkowski-Alexandrov theorem in convex geometry. This work gives an explicit method to construct such a polyhedron based on the variational principle, and formulates the solution to the optimal transport map as the unique optimum of a convex energy. In practice, the energy optimization can be carried out using Newton’s method, and each iteration constructs a power Voronoi diagram dynamically. We tested the proposal algorithms on 3D surfaces scanned from real life. Experimental results demonstrate the efficiency and efficacy of the proposed variational approach for the optimal transport map.     

As-rigid-as-possible spherical parametrization

In this paper, we present an efficient approach for parameterizing a genus-zero triangular mesh onto the sphere with an optimal radius in an as-rigid-as-possible (ARAP) manner, which is an extension of planar ARAP parametrization approach to spherical domain. We analyze the smooth and discrete ARAP energy and formulate our spherical parametrization energy from the discrete ARAP energy. The solution is non-trivial as the energy involves a large system of non-linear equations with additional spherical constraints. To this end, we propose a two-step iterative algorithm. In the first step, we adopt a local/global iterative scheme to calculate the parametrization coordinates. In the second step, we optimize a best approximate sphere on which parametrization triangles can be embedded in a rigidity-preserving manner. Our algorithm is simple, robust, and efficient. Experimental results show that our approach provides almost isometric spherical parametrizations with lowest rigidity distortion over state-of-the-art approaches.     

Optimal surface parameterization using inverse curvature map

Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.     

Content-aware texture mapping

Interactively mapping texture into a curved surface has wide applications. Apart from mapping texture to desired positions with low distortions, which has been considered by existing works, few work considered the intuitive need to distribute distortions according to the texture contents. In this work, we present a texture mapping method guided by importance map to preserve the shape of the prominent content. We formulate it as an importance-value-weighted parameterization. The mapping distortions are measured by LSCM+ energy, which is capable of decreasing the appearance of shrunken and fold-over triangles as well as shape preservation; and the weights are efficiently calculated by transforming the area integral into a line integral. To solve the parameterization, we employ the ‘L–M’ (Levenberg–Marquardt) method and alternately update the weights and the coordinates since they are dependent on each other. Finally, we show some examples to demonstrate the content-aware effect by comparison to traditional parameterization.     

Approximate straightest path computation and its application in parameterization

This paper proposes an approaching method to compute the straightest path between two vertices on meshes. An initial cutting plane is first constructed using the normal information of the source and destination vertices. Then an optimal cutting plane is iteratively created by comparing with previous path distance. Our study shows that the final straightest path based on this optimal cutting plane is more accurate and insensitive to the mesh boundary. Furthermore, we apply the straightest path result to compute the measured boundary in the parameter domain for mesh parameterization, and we obtain a new computing formula for vertex stretch in the planar parameterization. Experimental results show that our parameterization method can effectively reduce distortions.     

Rigidity controllable as-rigid-as-possible shape deformation

Shape deformation is one of the fundamental techniques in geometric processing. One principle of deformation is to preserve the geometric details while distributing the necessary distortions uniformly. To achieve this, state-of-the-art techniques deform shapes in a locally as-rigid-as-possible (ARAP) manner. Existing ARAP deformation methods optimize rigid transformations in the 1-ring neighborhoods and maintain the consistency between adjacent pairs of rigid transformations by single overlapping edges. In this paper, we make one step further and propose to use larger local neighborhoods to enhance the consistency of adjacent rigid transformations. This is helpful to keep the geometric details better and distribute the distortions more uniformly. Moreover, the size of the expanded local neighborhoods provides an intuitive parameter to adjust physical stiffness. The larger the neighborhood is, the more rigid the material is. Based on these, we propose a novel rigidity controllable mesh deformation method where shape rigidity can be flexibly adjusted. The size of the local neighborhoods can be learned from datasets of deforming objects automatically or specified by the user, and may vary over the surface to simulate shapes composed of mixed materials. Various examples are provided to demonstrate the effectiveness of our method.     

基于遗传算法的B样条曲线和Bézier曲线的最小二乘拟合

考虑用B样条曲线拟合平面有序数据使得最小二乘拟合误差最小.一般有两种考虑,一种是保持B样条基函数的节点不变,选择参数使得拟合较优.参数的选择方法包括均匀取值、累加弦长法、centripetal model、Gauss-Newton迭代法等.另一种则是先确定好参数值(一般用累加弦长法),然后再用.某一算法计算出节点,使得拟合较优.同时把两者统一考虑,用遗传算法同时求出参数、节点使得拟合在最小二乘误差意义下最优.与Gauss-Newton迭代法、Piegl算法相比,本方法具有较好的鲁棒性(拟合曲线与初始值无关)、较高的精度及控制顶点少等优点.实验结果说明采用遗传算法得到的曲线逼近效果更好.用遗传算法对Bezier曲线拟合平面有序数据也进行了研究.     

ARAP Revisited Discretizing the Elastic Energy using Intrinsic Voronoi Cells

As-rigid-as-possible (ARAP) surface modelling is widely used for interactive deformation of triangle meshes. We show that ARAP can be interpreted as minimizing a discretization of an elastic energy based on non-conforming elements defined over dual orthogonal cells of the mesh. Using the intrinsic Voronoi cells rather than an orthogonal dual of the extrinsic mesh guarantees that the energy is non-negative over each cell. We represent the intrinsic Delaunay edges extrinsically as polylines over the mesh, encoded in barycentric coordinates relative to the mesh vertices. This modification of the original ARAP energy, which we term iARAP, remedies problems stemming from non-Delaunay edges in the original approach. Unlike the spokes-and-rims version of the ARAP approach it is less susceptible to the triangulation of the surface. We provide examples of deformations generated with iARAP and contrast them with other versions of ARAP. We also discuss the properties of the Laplace-Beltrami operator implicitly introduced with the new discretization.     

Template-based 3D model fitting using dual-domain relaxation

We introduce a template fitting method for 3D surface meshes. A given template mesh is deformed to closely approximate the input 3D geometry. The connectivity of the deformed template model is automatically adjusted to facilitate the geometric fitting and to ascertain high quality of the mesh elements. The template fitting process utilizes a specially tailored Laplacian processing framework, where in the first, coarse fitting stage we approximate the input geometry with a linearized biharmonic surface (a variant of LS-mesh), and then the fine geometric detail is fitted further using iterative Laplacian editing with reliable correspondence constraints and a local surface flattening mechanism to avoid foldovers. The latter step is performed in the dual mesh domain, which is shown to encourage near-equilateral mesh elements and significantly reduces the occurrence of triangle foldovers, a well-known problem in mesh fitting. To experimentally evaluate our approach, we compare our method with relevant state-of-the-art techniques and confirm significant improvements of results. In addition, we demonstrate the usefulness of our approach to the application of consistent surface parameterization (also known as cross-parameterization).     

带尖锐特征的细分曲线参数化及曲线拟合

快速精确地估计曲线曲面参数具有广泛的应用。在前人研究的基础上,通过对细分过程及三次B样条细分矩阵的特征结构进行分析,将细分模式转换到其特征空间,给出了带尖锐特征的B样条细分曲线的参数化形式。并用于处理带尖锐特征的光滑曲线拟合问题。以曲率极大点作为初始拟合点。利用推导的参数化公式构造曲线的尖锐部分并方便误差估计。拟合点为曲线段端点,误差估计时不仅优化计算速度,而且在曲线分支距离过近或自交情况下避免错误匹配。     

Robust mesh editing using Laplacian coordinates

Shape deformation and editing are important for animation and game design. Laplacian surface based methods have been widely investigated and used in many works. In this paper we propose a robust mesh editing framework which improves traditional Laplacian surface editing. It consists of two procedures: skeleton based as-rigid-as-possible (ARAP) shape modeling and detail-preserving mesh optimization. Traditional ARAP shape modeling relies on the mesh quality. Degenerated mesh may adversely affect the deformation performance. A preprocessing step of mesh optimization can alleviate this problem. However, skinny triangles can still be generated during deformation, which adversely affect the editing performance. Thus our method performs Laplacian mesh deformation and optimization alternately in each iteration, which ensures mesh quality without noticeably increasing computational complexity or changing the shape details. This approach is more robust than those solely using Laplacian mesh deformation. An additional benefit is that the skeleton-based ARAP modeling can approximately preserve the volume of an object with large-scale deformations. The volume is roughly kept by leveraging the skeleton information and employing a carefully designed energy function to preserve the edge length. This method does not break the manifoldness of traditional ARAP methods or sacrifice speed. In our experiments, we show that (1) our method is robust even for degenerated meshes, (2) the deformation is natural in terms of recovering rotations, and (3) volumes are roughly kept even under large-scale deformations. The system achieves real time performance for surface meshes with 7k vertices.     

Blind image quality assessment with improved natural scene statistics model

No-reference (NR)/blind image quality assessment (IQA) metrics play an important role in the area of image processing. Natural scene statistics (NSS) model assumes that natural images possess certain regular statistical properties and is widely used in NR IQA metrics. Most existing NSS-based NR algorithms are achieved by measuring the variation of image statistics, which are characterized by the fitting parameters of NSS model, across different distortions. However, distortions not only change the image statistics, but also disturb the statistical regularity held by natural images. As a result, the distribution of distorted images can not well follow the NSS model. There exists fitting error between the real distribution of the distorted image and the fitted one under certain NSS model. In this paper, the statistical distributions of the distorted images are discussed in detail. We suggest to take the fitting errors into account as well as the fitting parameters for feature extraction, and propose a novel NR IQA algorithm. Experimental results on several image databases demonstrate that the proposed metric performs highly consistent with human visual perception.     

基于GIMT和弧长参数化的QG-Ball曲线近似合并

曲线近似合并作为 CAGD 中复杂曲线设计的一种有效技术,一直备受学者们的关注,并在 CAD/CAM 领域得到了广泛的应用。针对现有带形状参数的广义 Ball 曲线难以合并的问题,提出了一种基于广 义逆矩阵理论(GIMT)和弧长参数化的 QG-Ball 曲线近似合并方法。首先,利用曲线近似弧长参数化算法计算出 QG-Ball 曲线弧长等分对应的配置点列(亦称等分点)和配置点参数值;其次,基于所得等弧长配置点列及其参 数值,再结合广义逆矩阵理论和曲线拟合方法,便可以直接得到计算合并后 QG-Ball 曲线控制顶点的一个显式 表达式;最后,利用连续函数的 L2 范数定义了一个度量曲线合并效果的误差计算公式,并给出了一些具有代 表性的数值算例及其合并误差。实例结果表明,所提出的方法可以高效地实现 QG-Ball 曲线的近似合并,不仅 易于操作、误差计算简单,而且能方便地推广到其他曲线的近似合并。     

Parameterization‐Aware MIP‐Mapping

We present a method of generating mipmaps that takes into account the distortions due to the parameterization of a surface. Existing algorithms for generating mipmaps assume that the texture is isometrically mapped to the surface and ignore the actual surface parameterization. Our method correctly downsamples warped textures by assigning texels weights proportional to their area on a surface. We also provide a least‐squares approach to filtering over these warped domains that takes into account the postfilter used by the GPU. Our method improves texture filtering for most models but only modifies mipmap generation, requires no modification of art assets or rasterization algorithms, and does not affect run‐time performance.     

Parameterization and reconstruction from 3D scattered points basedon neural network and PDE techniques

Reverse engineering ordinarily uses laser scanners since they can sample 3D data quickly and accurately relative to other systems. These laser scanner systems, however, yield an enormous amount of irregular and scattered digitized point data that requires intensive reconstruction processing. Reconstruction of freeform objects consists of two main stages: parameterization and surface fitting. Selection of an appropriate parameterization is essential for topology reconstruction as well as surface fitness. Current parameterization methods have topological problems that lead to undesired surface fitting results, such as noisy self-intersecting surfaces. Such problems are particularly common with concave shapes whose parametric grid is self-intersecting, resulting in a fitted surface that considerably twists and changes its original shape. In such cases, other parameterization approaches should be used in order to guarantee non-self-intersecting behavior. The parameterization method described in this paper is based on two stages: 2D initial parameterization; and 3D adaptive parameterization. Two methods were developed for the first stage: partial differential equation (PDE) parameterization and neural network self organizing maps (SOM) parameterization. The Gradient Descent Algorithm (GDA) and Random Surface Error Correction (RSEC), both of which are iterative surface fitting methods, were developed and implemented     

Enhancement of spatially adaptive smoothing splines via parameterization of smoothing parameters

This paper considers the problem of estimating curve and surface functions when the structures of an unknown function vary spatially. Classical approaches such as using smoothing splines, which are controlled by a single smoothing parameter, are inefficient in estimating the underlying function that consists of different spatial structures. In this paper, we propose a blockwise method of fitting smoothing splines wherein the smoothing parameter λ varies spatially, in order to accommodate possible spatial nonhomogeneity of the regression function. A key feature of the proposed blockwise method is the parameterization of a smoothing parameter function λ(x) that produces a continuous spatially adaptive fit over the entire range of design points. The proposed parameterization requires two important ingredients: (1) a blocking scheme that divides the data into several blocks according to the degree of spatial variation of the data; and (2) a method for choosing smoothing parameters of blocks. We propose a block selection approach that is based on the adaptive thinning algorithm and a choice of smoothing parameters that minimize a newly defined blockwise risk. The results obtained from numerical experiments validate the effectiveness of the proposed method.     

Image distortion analysis using polynomial series expansion

In this paper, we derive a technique for analysis of local distortions which affect data in real-world applications. In the paper, we focus on image data, specifically handwritten characters. Given a reference image and a distorted copy of it, the method is able to efficiently determine the rotations, translations, scaling, and any other distortions that have been applied. Because the method is robust, it is also able to estimate distortions for two unrelated images, thus determining the distortions that would be required to cause the two images to resemble each other. The approach is based on a polynomial series expansion using matrix powers of linear transformation matrices. The technique has applications in pattern recognition in the presence of distortions.     

Iteration and optimization scheme for the reconstruction of 3D surfaces based on non-uniform rational B-splines

Curve or surface reconstruction is a challenging problem in the fields of engineering design, virtual reality, film making and data visualization. Non-uniform rational B-spline (NURBS) fitting has been applied to curve and surface reconstruction for many years because it is a flexible method and can be used to build many complex mathematical models, unlike certain other methods. To apply NURBS fitting, there are two major difficult sub-problems that must be solved: (1) the determination of a knot vector and (2) the computation of weights and the parameterization of data points. These two problems are quite challenging and determine the effectiveness of the overall NURBS fit. In this study, we propose a new method, which is a combination of a hybrid optimization algorithm and an iterative scheme (with the acronym HOAAI), to address these difficulties. The novelties of our proposed method are the following: (1) it introduces a projected optimization algorithm for optimizing the weights and the parameterization of the data points, (2) it provides an iterative scheme to determine the knot vectors, which is based on the calculated point parameterization, and (3) it proposes the boundary-determined parameterization and the partition-based parameterization for unorganized points. We conduct numerical experiments to measure the performance of the proposed HOAAI with six test problems, including a complicated curve, twisted and singular surfaces, unorganized data points and, most importantly, real measured data points from the Mashan Pumped Storage Power Station in China. The simulation results show that the proposed HOAAI is very fast, effective and robust against noise. Furthermore, a comparison with other approaches indicates that the HOAAI is competitive in terms of both accuracy and runtime costs.     

Local matrix adaptation in topographic neural maps

The self-organizing map (SOM) and neural gas (NG) and generalizations thereof such as the generative topographic map constitute popular algorithms to represent data by means of prototypes arranged on a (hopefully) topology representing map. Most standard methods rely on the Euclidean metric, hence the resulting clusters tend to have isotropic form and they cannot account for local distortions or correlations of data. For this reason, several proposals exist in the literature which extend prototype-based clustering towards more general models which, for example, incorporate local principal directions into the winner computation. This allows to represent data faithfully using less prototypes. In this contribution, we establish a link of models which rely on local principal components (PCA), matrix learning, and a formal cost function of NG and SOM which allows to show convergence of the algorithm. For this purpose, we consider an extension of prototype-based clustering algorithms such as NG and SOM towards a more general metric which is given by a full adaptive matrix such that ellipsoidal clusters are accounted for. The approach is derived from a natural extension of the standard cost functions of NG and SOM (in the form of Heskes). We obtain batch optimization learning rules for prototype and matrix adaptation based on these generalized cost functions and we show convergence of the algorithm. The batch optimization schemes can be interpreted as local principal component analysis (PCA) and the local eigenvectors correspond to the main axes of the ellipsoidal clusters. Thus, this approach provides a cost function associated to proposals in the literature which combine SOM or NG with local PCA models. We demonstrate the behavior of matrix NG and SOM in several benchmark examples and in an application to image compression.     

基于匹配与平差的景象镶嵌方法

在概括介绍消附相邻景象辐射差异常用算法的基础上,为有效消除相邻景象间几何位置的差异,提出了一种将影象匹配与平差理论相结合的影象镶嵌新算法,该方法是在镶嵌影象的重叠部分进行基于特征的影象匹配,即首先获取一序列同名点,且这些同名点的匹配精度在一个象元以内;然后用正形多项式模拟这些同名点间的几何位置差异,再按最小二乘法原理对右边影象进行平差改正,以实现两影象空间位置上的无缝连接。实验结果表明,庐镶嵌方法具有良好的效果。