Showing papers on "Piecewise published in 1994"

Ideal spatial adaptation by wavelet shrinkage

Abstract: SUMMARY With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels; however, it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone. We describe a new principle for spatially-adaptive estimation: selective wavelet reconstruction. We show that variable-knot spline fits and piecewise-polynomial fits, when equipped with an oracle to select the knots, are not dramatically more powerful than selective wavelet reconstruction with an oracle. We develop a practical spatially adaptive method, RiskShrink, which works by shrinkage of empirical wavelet coefficients. RiskShrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so. A new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n, where n is the sample size. Moreover no estimator can give a better guarantee than this. Within the class of spatially adaptive procedures, RiskShrink is essentially optimal. Relying only on the data, it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle. In contrast, it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone.

Elliptic Problems in Domains with Piecewise Smooth Boundaries

Abstract: Discusses general elliptic boundary value problems in domains with boundaries containing conical points and edges of different dimensions. The main topic is the solvability of such problems and the existence of asymptotic formulas for solutions near singularities. Applications to the mechanics of so

Piecewise smooth surface reconstruction

Abstract: We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering—the automatic generation of CAD models from physical objects. Novel aspects of the method are its ability to model surfaces of arbitrary topological type and to recover sharp features such as creases and corners. The method has proven to be effective, as demonstrated by a number of examples using both simulated and real data.A key ingredient in the method, and a principal contribution of this paper, is the introduction of a new class of piecewise smooth surface representations based on subdivision. These surfaces have a number of properties that make them ideal for use in surface reconstruction: they are simple to implement, they can model sharp features concisely, and they can be fit to scattered range data using an unconstrained optimization procedure.

Construction of piecewise Lyapunov functions for stabilizing switched systems

Abstract: This paper discusses the problem of stabilizing a pair of switched linear systems. A control law is developed using a Lyapunov function having a piecewise continuous derivative. A Lyapunov function yielding a stable switching rule is shown to exist as long as there exists a stable convex combination of the system matrices. The use of this stable combination for other control strategies is explored. >

An Alternative Model for Mixtures of Experts

Abstract: We propose an alternative model for mixtures of experts which uses a different parametric form for the gating network. The modified model is trained by the EM algorithm. In comparison with earlier models--trained by either EM or gradient ascent--there is no need to select a learning stepsize. We report simulation experiments which show that the new architecture yields faster convergence. We also apply the new model to two problem domains: piecewise nonlinear function approximation and the combination of multiple previously trained classifiers.

An image-enhancement technique for electrical impedance tomography

Abstract: We propose an idea for reconstructing 'blocky' conductivity profiles in electrical impedance tomography. By 'blocky' profiles, we mean functions that are piecewise constant, and hence have sharply defined edges. The method is based on selecting a conductivity distribution that has the least total variation from all conductivities that are consistent with the measured data. We provide some motivation for this approach and formulate a computationally feasible problem for the linearized version of the impedance tomography problem. A simple gradient descent-type minimization algorithm, closely related to recent work on noise and blur removal in image processing via non-linear diffusion is described. The potential of the method is demonstrated in several numerical experiments.

General noise analysis of nonlinear microwave circuits by the piecewise harmonic-balance technique

Abstract: This paper presents a self-consistent set of algorithms for the numerical computation of noise effects in forced and autonomous nonlinear microwave circuits. The analysis relies upon the piecewise harmonic-balance method, and thus retains all the peculiar advantages of this technique, including general-purposeness in the widest sense. The noise simulation capabilities include any kind of forced or autonomous nonlinear circuit operated in a time-periodic large-signal steady state, as well as microwave mixers of arbitrary topology. The limitations of the traditional frequency-conversion approach to noise analysis are overcome. The analysis takes into account the thermal noise generated in the passive subnetwork, the noise contributions of linear and nonlinear active devices, and the noise injected by sinusoidal driving sources of known statistical properties. The nonlinear noise models of two representative families of microwave devices (FET's/HEMT's and Schottky-barrier diodes) are discussed in detail, and several applications are illustrated. >

Surface approximation and geometric partitions

Abstract: Motivated by applications in computer graphics, visualization, and scientic compu- tation, we study the computational complexity of the following problem: given a set S of n points sampled from a bivariate function f(x;y) and an input parameter "> 0, compute a piecewise-linear function (x;y) of minimum complexity (that is, an xy-monotone polyhedral surface, with a mini- mum number of vertices, edges, or faces) such thatj(xp;yp) i zp j" for all (xp;yp;zp)2 S.W e give hardness evidence for this problem, by showing that a closely related problem is NP-hard. The main result of our paper is a polynomial-time approximation algorithm that computes a piecewise- linear surface of size O(Ko logKo), where Ko is the complexity of an optimal surface satisfying the constraints of the problem. The technique developed in our paper is more general and applies to several other problems that deal with partitioning of points (or other objects) subject to certain geometric constraints. For instance, we get the same approximation bound for the following problem arising in machine learning: givenn \red" andm \blue" points in the plane, nd a minimum number of pairwise disjoint triangles such that each blue point is covered by some triangle and no red point lies in any of the triangles.

On generating topologically consistent isosurfaces from uniform samples

Abstract: A set of sample points of a function of three variables may be visualized by defining an interpolating functionf of the samples and examining isosurfaces of the formf(x, y, z)=t for various values oft. To display the isosurfaces on a graphics device, it is desirable to approximate them with piecewise triangular surfaces that (a) are geometrically good approximations, (b) are topologically consistent, and (c) consist of a small number of triangles. By topologically consistent we mean that the topology of the piecewise triangular surface matches that of the surfacef(x, y, z)=t, i.e., the interpolantf determines both the geometry and the topology of the piecewise triangular surface. In this paper we provide an efficient algorithm for the case in whichf is the piecewise trilinear interpolant; for this case existing methods fail to satisfy all three of the above conditions simultaneously.

Piecewise polynomial regression trees

Abstract: A nonparametric function estimation method called SUPPORT ("Smoo- thed and Unsmoothed Piecewise-Polynomial Regression Trees'') is described. The estimate is typically made up of several pieces, each piece being obtained by fitting a polynomial regression to the observations in a subregion of the data space. Partitioning is carried out recursively as in a tree-structured method. If the estimate is required to be smooth, the polynomial pieces may be glued together by means of weighted averaging. The smoothed estimate is thus obtained in three steps. In the first step, the regressor space is recursively partitioned until the data in each piece are adequately fitted by a polynomial of a fixed order. Partitioning is guided by analysis of the distributions of residuals and cross-validation estimates of prediction mean square error. In the second step, the data within a neighborhood of each partition are fitted by a polynomial. The final estimate of the regression function is obtained by averaging the polynomial pieces, using smooth weight functions each of which diminishes rapidly to zero outside its associated partition. Estimates of derivatives of the regression function may be obtained by similar averaging of the derivatives of the polynomial pieces. The advantages of the proposed estimate are that it possesses a smooth analytic form, is as many times differentiable as the family of weight functions are, and has a decision tree representation. The asymptotic properties of the smoothed and unsmoothed estimates are explored under appropriate regularity conditions. Examples comparing the accuracy of SUPPORT to other methods are given.

Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L 2 -norm

Abstract: Exact numerical convergence factors for any multigrid cycle can be predicted by local mode (Fourier) analysis. For general linear elliptic partial differential equation (PDE) systems with piecewise smooth coefficients in general domains discretized by uniform grids, it is proved that, in the limit of small meshsizes, these predicted factors are indeed obtained, provided the cycle is supplemented with a proper processing at and near the boundaries. That processing, it is proved, costs negligible extra computer work. Apart from mode analysis, a coarse grid approximation (CGA) condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly. The present part studies the $L_2 $ convergence in one cycle for equations with constant coefficients. In the sequel [Brandt, Rigorous quantitative analysis of multigrid, II: Extensions and practical implications, manuscript] extensions are discussed to many cycles (asymptotic convergence), to more levels with arbitrary cycle ty...

Curves and surfaces in geometric design

Abstract: This volume documents the results and presentations, related to aspects of geometric design, of the Second International Conference on Curves and Surfaces, held in Chamonix in 1993. The papers represent directions for future research and development in many areas of application. From the table of contents: - Object Oriented Spline Software - An Introduction to Pade Approximations - Zonoidal Surfaces - Projective Blossoms and Derivatives - Piecewise Polynomial Approximation of Spheres - A Geometrical Approach to Interpolation on Quadric Surfaces

A complete linear discretization for calculating the magnetic field using the boundary element method

Abstract: An analytic solution is derived for the magnetic field generated by current sources located in a piecewise homogeneous volume conductor. A linear discretization approach is used, whereby the surface potential is assumed to be a piecewise linear function over each tessellated surface defining the regions of differing conductivity. The magnetic field is shown to be a linear combination of vector functions which are strictly dependent on the geometry of the problem, the surface tesselation, and the observation point. >

Optimal sensor location for robust control of distributed parameter systems

Abstract: In this paper, we discuss an approach to sensor/estimator design for robust control of distributed parameter systems. This approach involves using minmax compensator design and piecewise approximates to the optimal feedback gains. To illustrate, we present the results for a nonlinear hybrid partial and ordinary differential equation system. >

Method and apparatus for representing image data using polynomial approximation method and iterative transformation-reparametrization technique

Abstract: A piecewise parametric polynomial curve-fitting method using an iterative transformation-reparametrization technique is used to compress information describing lines, such as those formed by handwritten lines, for storage in a compressed form in a computer. The curve-fitting method is applied iteratively with adaptive segmenting of curve segments to optimize piecewise approximations of complex curves. Each piecewise segment is iteratively lengthened, parameterized with an updatable parametrization table, and approximated using a cosine-type transform. To minimize approximation errors, both the accuracy and the trend of the approximation errors are monitored. In order to match end-point positions of the piecewise approximation segments, the cosine coefficients representing each piecewise segment are modified in view of the edge conditions so the segments properly abut one another upon reconstruction.

Contact lens and a method for manufacturing contact lens

Abstract: A contact lens (30) having a smooth surface (55, 65) is made through the use of a computer implementing a spline approximation of corneal topography (145). Piecewise polynomials approximating the corneal topography (145) have equal first and second derivatives where they join. A curve (50) representing the central optical portion of the lens and the piecewise polynomial (70) adjacent to the central optical portion curve (50) have an equal first derivative where they join. A contact lens (30) is cut corresponding to the smooth surface (55, 65) defined by the piecewise polynomials.

5-axis-freeform surface milling using piecewise ruled

Abstract: This paper presents a 5-axis side milling scheme for freeform surfaces based on automatic piecewise ruled surface approximation. With this scheme, resulting surface finish is accurate and pleasing, and has a smaller scallop height compared to ball-end milling. The ruled surface approximation can be made arbitrarily precise resulting in an overall fast milling operation that satisfies tight tolerances, and smoother surface finish. The class of surfaces that can take advantage of this type of 5-axis milling operation includes both convex and saddle-like (hyperbolic) shapes.

Piecewise-polynomial approximation of functions fromH ℓ((0, 1) d ), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator

Abstract: For the selfadjoint Schrodinger operator −Δ−αV on ℝ2 the number of negative eigenvalues is estimated. The estimates obtained are based upon a new result on the weightedL2-approximation of functions from the Sobolev spaces in the cases corresponding to the critical exponent in the embedding theorem.

Multilevel algorithms considered as iterative methods on semidefinite systems

Abstract: For the representation of piecewise d-linear functions instead of the usual finite element basis, a generating system is introduced that contains the nodal basis functions of the finest level and of all coarser levels of discretization. This approach enables the author to work directly with multilevel decompositions of a function.For a partial differential equation, the Galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the one-dimensional (1D) case only about twice, in the two-dimensional (2D) case about $4/3$ times, and in the three-dimensional (3D) case about $8/7$ times as many unknowns as the usual system. Furthermore, the semidefinite system possesses not just one, but many solutions. However, the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem. The author shows that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite sy...

On the gibbs phenomenon v: recovering exponential accuracy from collocation point values of a piecewise analytic function

Abstract: The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or an highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.

A New Set of Equations for Very Shallow Water and Partially Dry Areas Suitable to 2D Numerical Models

Abstract: When a 2D model is used to compute the flood propagation over an initially dry area some difficulties arise due to the changes in the computational domain as large parts of it are flooded and dried during the phenomenon. A second problem concerns the representation of soil surface: usually it is described as a piecewise constant or a piecewise linearly varying surface hence neglecting the ground unevenness. For very small water depths, unevenness plays instead a very important role. In this paper, new dynamic and mass balance equations for 2D free surface flows are presented which, by means of a conceptual sub-model, account for the ground unevenness effects and allow a very realistic description of flooding and drying phenomena. It is also shown that for a very smooth ground surface or for relatively large water depths, the new equations reduce to the well known De S. Venant equations.

Unique reconstruction of piecewise-smooth images by minimizing strictly convex nonquadratic functionals

Abstract: We propose the minimization of a nonquadratic functional or, equivalently, a nonlinear diffusion model to smooth noisy image functionsg:Ω ⊂R n →R while preserving significant transitions of the data. The model is chosen such that important properties of the conventional quadratic-functional approach still hold: (1) existence of a unique solution continuously depending on the datag and (2) stability of approximations using the standard finite-element method. Relations with other global approaches for the segmentation of image data are discussed. Numerical experiments with real data illustrate this approach.

Robust priors for smoothing and image restoration

Abstract: The Bayesian method for restoring an image corrupted by added Gaussian noise uses a Gibbs prior for the unknown clean image. The potential of this Gibbs prior penalizes differences between adjacent grey levels. In this paper we discuss the choice of the form and the parameters of the penalizing potential in a particular example used previously by Ogata (1990,Ann. Inst. Statist. Math.,42, 403–433). In this example the clean image is piecewise constant, but the constant patches and the step sizes at edges are small compared with the noise variance. We find that contrary to results reported in Ogata (1990,Ann. Inst. Statist. Math.,42, 403–433) the Bayesian method performs well provided the potential increases more slowly than a quadratic one and the scale parameter of the potential is sufficiently small. Convex potentials with bounded derivatives perform not much worse than bounded potentials, but are computationally much simpler. For bounded potentials we use a variant of simulated annealing. For quadratic potentials data-driven choices of the smoothing parameter are reviewed and compared. For other potentials the smoothing parameter is determined by considering which deviations from a flat image we would like to smooth out and retain respectively.

Numerical approximation of the solutions of delay differential equations on an infinite interval using piecewise constant arguments

Abstract: The idea of approximating the solutions of delay differential equations by equations with piecewise constant arguments (EPCA) has been suggested by Gyijri [l], who proved convergence of the method for linear and nonlinear delay equations on compact intervals, and under certain conditions also on the half line. The approximating equations with piecewise constant argument can, in turn, be solved by use of difference equations. The latter then also provide an approximation of the original delay equation. (The theory of EPCA was initiated and studied by Cooke and Wiener in [2] and [3].) In this paper, we begin to investigate the question of whether in this method of approximation the asymptotic dynamics of the delay differential equations are preserved. A number of authors have dealt with the problem of showing that discretization of ordinary differential equations does not significantly alter the basic qualitative features. Readers may refer to the papers of Kloeden and Lorenz [4] and Beyn [5] for some of this work. For delay differential equations, questions of this sort have been studied by Cryer [6], Barwell [7], Zennaro [8], and later authors. The paper of Strehmel et al. [9] contains an up-to-date list of references. In our method of approximation, the delay equation is first replaced by an EPCA, and then by a difference equation, and our objective is to relate the qualitative dynamics of these three equations. We obtain results for autonomous and nonautonomous equations with one or many delays. For autonomous equations, the resulting difference algorithm is just a simple Euler scheme, but for nonautonomous equations, it includes other possibilities. In order to make the method and the nature of results as clear as possible, we begin with a particular test equation, k(t) = -pz(t 7), p > 0, 7 > 0.

Tree-structured proportional hazards regression modeling.

Abstract: A method for fitting piecewise proportional hazards models to censored survival data is described. Stratification is performed recursively, using a combination of statistical tests and residual analysis. The bootstrap is employed to keep the probability of a Type I error (the error of discovering two or more strata when there is only one) of the method close to a predetermined value. The proposed method can thus also serve as a formal goodness-of-fit test for the proportional hazards model. Real and simulated data are used for illustration.

Surface Reconstruction from Unorganized Points (PhD Thesis)

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.

Stability of higher order triangular hood-taylor methods for the stationary stokes equations

Abstract: We prove the stability for the approximation of the stationary Stokes equations by means of piecewise continuous velocities of degree k+1 and piecewise continuous pressures of degree k for k≥1. The necessary and sufficient condition required on the triangulation is that it contains at least three triangles. The theorem is compared with previous results.