Parabolic curves of evolving surfaces

In this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1-parameter family. We concentrate on the parabolic set and its image under the Gauss map, but the same techniques also classify the changes in the dual of the surface. All these have significance for computer vision, for example through their connection with specularities and apparent contours. With the aid of our complete classification, which includes all the phenomena associated with multi-contact tangent planes as well as those associated with parabolic sets, we re-examine examples given by J. Koenderink in his book (1990) under the title of Morphological Scripts.We also explain some of the connections between parabolic sets and ridges of a surface, where principal curvatures achieve turning values along lines of curvature.The point of view taken is the analysis of the contact between surfaces and their tangent planes. A systematic investigation of this yields the results using singularity theory. The mathematical details are suppressed here and appear in Bruce et al. (1993).The third author was supported by the Esprit grant VIVA while this paper was in preparation.     

Anisotropic mesh adaptation for evolving triangulated surfaces

Dynamic surfaces arise in many applications, such as free surfaces in multiphase flows and moving interfaces in fluid–solid interaction. In many engineering applications, an explicit surface triangulation is often used to represent dynamic surfaces, posing significant challenges in adapting their meshes, especially if large curvatures and sharp features may dynamically emerge or vanish as the surfaces evolve. In this paper, we present an anisotropic mesh adaptation technique to meet these challenges. Our technique strives for optimal aspect ratios of the triangulation to reduce positional errors and to capture geometric features of dynamic surfaces based on a novel extension of the quadrics. Our adaptation algorithm combines the operations of vertex redistribution, edge flipping, edge contraction, and edge splitting. Experimental results demonstrate the effectiveness of our anisotropic adaptation technique for static and dynamic surfaces.     

An Eulerian approach to transport and diffusion on evolving implicit surfaces

In this article we define a level set method for a scalar conservation law with a diffusive flux on an evolving hypersurface Γ(t) contained in a domain W ì \mathbb Rⁿ⁺¹{\Omega \subset \mathbb R^{n+1}} . The partial differential equation is solved on all level set surfaces of a prescribed time dependent function Φ whose zero level set is Γ(t). The key idea lies in formulating an appropriate weak form of the conservation law with respect to time and space. A major advantage of this approach is that it avoids the numerical evaluation of curvature. The resulting equation is then solved in one dimension higher but can be solved on a fixed grid. In particular we formulate an Eulerian transport and diffusion equation on evolving implicit surfaces. Using Eulerian surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations. The finite element method is applied to the weak form of the conservation equation yielding an Eulerian Evolving Surface Finite Element Method. The computation of the mass and element stiffness matrices, depending only on the gradient of the level set function, are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.     

Octree-based, GPU implementation of a continuous cellular automaton for the simulation of complex, evolving surfaces

Presently, dynamic surface-based models are required to contain increasingly larger numbers of points and to propagate them over longer time periods. For large numbers of surface points, the octree data structure can be used as a balance between low memory occupation and relatively rapid access to the stored data. For evolution rules that depend on neighborhood states, extended simulation periods can be obtained by using simplified atomistic propagation models, such as the Cellular Automata (CA). This method, however, has an intrinsic parallel updating nature and the corresponding simulations are highly inefficient when performed on classical Central Processing Units (CPUs), which are designed for the sequential execution of tasks. In this paper, a series of guidelines is presented for the efficient adaptation of octree-based, CA simulations of complex, evolving surfaces into massively parallel computing hardware. A Graphics Processing Unit (GPU) is used as a cost-efficient example of the parallel architectures. For the actual simulations, we consider the surface propagation during anisotropic wet chemical etching of silicon as a computationally challenging process with a wide-spread use in microengineering applications. A continuous CA model that is intrinsically parallel in nature is used for the time evolution. Our study strongly indicates that parallel computations of dynamically evolving surfaces simulated using CA methods are significantly benefited by the incorporation of octrees as support data structures, substantially decreasing the overall computational time and memory usage.     

Constructing PDE-based surfaces bounded by geodesics or lines of curvature

In order to explore a new approach to construct surfaces bounded by geodesics or lines of curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First, the corresponding biharmonic surface can naturally be derived under some degenerative conditions; second, the parameters in the PDE implicate some physical meaning, such as elasticity or rigidity; third, there are only a few parameters that need to be evaluated, and hence the computation is simple. In addition, this paper constructs two adjacent surfaces with C¹ continuity whose common boundary is the same given curve as well as respective geodesic (or line of curvature). Examples show that this method to construct PDE-based surfaces bounded by geodesics or lines of curvature is easy and effective.     

Differential geometry properties of lines of curvature of parametric surfaces and their visualization

We present algorithms for computing the differential geometry properties of lines of curvature of parametric surfaces. We derive a unit tangent vector, curvature vector, binormal vector, torsion, and algorithms to evaluate their higher-order derivatives of lines of curvature of parametric surfaces. Among these quantities, it is shown that the curvature and its first derivative of the lines of curvature lend a hand for the formation of curved plates in shipbuilding. We also visualize the twist of lines of curvature, which enables us to observe how much the osculating plane of the line of curvature turns about the tangent vector.     

Properties of Ridges and Cores for Two-Dimensional Images

Pizer and Eberly introduced the core as the analogue of the medial axis for greyscale images. For two-dimensional images, it is obtained as the ridge of a medial function defined on 2 + 1-dimensional scale space. The medial function is defined using Gaussian blurring and measures the extent to which a point is in the center of the object measured at a scale. Numerical calculations indicate the core has properties quite different from the medial axis. In this paper we give the generic properties of ridges and cores for two-dimensional images and explain the discrepancy between core and medial axis properties. We place cores in a larger relative critical set structure, which coherently relates disjoint pieces of core. We also give the generic transitions which occur for sequences of images varying with a parameter such as time. The genericity implies the stability of the full structure in any compact viewing area of scale space under sufficiently small L2 perturbations of the image intensity function. We indicate consequences for finding cores and also for adding markings to completely determine the structure of the medial function.     

Terrain modelling with B-spline type surfaces defined on curved knot lines

B-spline type surfaces on curved knot lines are useful for interpolation of scattered data while preserving desirable terrain breaks or discontinuities which are not necessarily parallel to a given straight line. This paper considers data scattered along parallel scan profiles. It is shown how the large composite linear system which arises for curved knot lines in two directions may be decomposed into smaller linear systems with subsequent savings in storage and processing time.     

An algorithm for finding the intersection lines between two B-spline surfaces

A divide-and-conquer algorithm is presented for finding all the intersection lines between two B-spline surfaces. Each surface is organized as an $\text{n-}\text{branch}$ tree. For each intersection line, an initial point is detected after a depth-first search along one tree, ie the host tree. Extrapolation methods are then used to trace the entire length of the line, thus the line appears naturally in a continuous form. Efficiency of the algorithm is achieved by the employment of an adaptive division strategy and by the careful choice of the representation basis of the patches on both surfaces.     

Implicitization,parameterization and singularity computation of Steiner surfaces using moving surfaces

A Steiner surface is a quadratically parameterizable surface without base points. To make Steiner surfaces more applicable in Computer Aided Geometric Design and Geometric Modeling, this paper discusses implicitization, parameterization and singularity computation of Steiner surfaces using the moving surface technique. For implicitization, we prove that there exist two linearly independent moving planes with total degree one in the parametric variables. From this fact, the implicit equation of a Steiner surface can be expressed as a 3×3 determinant. The inversion formula and singularities for the Steiner surface can also be easily computed from the moving planes. For parameterization, we first compute the singularities of a Steiner surface in implicit form. Based on the singularities, we can find some special moving planes, from which a quadratic parameterization of the Steiner surface can be retrieved.     

基于MOIA的脉搏波峰值并行搜索的研究

首先介绍了免疫算法以及无创血压的测量原理,提出了将多目标优化免疫算法（MOIA）应用于识别脉搏波峰值的思想,并详细研究了针对血压测量免疫算法的亲和度计算、抗体浓度控制和克隆选择、交叉及变异操作的方法,分析指出了免疫算法在血压测量中应用的优点。     

基于脊线跟踪的指纹细节特征提取

提出了一种基于脊线跟踪的指纹细节特征提取方法。该方法用脊线跟踪的方法来寻找和验证特征点,用相对于中心点的位置来定位特征点,因此能排除按捺指纹时产生的因为图像旋转、扭曲和变形造成的干扰,大大提高了指纹识别时的精度。     

A study on shuffle, stopwatches and independently evolving clocks

We show that stopwatch automata are equivalent with timed shuffle expressions, an extension of timed regular expressions with the shuffle operation. Since the emptiness problem is undecidable for stopwatch automata, and hence also for timed shuffle expressions, we introduce a decidable subclass of stopwatch automata called partitioned stopwatch automata. We give for this class an equivalent subclass of timed shuffle expressions and investigate closure properties by showing that partitioned stopwatch automata are closed under union, concatenation, star, shuffle and renaming, but not under intersection. We also show that partitioned stopwatch automata are equivalent with distributed time-asynchronous automata, which are asynchronous compositions of timed automata in which time may evolve independently.     

On-line trust: concepts, evolving themes, a model

Trust is emerging as a key element of success in the on-line environment. Although considerable research on trust in the offline world has been performed, to date empirical study of on-line trust has been limited. This paper examines on-line trust, specifically trust between people and informational or transactional websites. It begins by analysing the definitions of trust in previous offline and on-line research. The relevant dimensions of trust for an on-line context are identified, and a definition of trust between people and informational or transactional websites is presented. We then turn to an examination of the causes of on-line trust. Relevant findings in the human–computer interaction literature are identified. A model of on-line trust between users and websites is presented. The model identifies three perceptual factors that impact on-line trust: perception of credibility, ease of use and risk. The model is discussed in detail and suggestions for future applications of the model are presented.     

The evolving tree-analysis and applications

In this paper, we enhance and analyze the Evolving Tree (ETree) data analysis algorithm. The suggested improvements aim to make the system perform better while still maintaining the simple nature of the basic algorithm. We also examine the system's behavior with many different kinds of tests, measurements and visualizations. We compare the ETree's performance against classical data analysis methods and very similar modern systems. We find that the ETree is a suitable method for unsupervised analysis of huge data sets.     

抽取时频分布图像脊-Hough变换算法及其性能分析

在分析了双线性时频分布-Hough变换算法对多分量Chirp信号检测与参数估计原理的存在问题后，提出了将多分量Chirp信号的双线性时频分布作为图像，抽取图像脊后，再进行Hough变换的改进方法。仿真试验结果表明，改进的方法不仅解决了WVD的交叉项引起Hough变换的伪尖峰对检测所造成的误判，而且缩短了双线性时频分布因平滑交叉项导致时频聚集性下降而引起的Hough变换增加的时间，从而提高了Chirp信号检测与参数估计的准确性，也进一步改善了该算法的抗噪声性能。     

基于脊线跟踪的指纹核心点检测

准确、可靠地检测指纹核心点对于指纹的分类和匹配有重要的意义。针对指纹图像核心点提取中准确判断和精确定位的难题,介绍了一种比较好的核心点检测算法。根据核心点是纹线曲率最大的点的定义,利用脊线跟踪的方法求出核心点。实验结果证明该方法能够从指纹图像中较精确、可靠地提取出核心点。并具有较好的鲁棒性。     

Self-intersections of offsets of quadratic surfaces: Part II, implicit surfaces

The paper investigates self-intersections of offsets of implicit quadratic surfaces. The quadratic surfaces are the simplest curved objects, referred to as quadrics, and are widely used in mechanical design. In an earlier paper, we have investigated the self-intersections of offsets of explicit quadratic surfaces, such as elliptic paraboloid, hyperbolic paraboloid and parabolic cylinder, since not only are they used in mechanical design, but also any regular surface can be locally approximated by such explicit quadratic surfaces. In this paper, we investigate the rest of the quadrics whose offsets may degenerate, i.e. the implicit quadratic-surfaces (ellipsoid, hyperboloid, elliptic cone, elliptic cylinder and hyperbolic cylinder). We found that self-intersection curves of offsets of all the implicit quadratic surfaces are planar implicit conics and their corresponding curve on the progenitor surface can be expressed as the intersection curve between an ellipsoid, whose semi-axes are proportional to the offset distance, and the implicit quadratic surfaces themselves.