Generalized shape operators on polyhedral surfaces

This work concerns the approximation of the shape operator of smooth surfaces in R³ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions. The estimates are confirmed by numerical experiments.     

Ridges, crests and sub-parabolic lines of evolving surfaces

The ridge lines on a surface can be defined either via contact of the surface with spheres, or via extrema of principal curvatures along lines of curvature. Certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications. There is a related concept of sub-parabolic line on a surface, also defined via extrema of principal curvatures.In this paper we study in detail the structure of the ridge lines, crest lines and sub-parabolic lines on a generic surface, and on a surface which is evolving in a generic (one-parameter) family. The mathematical details of this study are in Bruce et al. (1994c).Supported by the Esprit grant VIVA.     

Distributed learning and cooperative control for multi-agent systems

This paper presents an algorithm and analysis of distributed learning and cooperative control for a multi-agent system so that a global goal of the overall system can be achieved by locally acting agents. We consider a resource-constrained multi-agent system, in which each agent has limited capabilities in terms of sensing, computation, and communication. The proposed algorithm is executed by each agent independently to estimate an unknown field of interest from noisy measurements and to coordinate multiple agents in a distributed manner to discover peaks of the unknown field. Each mobile agent maintains its own local estimate of the field and updates the estimate using collective measurements from itself and nearby agents. Each agent then moves towards peaks of the field using the gradient of its estimated field while avoiding collision and maintaining communication connectivity. The proposed algorithm is based on a recursive spatial estimation of an unknown field. We show that the closed-loop dynamics of the proposed multi-agent system can be transformed into a form of a stochastic approximation algorithm and prove its convergence using Ljung’s ordinary differential equation (ODE) approach. We also present extensive simulation results supporting our theoretical results.     

Characterizing three-dimensional surface structures from visualimages

A new technique for computing intrinsic surface properties is presented. Intrinsic surface properties are those properties of a surface that are not affected by the choice of the coordinate system, the position of the viewer relative to the surface, and the particular parametric representation used to describe the imaged surface. Since intrinsic properties are characteristics of a surface, they are ideal for the purposes of representation and recognition. The intrinsic properties of interest are the principal curvatures, the Gaussian curvatures, and the lines of curvature. It is proposed that a structured-light sensing configuration where a grid pattern is projected to encode the imaged surfaces for analysis be adopted. At each stripe junction, the curvatures of the projected stripes on the imaged surface are computed and related to those of the normal sections that share the same tangential directional as the projected curves. The principal curvatures and their directions at the stripe junction under consideration are then recovered using Euler's theorem. Results obtained using both synthetic and real images are presented     

Comparing scalar and vector forms of the finite element method by example of an elliptic cylinder

An algorithm for constructing the stiffness matrix of a quadrangular curvilinear finite element in the form of a fragment of the middle surface of an elliptic cylinder with 18 degrees of freedom in the node is given. Implementation of the finite-difference procedure includes two variants of approximation of the sought variables, i.e., a scalar one and a vector one. Numerical examples are given and prove that the vector approximation has principal advantages over the scalar approximation for arbitrary shells with significant gradients of curvature of the lines of the middle surface or allowing for displacement as a rigid body.     

Curvature-Domain Shape Processing

We propose a framework for 3D geometry processing that provides direct access to surface curvature to facilitate advanced shape editing, filtering, and synthesis algorithms. The central idea is to map a given surface to the curvature domain by evaluating its principle curvatures, apply filtering and editing operations to the curvature distribution, and reconstruct the resulting surface using an optimization approach. Our system allows the user to prescribe arbitrary principle curvature values anywhere on the surface. The optimization solves a nonlinear least‐squares problem to find the surface that best matches the desired target curvatures while preserving important properties of the original shape. We demonstrate the effectiveness of this processing metaphor with several applications, including anisotropic smoothing, feature enhancement, and multi‐scale curvature editing.     

Multi-agent source seeking via discrete-time extremum seeking control

Recent developments in extremum seeking theory have established a general framework for the methodology, although the specific implementations, particularly in the context of multi-agent systems, have not been demonstrated. In this work, a group of sensor-enabled vehicles is used in the context of the extremum seeking problem using both local and global optimisation algorithms to locate the extremum of an unknown scalar field distribution. For the former, the extremum seeker exploits estimates of gradients of the field from local dithering sensor measurements collected by the mobile agents. It is assumed that a distributed coordination which ensures uniform asymptotic stability with respect to a prescribed formation of the agents is employed. An inherent advantage of the frameworks is that a broad range of nonlinear programming algorithms can be combined with a wide class of cooperative control laws to perform extreme source seeking. Semi-global practical asymptotically stable convergence to local extrema is established in the presence of field sampling noise. Subsequently, global extremum seeking with multiple agents is investigated and shown to give rise to robust practical convergence whose speed can be improved via computational parallelism. Nonconvex field distributions with local extrema can be accommodated within this global framework.     

Axis estimation and grouping of rotationally symmetric object segments

This paper proposes a new method for estimating the symmetric axis of a pottery from its small fragment using surface geometry. Also, it provides a scheme for grouping such fragments into shape categories using distribution of surface curvature. For automatic assembly of pot from broken sherds, axis estimation is an important task and when a fragment is small, it is difficult to estimate axis orientation since it looks like a patch of a sphere and conventional methods mostly fail. But the proposed method provides fast and robust axis estimation by using multiple constraints. The computational cost is also too lowered. To estimate the symmetric axis, the proposed algorithm uses three constraints: (1) The curvature is constant on a circumference C_H. (2) The curvature is invariant in any scale. (3) Also the principal curvatures does not vary on C_H. C_H is a planar circle which is one of all the possible circumferences of a pottery or sherd. A hypothesis test for axis is performed using maximum likelihood. The variance of curvature, multi-scale curvature and principal curvatures is computed in the likelihood function. We also show that the principal curvatures can be used for grouping of sherds. The grouping of sherds will reduce the computation significantly by omitting impossible configurations in broken pottery assembly process.     

A new feature-preserving mesh-smoothing algorithm

We present a novel mesh denoising and smoothing method in this paper. Our approach starts by estimating the principal curvatures and mesh saliency value for each vertex. Then, we calculate the uniform principal curvature of each vertex based on the weighted average of local principal curvatures. After that, we use the weighted bi-quadratic Bézier surface to fit the neighborhood of each vertex using the least-square method and obtain the new vertex position by adjusting the parameters of the fitting surface. Experiments show that our smoothing method preserves the geometric feature of the original mesh model efficiently. Our approach also prevents the volume shrinkage of the input mesh and obtains smooth boundaries for non-closed mesh models.     

Volumetric Shapes of Solids of Revolution from a Single-View Range Image

Many man-made objects such as industrial parts are partially constructed of surfaces of revolution, as well as planar surfaces. We have studied the problem of finding and recovering solids of revolution in range data which are potentially useful for modeling and recognizing 3D objects. We propose an approach to the problem which is based on the fact that at least one of two focal surfaces for a surface of revolution degenerates into the axis of rotation. First, by computing the surface normal and principal curvatures, the centers of principal curvature which construct the focal surfaces are obtained for each point in the range image. Then, using the Hough transform, the axes of rotation are detected by finding the centers of principal curvature which lie on straight lines in space. Finally, the solid of revolution is completely determined by estimating the radius function of cross-section along each rotational axis. The proposed method can be used even in situations where occlusion or truncation is a problem because it does not require the visibility of entire surfaces. Experiments have been successfully carried out with real range data obtained from laser rangefinders.     

Zero crossings on lines of curvature

We investigate how much information can be found about the geometry of an object from an image when the general form of the reflection function is known but its specific form is not. We prove theorems showing that the zero crossings of the second directional derivatives occur near the extrema of the curvature along the principal directions of curvature. We next rederive and extend results of Koenderink and van Doorn showing that most extrema of the image intensity lie on parabolic lines. We prove that the directions of the isophotes (the lines of constant image intensity) always lie along the directions of curvature at parabolic lines and hence are photometric invariants. We prove that isophotes which are brighter (or dimmer) than their neighbours must necessarily be parabolic lines.     

Representation of Dupin cyclides using quaternions

Dupin cyclides are surfaces characterized by the property that all their curvature lines are circles or lines. Spheres, circular cylinders, cones and tori are particular examples. We introduce a bilinear rational Bézier-like formula with quaternion weights for parametrizing principal patches of Dupin cyclides. The proposed construction is not affine invariant but it is Möbius invariant, has lower degrees compared with the standard representation, and it is convenient for offsetting. Several important properties of Dupin cyclides can be recovered in terms of closed quaternion formulas: implicit equation, principal curvatures, representation as canal surfaces. Advantages of this approach are demonstrated by deriving a new formula for the Willmore energy of a principal patch.     

Extraction of ridge and valley lines from unorganized points

Given an unstructured point set, we use an MLS (moving least-squares) approximation to estimate the local curvatures and their derivatives at a point by means of an approximating surface. Then, we compute neighbor information using a Delaunay tessellation. Ridge and valley points can then be detected as zero-crossings, and connected using curvature directions. We demonstrate our method on several large point-sampled models, rendered by point-splatting, on which the ridge and valley lines are rendered with line width determined from curvatures.     

曲面处理的曲率域能量优化方法研究*

鉴于曲率在几何处理中的重要性,提出了一种基于曲率域的三维几何处理方法,提供了便于曲面形状编辑、过滤及合成的几何工具。对于给定曲面,估计其主曲率,过滤和编辑其曲率分布,完成曲面空间域到曲率域的映射。采用能量优化处理方法对非线性最小二乘曲面问题进行求解,获得最优匹配于期望曲率的曲面,同时保持原始曲面形状的重要几何属性及特征。实例证明,该方法实用性好,对各向异性光顺、特征增强、多尺度曲率编辑的处理效果理想。     

Surface Analysis Methods

Parametric polynomial surfaces defined to satisfy given interpolatory or boundary constraints often suffer extraneous undesired features because of poor control over their many degrees of freedom. Current techniques for detecting and remedying such unexpected surface characteristics are primitive and inadequate. This article describes several surface analysis tools useful in detecting such anomalous surface features. These include standard techniques such as contouring and high-resolution shaded image displays based on direct ray tracing, and some novel methods such as maps of the principal curvatures, the integration of lines of curvature to show the variation ofthe principal directions, and the determination of geodesic paths on the surface.     

Discrete surface Ricci flow

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.     

Adaptive design for reference velocity recovery in motion coordination

We study a coordination problem where the objective is to steer a group of agents to a formation that translates with a prescribed reference velocity. Unlike existing designs which assume that the reference velocity information is available to each agent, we consider the situation where this information is available only to a leader. We then develop an adaptive design with which the other agents reconstruct the reference velocity and recover the desired formation. This design relies only on relative distance information with respect to neighbouring agents and, thus, can be implemented in a decentralized fashion.     

Formations on directed cycles with bearing‐only measurements

This paper studies a bearing‐only–based formation control problem for a group of single‐integrator agents with directed cycle sensing topology. In a 2‐dimensional space, a necessary and sufficient condition for the set of desired bearing vectors to be feasible is derived. Then, we propose a bearing‐only control law for every agent and prove that the formation asymptotically converges to a formation specified by a set of feasible desired bearing vectors. Analysis of the equilibrium formations in the plane for a 3‐agent system and subsequent extension to an n‐agent system is provided. We further extend the analysis on directed triangular formation into a 3‐dimensional space. Finally, simulations validate the theoretical results.