Showing papers on "Piecewise published in 1992"

Surface reconstruction from unorganized points

Abstract: This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects. Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners. The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data. Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.

Control and stabilization of nonholonomic dynamic systems

Abstract: A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested. >

Exponential stabilization of mobile robots with nonholonomic constraints

Abstract: An exponentially stable controller for a two-degree-of-freedom robot with nonholonomic constraints is presented. Although this type of system is open-loop controllable, this system has been shown to be nonstabilizable via pure smooth feedback. A particular class of piecewise continuous controllers is shown to exponentially stabilize the mobile robot about the origin. This controller has the characteristic of not requiring infinite switching like other approaches, such as the sliding controller. Simulation results are presented. >

Multilayer Feedforward Networks with a Non-Polynomial Activation Function Can Approximate Any Function

Abstract: Several researchers characterized the activation function under which multilayer feedforwardnetworks can act as universal approximators. We show that most of all the characterizationsthat were reported thus far in the literature are special cases of the followinggeneral result: a standard multilayer feedforward network with a locally bounded piecewisecontinuous activation function can approximate any continuous function to any degree ofaccuracy if and only if the network's activation function is not a polynomial. We alsoemphasize the important role of the threshold, asserting that without it the last theoremdoes not hold.

Discontinuity meshing for accurate radiosity

Abstract: An algorithm for compactly and accurately capturing the illumination of a diffuse polyhedral environment caused by an area light source is presented. The algorithm constructs a discontinuity mesh that explicitly represents discontinuities in the radiance function as boundaries between mesh elements. A piecewise quadratic interpolant is used to approximate the radiance function, preserving the discontinuities associated with the edges in the mesh. >

Viscous limits for piecewise smooth solutions to systems of conservation laws

Abstract: In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.

Using iterated function systems to model discrete sequences

Abstract: Two iterated function system (IFS) models are explored for the representation of single-valued discrete-time sequences: the self-affine fractal model and the piecewise self-affine fractal model. Algorithms are presented, one of which is suitable for a multiprocessor implementation, for identification of the parameters of each model. Applications of these models to a variety of data types are given where signal-to-noise ratios are presented, quantization effects of the model parameters are investigated, and compression ratios are computed. >

State-of-the-art harmonic-balance simulation of forced nonlinear microwave circuits by the piecewise technique

Abstract: The theoretical foundations and the numerical performance of an advanced nonlinear circuit simulator based on the piecewise harmonic-balance (HB) technique are discussed. The exact computation of the Jacobian matrix for Newton-iteration based HB simulation and the related conversion-matrix technique for fast mixer analysis are formulated in a general form. Convergence problems at high drive levels are solved by a parametric formulation of the device models coupled with an advanced norm-reducing iteration. A physics-based approximation is shown to allow the HB equations to be effectively decoupled in many practical cases, bringing large-sized jobs, such as pulsed-RF analysis, within the reach of ordinary workstations. The exact Jacobian is used in conjunction with an exact formula for the gradient of the objective function, to implement an efficient broadband nonlinear circuit optimization capability. Examples are presented. >

An introduction to motion planning under multirate digital control

Abstract: The authors propose digital control methods for steering real analytic controllable systems between arbitrary state configurations. The main idea is to achieve a multirate sampled procedure to perform motions in all the directions of controllability under piecewise constant controls. When it is applied to non-holonomic control systems without drift, the procedure simplifies. In particular, it results in exact steering on chained systems recently introduced in the motion planning literature. A classical example is reported. >

A volume density optical model

Abstract: A simple, but accurate, formal volume density optical model is developed for volume rendering scattered data or scalar fields from the finite element method, as opposed to scanned data sets where material classification is involved. The model is suitable either for ray tracing or projection methods and allows maximum flexibility in setting color and opacity. An expression is derived for the light intensity along a ray in terms of six userspecified transfer functions, three for optical density and three for color. Closed form solutions under several different assumptions are presented including a new exact result for the case that the transfer functions vary piecewise linearly along a ray segment within a cell.

Curve and surface design

Abstract: Part I. Curve Design: Properties of Minimal Energy Splines G. Brunnett Minimal Energy splines with Various End Constraints E. Jou and W. Han Interval Weighted Tau- splines D. Lasser and H. Hagen Curve and surface Interpolation using Quintic Weight Tau-splines D. Neuser Weighted splines Based on Piecewise Polynomial Weighted Functions K. Salkauskas Algorithms for Geometric spline Curves M. Eck On the Problem of Determining the distance Between parametric curves F. Fritsch and G. Nielson Part II. Non-Tensor Product Surfaces: A survey of scattered Data Fitting Using Triangular Interpolants T. De Rose Free-form surfaces from Partial Differential Equations M. I. G. Bloor and M. J. Wilson Modeling with Box spline surfaces M. Daehlen.

Exponential control law for a mobile robot: extension to path following

Abstract: The authors present an exponentially stable controller for a two-degree-of-freedom robot with nonholonomic constraints. Although this system is controllable, it has been shown to be nonstabilizable via smooth state feedback. A particular class of piecewise continuous controller which exponentially stabilizes the robot about the origin was previously proposed by the authors (1991). This approach is extended to stabilize about an arbitrary position and orientation, and to track a sequence of points. This feedback law is naturally combined with path planning when the desired path to be followed can be composed of a sequence of straight lines and circle segments, i.e. shortest paths of bounded curvature in the plane. >

Simple sigmoid-like activation function suitable for digital hardware implementation

Abstract: A simple sigmoid-like second-order piecewise activation function suitable for direct digital hardware implementation is presented. Simulation results on the uses of the second-order function and the bipolar sigmoid function for training multilayer feedforward networks using the backpropagation algorithm show that they have similar generalisation properties while the second-order function has a slight advantage in convergence speed.< >

Computerized method for decomposing a geometric model of surface or volume into finite elements

Abstract: A computerized process for defining finite elements in a surface or volume for ultimately predicting a physical characteristic of the surface or volume. For the surface, for example, the process includes a first step of inputting surface boundary point coordinates of a geometric model of the surface to a computer system, the computer system including an image display screen displaying the geometric model. The process further includes preparing the boundary edges of surface by generating piecewise geometrically smooth bezier curves between boundary points and converting the bezier curves to cubic interpolation polynomials and defining evenly spaced points on each cubic interpolation polynomial and decomposing the surface with divider curves, if the surface is not already 3, 4, or 5 sided, into 3, 4, and 5 sided primitives. Thereafter, a determination is made of the largest acceptable element size and the number of elements disposed along each edge of each primitive. The next steps are readjusting one of the divider curves to match a closest even element vertices and mapping in 3 and 5 sided clusters and then preparing remaining 4 sided primitives for decomposition into 4 sided elements and mapping in elements and element patches. Thereafter, method includes the steps of optimising the elements in the mesh and writing the resulting mesh to a storage file.

The complete canonical piecewise-linear representation. I. The geometry of the domain space

Abstract: Continuous piecewise-linear functions from R/sup n/ to R/sup m/ are analyzed in terms of the dimensions of their domain space and of degenerate kth-order intersections of region boundaries. The theory developed demonstrates how these two quantities are connected. Moreover, the exact number of independent parameters is demonstrated for boundary configurations containing degenerate intersections of arbitrary orders. >

Identifying linear reduced-order models for systems with arbitrary initial conditions using Prony signal analysis

Abstract: A method of identifying reduced-order linear models for systems operating in the neighborhood of an equilibrium point is presented. The method is based on Prony signal analysis, which has recently received considerable attention in the study of power system electromechanical oscillations. Prior to the application of the input test signal, the system can be in a transient state. The system input test signal is piecewise continuous and allows several Prony analyses to be performed during a transient, with each analysis conducted between input discontinuities. Results of these Prony analyses can be combined in various ways to obtain system eigenvalues, transfer-function residues, and initial condition residues. Two examples are given to illustrate the use of the method. >

Generative modeling: a symbolic system for geometric modeling

Abstract: This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higher-dimensioned shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as “reparameterize this curve by arclength”, “compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces”, or “solve this constraint or ODE system”. The system has been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds multidimensional parametric functions. The language is baaed on a set of symbolic operators on continuous, piecewise differentiable parametric functions. We present several shape examples to show bow conveniently shapes can be specified in the system. We also discuss the kinds of operators useful in a geometric modeling system, including arithmetic operators, vector and matrix operators, integration, differentiation, constraint solution, and constrained minimisation. Associated with each operator are several methods, which compute properties about the parametric functions represented with the operators. We show how many powerful rendering and analytical operations can be supported with only three methods: evaluation of the parametric function at a point, symbolic dlfferentiation of the parametric function, and evacuation of an inclusion function for the parametric function. Like CSG, and unlike most other geometric modeling approaches, 3Ms modeling approach is closed, meaning that further modeling operations cart be applied to any results of modeling operations, yielding valid models. Because of this closure property, the symbolic operators can be composed very flexibly, allowing the construction of higher-level operators without changing the underlying implementation of the system. Because the modeling operations are described symbolically, specified models can capture the designer’s intent without approximation error.

Semi-Lagrangian Piecewise Biparabolic Scheme for Two-Dimensional Horizontal Advection of a Passive Scalar

Abstract: A conservative semi-Lagrangian algorithm with a snall computational diffusion is presented that may be applied to advection of passive scalars in numerical models of the atmosphere. The technique is preferable for the horizontal semistaggered grids where a scalar point is surrounded by four velocity points. It is a coupling of the semi-Lagrangian approach and the piecewise parabolic method (PPM). Unlike the original PPM when applied to the advection of passive scalars, the new scheme is a fully two-dimensional algorithm. Also, it is not restricted by the linear stability condition. This paper describes the two steps comprising the two-dimensional algorithm. The first one is the interpolation pressure for getting the piecewise biparabolic function, and the second is a conservative remapping from the original grid to the grid made by the departure domains. Several test integrations are presented in which the described scheme performs very successfully.

Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations

Abstract: Fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise Hermite bicubics is employed for the approximate solution of Poisson’s ...

Solving the Ginzburg-Landau equations by finite-element methods

Abstract: We consider finite-element methods for the approximation of solutions of the Ginzburg-Landau equations of superconductivity. The methods are based on a discretization of the Euler-Lagrange equations resulting from the minimization of the free-energy functional. The discretization is effected by requiring the approximate solution to be a piecewise polynomial with respect to a grid. The magnetization versus magnetic field curves obtained through the finite-element methods agree well with analogous calculations obtained by other schemes. We demonstrate, both by analyzing the algorithms and through computational experiments, that finite-element methods can be very effective and efficient means for the computational simulation of superconductivity phenomena and therefore could be applied to determine macroscopic properties of inhomogeneous, anisotropic superconductors.

On the approximation of invariant measures

Abstract: Given a discrete dynamical system defined by the map τ:X →X, the density of the absolutely continuous (a.c.) invariant measure (if it exists) is the fixed point of the Frobenius-Perron operator defined on L1(X). Ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator. T. Y. Li proved the convergence of the scheme for expanding maps of the interval. G. Keller and M. Blank extended this result to piecewise expanding maps of the cube in ℝn. We show convergence of a variation of Ulam's scheme for maps of the cube for which the Frobenius-Perron operator is quasicompact. We also give sufficient conditions onτ for the existence of a unique fixed point of the matrix approximation, and if the fixed point of the operator is a function of bounded variation, we estimate the convergence rate.

Elementary quantum models with position-dependent mass

Abstract: The practical availability of semiconductor heterostructure gives a new interest to elementary quantum models for particles in piecewise flat potentials, with the new feature of a position-dependent mass. After sketching the basics of the theory, and writing the generalized Hamiltonian and continuity conditions to be used, the simplest problems, namely, that of a step, barrier and lattice in potential and mass are treated and their novel aspects stressed.

Convexity preserving piecewise rational interpolation for planar curves

Abstract: This paper uses a piecewise ratonal cubic interpolant to solve the problem of shape preserving interpolation for plane curves; scalar curves are also considered as a special case. The results derived here are actually the extensions of the convexity preserving results of Delbourgo and Gregory [Delbourgo and Gregory'85] who developed a shape preserving interpolation scheme for scalar curves using the same piecewise rational function. They derived the ocnstraints, on the shape parameters occuring in the rational function under discussion, to make the interpolant preserve the convex shape of the data. This paper begins with some preliminaries about the rational cubic interpolant. The constraints consistent with convex data, are derived in Sections 3. These constraints are dependent on the tangent vectors. The description of the tangent vectors, which are consistent and dependent on the given data, is made in Section 4. the convexity preserving results are explained with examples in Section 5.