Showing papers on "Piecewise published in 1988"
TL;DR: In this article, the authors show the virtues of monotone splines through a number of statistical applications, including response variable transformation in nonlinear regression, transformation of variables in multiple regression, principal components and canonical correlation.
Abstract: Piecewise polynomials or splines extend the advantages of polynomials to include greater flexibility, local effects of parameter changes and the possibility of imposing useful constraints on estimated functions. Among these constraints is monotonicity, which can be an important property in many curve estimation problems. This paper shows the virtues of monotone splines through a number of statistical applications, including response variable transformation in nonlinear regression, transformation of variables in multiple regression, principal components and canonical correlation, and the use of monotone splines to model a dose-response function and to perform item analysis. Computational and inferential issues are discussed and illustrated.
830 citations
TL;DR: It is shown that when the environment is piecewise linear, it provides a powerful constraint on the kind of matches that exist between two images of the scene when the camera motion is unknown, and that this constraint can be recovered from an estimate of the matrix of this collineation.
Abstract: We show in this article that when the environment is piecewise linear, it provides a powerful constraint on the kind of matches that exist between two images of the scene when the camera motion is unknown. For points and lines located in the same plane, the correspondence between the two cameras is a collineation. We show that the unknowns (the camera motion and the plane equation) can be recovered, in general, from an estimate of the matrix of this collineation. The two-fold ambiguity that remains can be removed by looking at a second plane, by taking a third view of the same plane, or by using a priori knowledge about the geometry of the plane being looked at. We then show how to combine the estimation of the matrix of collineation and the obtaining of point and line matches between the two images, by a strategy of Hypothesis Prediction and Testing guided by a Kalman filter. We finally show how our approach can be used to calibrate a system of cameras.
571 citations
TL;DR: A numerical technique that approximates an implicit surface with a polygonal representation, so that the roots to the function need not be solved each time the surface is rendered.
532 citations
TL;DR: In this paper, an extension of the Kirchhoffer's formula for radiation from a closed surface to surfaces moving at speeds below the wave propagation speed is presented. But it is only applied to surfaces that are piecewise smooth.
322 citations
TL;DR: In this paper, simple periodic as well as more complex behaviors are shown to occur in the human pupil light reflex with piecewise constant mixed and delayed feedback, and the effect of uncontrollable fluctuations in the parameters on the observability of complex dynamics in this system is discussed.
Abstract: Simple periodic as well as more complex behaviors are shown to occur in the human pupil light reflex with piecewise constant mixed and delayed feedback. The output of an infrared video pupillometer, an analog voltage proportional to pupil area, is processed by an electronic comparator which synthesizes the piecewise constant feedback. The system is described by a nonlinear delay differential equation which has been previously shown analytically to exhibit periodic and aperiodic behavior. After parameter estimation from the data, it is found that the observed simple periodic behaviors correlate well with the model behaviors. Although more complex behavior can be observed for parameter values which gave complicated dynamics in the model, there is not a one-to-one correspondence between the observed and predicted results. The effect of uncontrollable fluctuations in the parameters on the observability of complex dynamics in this system is discussed.
195 citations
TL;DR: In this article, the first in a series devoted to the analysis of the regularity of the solution of elliptic partial differential equations with piecewise analytic data is presented, and the present paper analyzes the...
Abstract: This paper is the first in a series devoted to the analysis of the regularity of the solution of elliptic partial differential equations with piecewise analytic data. The present paper analyzes the...
175 citations
TL;DR: In this article, an unsplit, higher order Godunov method for scalar conservation laws in two dimensions is presented. But the method is not suitable for the special case of sharp fronts without any significant distortion.
159 citations
TL;DR: The numerical solution proposed here is obtained by modifying the recursion and using a simple piece-wise constant approximation to the density functions, yielding a bound on the maximum error growth, and a characterization of the situations with potential for large errors.
146 citations
TL;DR: A novel strategy (which the authors call "minimum model error'* estimation) for postexperiment optimal state estimation of discretely measured dynamic systems is developed and illustrated for a simple example and shows it to be quite accurate for state estimation for a poorly modeled dynamic system.
Abstract: A novel strategy (which we call "minimum model error'* estimation) for postexperiment optimal state estimation of discretely measured dynamic systems is developed and illustrated for a simple example. The method is especially appropriate for postexperiment estimation of dynamic systems whose presumed state governing equations are known to contain, or are suspected of containing, errors. The hew method accounts for errors in the system dynamic model equations in a rigorous manner. Specifically, the dynamic model error terms in the proposed method do not require the usual Kalman filter-smoother process noise assumptions of zero-mean, symmetrically distributed random disturbances, nor do they require representation by assumed parameterized time series (such as Fourier series); Instead, the dynamic model error terms require no prior assumptions other than piecewise continuity. Estimates of the state histories, as well as the dynamic model errors, are Obtained as part of the solution of a two-point boundary value problem. The state estimates are continuous and optimal in a global sense, yet the algorithm processes the measurements sequentially. The example demonstrates the method and shows it to be quite accurate for state estimation of a poorly modeled dynamic system.
120 citations
TL;DR: The method is based on minimax approximation techniques which allow the development of a detailed error analysis and by combining subdivision with the degree reduction algorithm, a piecewise defined approximation can be generated, which is within some preset error tolerance of the original curve.
Abstract: An algorithm for generating an ( n − 1 )st degree approximation to an nth degree Bezier curve is presented. The method is based on minimax approximation techniques which allow the development of a detailed error analysis. By combining subdivision with the degree reduction algorithm, a piecewise defined approximation can be generated, which is within some preset error tolerance of the original curve. The number of subdivisions required can be determined a priori and by iterating the scheme, a piecewise degree d approximation can be generated.
82 citations
TL;DR: In this paper, the stability of piecewise deterministic linear systems driven by an underlying finite Markov chain is analyzed and necessary and sufficient conditions for moment stability are obtained by means of an explicit formula for the corresponding Liapunov exponent.
05 Jun 1988
TL;DR: A curvature-based approach to motion analysis of nonrigid surfaces and a method to isolate regions of the surface where the homothetic assumption is violated are presented.
Abstract: A description is given of a curvature-based approach to motion analysis of nonrigid surfaces. Based on changes in the mean and Gaussian curvatures during the motion, it is possible to classify the motion of a surface at each point as rigid, isometric, homothetic, conformal, and general (nonconformal). The general theory of curvature changes for all types of motions is presented. Then the formula specifying changes in the Gaussian curvature during homothetic motion is derived. Also, two special cases of nonrigid motion are considered. The first case deals with piecewise rigid motion. An algorithm is presented which separates a piecewise rigid surface into its rigid parts. The second case deals with homothetic motion. A method is given to isolate regions of the surface where the homothetic assumption is violated. Both algorithms are tested on simulated data and results. >
TL;DR: By splitting each tetrahedron in an appropriate way, it is shown how the problem of producing a piecewise quadratic C 1 interpolant to positional and gradient information at the vertices of a tessellation in R 3 of Tetrahedra may be solved.
TL;DR: In this paper, it was shown that linear functionals (e.g., inner products or point functionals) of solutions of second kind Fredholm integral equations can be found within O(1/n 2r ) by very simple computational schemes using the Nystrom (or quadrature) method based on piecewise polynomials of order r on an $n + 1$ point mesh.
Abstract: The authors show that linear functionals (e.g., inner products or point functionals) of solutions of second kind Fredholm integral equations may be found within $O({1} / {n^{2r}})$ by very simple computational schemes using the Nystrom (or quadrature) method based on piecewise polynomials of order r on an $n + 1$ point mesh. Our theory covers a class of nonsmooth Wiener–Hopf–type equations, and applications include the calculation of crack energy and stress intensity factors in certain linear elastic fracture problems, and the use of the boundary integral method to compute various superconvergent solutions to boundary value problems in polygonal domains. A proof of the uniform convergence of Nystrom’s method for certain boundary integral equations on polygonal domains is also included.
01 Jun 1988
TL;DR: This chapter discusses low-dimensional polynomial systems, which are solved easily and reliably by the covering method, and how to save searching for large empty regions simply by choosing the next box near a box containing a solution point.
Abstract: Publisher Summary This chapter discusses covering methods for the enclosure of the solution set of a finite-dimensional system of nonlinear equations. The chapter discusses low-dimensional polynomial systems, which are solved easily and reliably by the covering method. There is no problem in solving polynomial systems piecewise when the slopes are calculated as in Neumaier. This makes the method suitable for problems in a computer-aided geometrical design. The situation may be different for high-dimensional systems, particularly if these involve functions not defined for all values of the variables; because of the overestimation, the method may generate an exponential number of boxes. In this case, a natural approach may be to combine the covering method with continuation techniques to save searching for large empty regions simply by choosing the next box near a box containing a solution point, taking into account the direction in which the curve leaves the current box.
TL;DR: A method is obtained for the piecewise analytical description of the hypersurface defined by a trivariate function and of the isosurface of such a function.
TL;DR: An adaptive local refinement finite-element procedure for solving initial boundary value problems for vector systems of parabolic partial differential equations on two-dimensional rectangular regions and several aspects of the refinement algorithm and the dynamic tree data structure that is used to store the mesh, solution, and error estimate are discussed.
Abstract: We discuss an adaptive local refinement finite-element procedure for solving initial boundary value problems for vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The differential equations are discretized in space using piecewise bilinear finite-element approximations. An estimate of the spatial discretization error of the solution is obtained using piecewise cubic polynomials that employ nodal superconvergence to gain computational efficiency. The resulting system of ordinary differential equations for the finite-element solution and error estimate are integrated in time using existing software for stiff differential systems. The spatial error estimate is used to locally refine the finite-element mesh in order to satisfy a prescribed error tolerance. We discuss several aspects of the refinement algorithm and the dynamic tree data structure that is used to store the mesh, solution, and error estimate. A code that is based on our methods is applied to several examples in order to demonstrate the effectiveness of the error-estimation procedure and adaptive algorithms.
TL;DR: In this paper, an algebraic analysis of the space H of all entire functions spanned by the integer translates of an EB-spline is presented, which is based on the observation that the structure of Z is relatively simple when Z is covered by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process.
Abstract: Exponential box splines (EB-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space Z spanned by exponential polynomials This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients The main part of this paper is devoted to algebraic analysis of the space H of all entire functions spanned by the integer translates of an EB-spline This investigation relies on a detailed description of Z and its discrete analog S° The approach taken here is based on the observation that the structure of Z is relatively simple when Z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case We also identify and study in detail several types of singularities which occur only for complex EB-splines The first is when the Fourier transform of the EB-spline vanishes at some critical points, the second is when Z cannot be embedded in y and the third is when H is a proper subspace of Z We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh
TL;DR: Oscillatory, nonoscillatory, and periodic solutions of re-tarded functional differential equations are investigated in this article, which concerns a system of two first order linear equations with piecewise constant argument.
Abstract: Oscillatory, nonoscillatory, and periodic solutions of re-tarded functional differential equations are investigated. The study concerns a system of two first order linear equations with piecewise constant argument.
TL;DR: This paper deals with approximation techniques for the optimal stopping of a piecewise-deterministic Markov process, and builds ε-optimal stopping times for the original problem using the payoff function of the discretized process.
Abstract: This paper deals with approximation techniques for the optimal stopping of a piecewise-deterministic Markov process (P.D.P.). Such processes consist of a mixture of deterministic motion and random jumps. In the first part of the paper (Section 3) we study the optimal stopping problem with lower semianalytic gain function; our main result is the construction of e-optimal stopping times. In the second part (Section 4) we consider a P.D.P. satisfying some smoothness conditions, and forN integer we construct a discretized P.D.P. which retains the main characteristics of the original process. By iterations of the single jump operator from ℝN to ℝN, each iteration consisting ofN one-dimensional minimizations, we can calculate the payoff function of the discretized process. We demonstrate the convergence of the payoff functions, and for the case when the state space is compact we construct e-optimal stopping times for the original problem using the payoff function of the discretized problem. A numerical example is presented.
TL;DR: A single-source single-sink dynamic network is considered where the link flows are real-valued measurable functions defined on a time interval and where storage is allowed at the nodes and an algorithm is presented which computes an optimal flow in O( | N | 4T4) time.
Abstract: A single-source single-sink dynamic network is considered where the link flows are real-valued measurable functions defined on a time interval and where storage is allowed at the nodes. Piecewise-constant link and storage capacities are given. A τ-maximum flow is a dynamic flow assignment that maximizes the total amount of commodity reaching the sink before time τ. The problem considered is that of computing a flow which is simultaneously τ-maximum for all τ. Such a flow solves a minimum-delay dynamic routing problem. An algorithm is presented which computes an optimal flow in O( | N | 4T4) time, where | N | is the number of nodes and T is the number of times that the capacities change. Previous polynomial-time algorithms have been given only for the case of constant capacities.
TL;DR: In this paper, three-dimensional Poisson problems containing boundary singularities are treated, and the forms of the solutions for certain problems of this type are derived, where the domains of the problems can be represented in terms either of spherical or of cylindrical-polar co-ordinates.
Abstract: Three-dimensional Poisson problems containing boundary singularities are treated. The forms of the solutions for certain problems of this type are derived, where the domains of the problems can be represented in terms either of spherical- or of cylindrical-polar co-ordinates. These singular forms are used to augment the basis of a standard piecewise polynomial Galerkin space, thus producing an augmented Galerkin technique which is suited to the context of a problem involving a singularity. Error estimates are derived.
07 Jun 1988
TL;DR: The author investigates systematic methods for the design of processor arrays using piecewise linear transformations and the proposed concept enables the efficient realization of more general classes of algorithms than the systolic concept.
Abstract: The author investigates systematic methods for the design of processor arrays. The proposed concept enables the efficient realization of more general classes of algorithms than the systolic concept. In particular, instance-dependent branching, instance-dependent processor configurations, and hierarchical formulations of imperative programs can be taken into account. The concept of a piecewise-regular dependence graph and that of its reduced description is given. The definition of piece-wise regular algorithms leads to their mapping onto piecewise regular systolic arrays using piecewise linear transformations. The hierarchical description of algorithms and dependence graphs and the corresponding transformations such as condensation, unfolding, clustering and unclustering are applied to partitioning problems (assignment, schedule segmentation, multidimensional mapping). >
Journal Article•
TL;DR: The monotonically and convexly constrained (MONCON) weighted v -spline interpolant is presented, which is the C 1 piecewise cubic solution to a constrained optimization problem that preserves the local monotonicity and local convexity of the data.
01 Dec 1988
TL;DR: This paper describes a class of Monte Carlo optimization problems for which unbiased derivative estimators of the infinitesimal perturbation analysis (IPA) type can be derived; and also a simple framework within which to establish unbiasedness.
Abstract: This paper describes a class of Monte Carlo optimization problems for which unbiased derivative estimators of the infinitesimal perturbation analysis (IPA) type can be derived; and also a simple framework within which to establish unbiasedness. Of central importance are systems with continuous, piecewise differentiable sample performance functions. Experience suggests that continuity is, in practice, almost necessary for IPA to work. "Piecewise" differentiable is a concession to the discrete nature of many applied probability models. We discuss a variety of examples, including both static and dynamic systems.
TL;DR: In this article, the authors developed a general paradigm for the analysis of non-rigid motion, which includes isometry, rigidity, and planarity, and applied it to turbulent motion.
Abstract: Determining 3D motion from a time-varying 2D image is an ill-posed problem; unless we impose additional constraints, an infinite number of solutions is possible. The usual constraint is rigidity, but many naturally occurring motions are not rigid and not even piecewise rigid. A more general assumption is that the parameters (or some of the parameters) characterizing the motion are approximately (but not exactly) constant in any sufficiently small region of the image. If we know the shape of a surface we can uniquely recover the smoothest motion consistent with image data and the known structure of the object, through regularization. This paper develops a general paradigm for the analysis of nonrigid motion. The variational condition we obtain includes many previously studied constraints as `special cases'. Among them are isometry, rigidity and planarity. If the variational condition is applied at multiple scales of resolution, it can be applied to turbulent motion. Finally, it is worth noting that our theory does not require the computation of correspondence (optic flow or discrete displacements), and it is effective in the presence of motion discontinuities.
TL;DR: A piecewise parametric production model that may be used as a reference technology for efficiency gauging and it is shown that other piecewise reference technologies used in efficiency measuring are obtained as special cases of this model.
Abstract: In this paper we introduce a piecewise parametric production model that may be used as a reference technology for efficiency gauging. We show that other piecewise reference technologies used in efficiency measuring are obtained as special cases of our model.
TL;DR: In this article, the set of all invariant measures of topologically transitive subsets with topop (L)>0 of a continuous piecewise monotonic transformation on [0, 1] with the weak topology is given.
Abstract: We endow the set of all invariant measures of topologically transitive subsetsL withhtop (L)>0 of a continuous piecewise monotonic transformation on [0, 1] with the weak topology. We show that the set of periodic orbit measures is dense, that the sets of ergodic, of nonatomic, and of measures with supportL are denseGδ-sets, that the set of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGδ-set.