Freeform Shape Representations for Efficient Geometry Processing

The most important concepts for the handling and storage of freeform shapes in geometry processing applications are parametric representations and volumetric representations. Both have their specific advantages and drawbacks. While the algebraic complexity of volumetric representations is independent from the shape complexity, the domain of a parametric representation usually has to have the same structure as the surface itself (which sometimes makes it necessary to update the domain when the surface is modified). On the other hand, the topology of a parametrically defined surface can be controlled explicitly while in a volumetric representation, the surface topology can change accidentally during deformation. A volumetric representation reduces distance queries or inside/outside tests to mere function evaluations but the geodesic neighborhood relation between surface points is difficult to resolve. As a consequence, it seems promising to combine parametric and volumetric representations to effectively exploit both advantages. In this talk, a number of projects are presented and discussed in which such a combination leads to efficient and numerically stable algorithms for the solution of various geometry processing tasks. Applications include global error control for mesh decimation and smoothing, topology control for level‐set surfaces, and shape modeling with unstructured point clouds.     

A Framework for Synchronized Editing of Multiple Curve Representations

Editing curves and surfaces is difficult in part because their mathematical representations rarely correspond to most people's idea of a curve or surface. The implementation (and hence, behavior) of most manipulation tools is intertwined with a particular curve or surface representation; this can make reimplementing the tool with a different representation problematic. A system using a single representation must therefore either limit the types of tools available or convert existing tools to work on the system's representation.
In this paper we present a framework for editing curves or surfaces which supports multiple representations and ensures that they stay synchronized. As a proof of concept, we have created a curve editor which contains several tools each of which manipulate one of three different curve representations: polylines, NURBs, and multi-resolution B-splines.     

On non-commutative operator graphs generated by covariant resolutions of identity

We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.     

NP-hardness of the stable matrix in unit interval family problem in discrete time

We show that to determine if a family of matrices, each with parameters in the unit interval, contains a matrix with all eigenvalues inside the unit circle is an NP-hard problem. We also discuss how this problem is closely related to the widespread problem of power control in wireless systems.     

面向“15分钟生活圈”社区结构的表示学习

利用城市大数据发现社区结构是城市计算中重要的研究方向。有效表示面向“15分钟生活圈”社区的结构特征可以细粒度地评价生活圈社区周围的设施情况，有利于城市规划建设，创造宜居的生活环境。首先，定义了面向“15分钟生活圈”的城市社区结构，并采用表示学习方法获取生活圈社区的结构特征；然后，提出了生活圈社区结构的嵌入表示框架，框架中利用居民的出行轨迹数据确定兴趣点（POI）与居民区的关系，构建反映不同时段居民出行规律的动态活动图；最后，对构建的动态活动图采用自编码器进行表示学习得到生活圈社区潜在特征的向量表示，从而有效概括居民日常活动所形成的社区结构。针对生活圈社区便利性评价、相似性度量等应用，利用真实数据集进行了实验评估，结果表明，分POI类别的日周期的潜在表示方法优于星期周期的潜在表示方法，且前者的归一化折损累计增益（NDCG）比后者最少提升了24.28%，最多提升了60.71%，验证了所提方法的有效性。     

Curve-skeleton extraction using iterative least squares optimization

A curve skeleton is a compact representation of 3D objects and has numerous applications. It can be used to describe an object's geometry and topology. In this paper, we introduce a novel approach for computing curve skeletons for volumetric representations of the input models. Our algorithm consists of three major steps: 1) using iterative least squares optimization to shrink models and, at the same time, preserving their geometries and topologies, 2) extracting curve skeletons through the thinning algorithm, and 3) pruning unnecessary branches based on shrinking ratios. The proposed method is less sensitive to noise on the surface of models and can generate smoother skeletons. In addition, our shrinking algorithm requires little computation, since the optimization system can be factorized and stored in the pre-computational step. We demonstrate several extracted skeletons that help evaluate our algorithm. We also experimentally compare the proposed method with other well-known methods. Experimental results show advantages when using our method over other techniques.     

Structure-oriented contour representation and matching for engineering shapes

Conventional shape matching for engineering models primarily considers rigid shape similarity. They do not seek global shape similarity while considering large local deformations. However, engineering models created by some parametric-based design can involve large parametric changes. As a result, they do not share similarity in their global shape. Hence our goal is to develop shape representations for global matching of part models that can have large dissimilarity through stretching and/or bending.This paper presents a strategy of an integrated shape matching for contours of engineering drawings inspired by the divide-and-conquer paradigm. The original shape is decoupled into two levels of shape representations namely, higher-level structure and lower-level geometry. The higher-level structure matching is then achieved driven by optimal integrated solutions from matching of lower-level local geometry. Feature points are first extracted using curve evolution to attain the two levels of representations. In order to suit engineering semantics, a new significance function for a point is defined to suppress small features using discrete curve evolution. To conduct the integrated shape matching, a mechanism of using lookup tables is employed to associate these two levels of representations. Dynamic Time Warping and Elastic Matching are employed at different levels of shape representations in order to achieve the optimal integration.To demonstrate the advantages of the proposed work for engineering shapes, experiments for contour evolution, feature point registration, and shape-based similarity for retrieval are conducted. They are also compared with the existing methods. The experimental results show that the structure-oriented contour representation and matching are more meaningful and consistent from an engineering perspective.     

Symbolic parametrization of curves

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance.We investigate the transformation of an implicit representation of a plane algebraic curve into a parametric representation. Various methods for computing a rational parametrization, if one exists, are described. As a new idea we introduce the concept of working with classes of conjugate (singular or simple) points on curves. All the necessary operations, like determining the multiplicity and the character of the singular points or passing a linear system of curves through these points, can be applied to such classes of conjugate points. Using this idea one can parametrize a curve if one knows only one simple point on it. We do not propose any new method for finding such a simple point. By classical methods a rational point on a rational curve can be computed, if such a point exists. Otherwise, one can express the coordinates of such a point in an algebraic extension of degree 2 over the ground field.     

Point-tangent/point-normal B-spline curve interpolation by geometric algorithms

We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms. The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C² continuity. We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method. Examples are provided to demonstrate the effectiveness of the proposed algorithms.     

Shape evolution with structural and topological changes usingblending

This paper describes a framework for the estimation of shape from sparse or incomplete range data. It uses a shape representation called blending, which allows for the geometric combination of shapes into a unified model - selected regions of the component shapes are cut-out and glued together. Estimation of shape by this representation is realized using a physics-based framework, and it also includes a process for deciding how to adapt the structure and topology of the model to improve the fit. The blending representation helps avoid abrupt changes in model geometry during fitting by allowing the smooth evolution of the shape, which improves the robustness of the technique. We demonstrate this framework with a series of experiments showing its ability to automatically extract structured representations from range data given both structurally and topologically complex objects     

Text-enhanced network representation learning

Network representation learning called NRL for short aims at embedding various networks into lowdimensional continuous distributed vector spaces. Most existing representation learning methods focus on learning representations purely based on the network topology, i.e., the linkage relationships between network nodes, but the nodes in lots of networks may contain rich text features, which are beneficial to network analysis tasks, such as node classification, link prediction and so on. In this paper, we propose a novel network representation learning model, which is named as Text-Enhanced Network Representation Learning called TENR for short, by introducing text features of the nodes to learn more discriminative network representations, which come from joint learning of both the network topology and text features, and include common influencing factors of both parties. In the experiments, we evaluate our proposed method and other baseline methods on the task of node classification. The experimental results demonstrate that our method outperforms other baseline methods on three real-world datasets.     

A Geometric Algorithm to Compute Time-Optimal Trajectories for a Bidirectional Steered Robot

This paper addresses the problem of determining time-optimal trajectories, between two specified configurations, for a nonholonomic bidirectional steered robot. It presents an original geometric reasoning that is grounded on Pontryagin's maximum principle, which provides analytical solutions of this problem in a visually clear way and allows for an effective algorithm to compute the exact optimal trajectories between two arbitrarily specified configurations. The proposed geometric reasoning is based on the analysis of the switching functions of the optimal controller and the definition of a switching vector from which it is able to determine a unit vector rotating along a unit circle of an appropriate coordinate system. It is shown that simple geometric rules are sufficient to determine all possible rotations of this unit vector, from which the time-optimal trajectories can be uniquely determined. The proposed algorithm, which is based on this geometric reasoning, is guaranteed to be complete and has a low computational cost. Moreover, the proposed geometric representation provides an interesting insight into the structure of this class of nonholonomic systems, thereby offering a model for further studies.     

Genotype representations in grammatical evolution

Grammatical evolution (GE) is a form of grammar-based genetic programming. A particular feature of GE is that it adopts a distinction between the genotype and phenotype similar to that which exists in nature by using a grammar to map between the genotype and phenotype. Two variants of genotype representation are found in the literature, namely, binary and integer forms. For the first time we analyse and compare these two representations to determine if one has a performance advantage over the other. As such this study seeks to extend our understanding of GE by examining the impact of different genotypic representations in order to determine whether certain representations, and associated diversity-generation operators, improve GE’s efficiency and effectiveness. Four mutation operators using two different representations, binary and gray code representation, are investigated. The differing combinations of representation and mutation operator are tested on three benchmark problems. The results provide support for the use of an integer-based genotypic representation as the alternative representations do not exhibit better performance, and the integer representation provides a statistically significant advantage on one of the three benchmarks. In addition, a novel wrapping operator for the binary and gray code representations is examined, and it is found that across the three problems examined there is no general trend to recommend the adoption of an alternative wrapping operator. The results also back up earlier findings which support the adoption of wrapping.     

曲线插值的一种具有还圆性的细分方法

传统的线性四点插值细分方法不能表示圆等非多项式曲线,为了解决这种 问题,基于几何特性提出了一种带有一个参数的四点插值型曲线细分方法。细分过程中,过 相邻三插值点作圆,过相邻二插值点的圆弧有两个中点,将其加权平均得到新插值点,文中 给出了插值公式和算法描述。所给方法具有还圆性,可以实现保凸性。实例分析对比了本方 法与多种细分方法的差异,说明本方法是有效的,当参数取值较小时,曲线靠近控制多边形。     

五次PH曲线的Hermite插值

应用复分析和曲线积分方法研究了满足Hermite插值的五次PH曲线的构造,导出了其相应的Bézier表示.所得五次PH插值曲线不但具有连续的单位切矢和有向曲率,而且其弧长函数是原参数的多项式函数,具有精确的有理Offset代数表示和优美的几何解释,可灵活处理拐点.     

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.     

Multiple View Geometry of General Algebraic Curves

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the recovery of camera geometry from matching curves. We first show how one can compute, without any knowledge on the camera, the homography induced by a single planar curve. Then we continue with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus.We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d – 1). Moreover using this representation, we address the problem of homography recovery for planar curves, (ii) dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone.     

Quantum teleportation and Birman–Murakami–Wenzl algebra

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman–Murakami–Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley–Lieb projector and the Yang–Baxter gate. We describe quantum teleportation using the Temperley–Lieb projector and the Yang–Baxter gate, respectively, and study teleportation-based quantum computation using the Yang–Baxter gate. On the other hand, we exploit the extended Temperley–Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.