Projection Equations Research Articles

Fast Multilevel Augmentation Methods for Solving Hammerstein Equations

Fast multilevel augmentation methods based on projections for solving Hammerstein equations are developed. Each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace. A subspace at a level is obtained from the subspace at the previous level by adding a difference subspace. Accordingly, the projection equation at the present level is obtained by augmenting the projection equation at the previous level. Using this idea recursively, we solve the nonlinear equation (by an iteration method) only at an initial lower resolution level while obtaining its solution at a higher resolution level. With the help of the multiscale analysis, we separate the procedure of solving the nonlinear operator equation at a high level into two major components: (1) solving the nonlinear equation only at an initial lower level, and (2) compensating the error by solving a linear system at the high level. We prove that the proposed methods require only linear (up to a logarithmic factor) computational complexity and have the optimal convergence order. A relationship between the proposed method and the multigrid method is discussed. Two specific fast methods based on the Galerkin projection and the collocation projection are developed. Numerical results are presented to confirm the theoretical estimates, with comparisons to the two-grid method.

Exact Solutions for a Third‐Order KdV Equation with Variable Coefficients and Forcing Term

The general projective Riccati equation method and the Exp‐function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

An Estimator of Prediction Error Variance for Projection Equations

Abstract An estimator of prediction error variance for projection equations was derived using the first-order Taylor expansion in this study. The estimator, a modified estimator of the prediction error variance for a population mean regression model, was adapted for situations in which projection equations are applied to unsampled individuals. The estimator accounted for the errors associated with the response variable on the right side of a projection equation, as well as the errors associated with parameter estimation and serial correlations in data. The application of the estimator was demonstrated using the example of 140 trees with diameters measured annually from age 5 to 19. The results indicated that the estimator represented prediction error variance well and was useful for constructing prediction intervals for a given projection equation and providing information on the contributions of the variance components associated with different error sources. It was evident that the error associated with the prior observation of the response variable of a projection equation had an important impact on model predictions for forward projections. Overlooking this variance component may result in significant misestimation for the prediction errors of the projection equation.

A Dynamic Model for a Hybrid Articulated Robot

AbstractThe dynamic modeling of hybrid systems, consisting of flexible and rigid parts results in large partial differential equation systems (PDE). With the assumption of small deflections and the Ritz expansion the PDE can be approximated by an ordinary differential equation system (ODE) but the number of degrees of freedom is generally high.In this paper a hybrid articulated robot with 2 flexible links and 6 joints is under consideration. The joints are equipped with Harmonic Drive gears with high gear ratio but relative low stiffness. Therefore additionally degrees of freedom are introduced for the elastic deflection of the gears. The links are modeled with flexibility in two bending directions and in torsional sense.To be able to achieve structured equations the projection equation in subsystem representation is used. The projection equation is based on the momentum and the angular momentum equations of each single finite or infinitesimal body which are projected into the space of minimal coordinates and subsequently are summed up. Groups of bodies are collected to the so called subsystems with separated describing velocities. These subsystems are linked together with the kinematical chain. Because the robot is tree structured it is possible to obtain an explicit expression for the second derivatives of the minimal coordinates with a recursive scheme (O(n) efficiency).The robot is controlled with a feed forward controller and a linear PD joint controller. Simulation results and measured data are presented and compared. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new variable coefficient algebraic method and non-travelling wave solutions of nonlinear equations

In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the validity and the advantages of the method, (2+1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.

Application of Extended Projective Riccati Equation Method to (2+1)-Dimensional Broer–Kaup–Kupershmidt System

In this paper, extended projective Riccati equation method is presented for constructing more new exact solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the effect of the method, Broer–Kaup–Kupershmidt system is employed and Jacobi doubly periodic solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.

Closed-form solution for circle pose estimation using binocular stereo vision

A simple method for pose estimation of a circle, with known circle radius, on the basis of binocular stereo vision is presented. The method is based on the projective equation of a circle, into which the stereo vision constraints are incorporated. The estimated pose parameters are compared with the ground truth and excellent performance is reported.

A Neutral-Type Delayed Projection Neural Network for Solving Nonlinear Variational Inequalities

A neutral-type delayed projection neural network is proposed to deal with nonlinear variational inequalities. Compared with the existing delayed neural networks for linear variational inequalities, the proposed approach apparently has the larger application domain. By the theory of functional differential equation, a delay-dependent sufficient stability condition is derived. This stability condition is easily checked, and can guarantee that the proposed neural network is convergent to the solution of nonlinear variational inequality problem exponentially, which improves the existing stability criteria for the neutral-type delayed neural network. Moreover, many related problems, such as the projection equation and optimization problems, can also be dealt with by the proposed method. Finally, simulation examples are given to illustrate the satisfactory performance of the proposed method.

Analysis of a consistency recovery method for the 1D convection–diffusion equation using linear finite elements

AbstractFor residual‐based stabilization methods such as streamline‐upwind Petrov–Galerkin (SUPG) and finite calculus (FIC), the higher‐order derivatives of the residual that appear in the stabilization term vanish when simplicial elements are used. The sub‐grid scale method using orthogonal sub‐scales (OSS) attempts to recover the lost consistency by using a fine‐scale projected residual in the stabilization term. The FIC method may also be cast into an OSS form with very little manipulation using an auxiliary convective projection equation. This paper discusses the gain/loss by recovering the consistency of the discrete residual in the stabilization terms via the form that includes the convective projection (as in the OSS method). We present the von Neumann analysis of the FIC method with recovered consistency (FIC_RC) for the 1D convection–diffusion problem and we compare it with the standard Bubnov–Galerkin linear finite element method and FIC/SUPG methods. The transient analysis is done by examining the discrete dispersion relation of the stabilization methods. The spectral results for the semi‐discrete and fully discrete problem are presented with time integration done by the trapezoidal and second‐order backward differencing formula schemes. The effect of lumping the effective mass matrix T is considered relative to using a consistent form. The effect of refinement in space and time is also discussed. Finally, an optimal expression for the stabilization parameter for the FIC_RC method on a uniform grid and for the steady state is given and its performance in the transient mode is discussed. Copyright © 2008 John Wiley & Sons, Ltd.

Step size adjustment and extrapolation for time‐stepping schemes in non‐smooth dynamics

AbstractIn this paper we use step size adjustment and extrapolation methods to improve Moreau's time‐stepping scheme for the numerical integration of non‐smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non‐smooth systems in the smooth parts, we show how the time‐stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time‐stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright © 2008 John Wiley & Sons, Ltd.

Camera calibration using one perspective view of two arbitrary coplanar circles

We present a camera calibration technique to determine the focal length and the extrinsic parameters of a camera merely by using one perspective view of two coplanar circles with arbitrary radii and topological configuration. We believe our effort is valuable, in that the position of a camera and its focal length are frequently adjusted in many vision applications. In contrast to the iterative optimization technique in previous work, whose convergence relies heavily on proper initialization, we propose a closed-form solution on the basis of simple matrix operations, which turns out to be computationally efficient. It can also be used to initialize existing algorithms for higher accuracy and speed. Our method is based on the projective equation of a circle, first presented here, which naturally encodes the intrinsic and extrinsic camera parameters and the degenerate conic envelope spanned by the image of circular points. Extensive experiments with both synthetic data and real images verify the efficiency and robustness of our technique.

Extended general variational inequalities

In this work, we introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. Noor [M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed.

Variable Separated Solutions and Four-Dromion Excitations for (2+1)-Dimensional Nizhnik–Novikov–Veselov Equation

Using the mapping approach via the projective Riccati equations, several types of variable separated solutions of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation are obtained, including multiple-soliton solutions, periodic-soliton solutions, and Weierstrass function solutions. Based on a periodic-soliton solution, a new type of localized excitation, i.e., the four-dromion soliton, is constructed and some evolutional properties of this localized structure are briefly discussed.

Exact solutions to the sine-Gordon-type equations

In this paper, the sine-Gordon-type equations, including single sine-Gordon equation, double sine-Gordon equation and triple sine-Gordon equation are studied by introducing certain dependent variable transformations and the projective Riccati equation method. Several new exact solutions are obtained.

General model for moiré contouring, part 1: theory

In the current moiré literature, techniques to determine displacements, strains, and techniques to get geometrical parameters of surfaces using the shadow-projection moiré method are considered two separated branches of moiré. We have formulated a mathematical model that shows a deeper commonality between the two moiré applications: strain fields and surfaces are tensors of the second order. A direct consequence of this property is that a system of orthogonal grids is required in both cases when Cartesian tensors are utilized. The two systems of lines projected on a surface to get its contour are assimilated to parametric lines used in differential geometry to describe a surface. The classical moiré equations of projection and observation from infinity are extended to more general conditions of projection and observation. The use of four projectors (i.e., two groups of two projectors) in a mutually orthogonal system with one camera is shown to provide the necessary means to implement the model for high-accuracy contouring. Geometrical primitives are introduced to provide a simple and direct procedure to reduce all the measured values to a preselected coordinate system. Different views of the same surface are merged to the selected coordinate system directly without the need to introduce markers on the surface or utilize correlation methods to identify identical regions. Examples of the application of the new model of contouring to practical cases are presented in the companion paper “A general model for moiré contouring, part 2: applications.”

Analysis on the strip-based projection model for discrete tomography

Discrete tomography deals with image reconstruction of an object with finitely many gray levels (such as two). Different approaches are used to model the raw detector reading. The most popular models are line projection with a lattice of points and strip projection with a lattice of pixels/cells. The line-based projection model fits some applications but involves a major approximation since the x-ray beams of finite widths are simplified as line integrals. The strip-based projection model formulates projection equations according to the fractional areas of the intersection of each strip-shaped beam and the rectangular grid of an image to be reconstructed, so is more realistic in some applications. In this paper, we characterize the strip-based projection model and establish an equivalence between the system matrices generated by the strip-based and line-based projection models.

Equations of the Mayr Projection

In this study, the pseudo-cylindrical projection of Franz Mayr is examined in detail. The computation of one of Mayr's projection equations depends on the solution of an elliptical integral. It is this characteristic of the projection that most likely contributes to it being the neglected one among the group of the pseudo-cylindrical projections available today. Franz Mayr used the Legendre tables for the elliptical functions E and F and gave the plane coordinates within one-degree latitude intervals on the 90° meridian. The research reported here derives analytical expressions instead of using the elliptical integral and the interpolation between the table values. Four different solutions have been introduced for mapping applications. The distortion quantities are also presented and discussed.

Panorama Mosaic Optimization for Mobile Camera Systems

In order to enable the real-time image mosaic in mobile devices that have very limited computing power and memory, we propose an efficient image mosaic algorithm with integer arithmetic. The proposed algorithm is focused not only on the computational efficiency, but also on the performance improvement of image mosaic. For the efficient projection without transform estimation, we obtain successive images through the semitransparent view finder which shows the fixed amount of right part of previously captured image. We project each images onto cylindrical surface by the proposed projection equation and integer arithmetic. We align the projected images by hierarchical hexagon search and a matching measure that is robust to illumination changes caused by different exposures and vignette distortion. Finally, the images are naturally blended using color compensation and dynamic programming based stitching. The proposed mosaic algorithm is embedded and tested in mobile phone systems. According to various tests and experiments, the proposed algorithm shows good panorama composition in real-time compared with the other PC-based methods.

Mapping on the HEALPix grid

The natural spherical projection associated with the Hierarchical Equal-Area and Isolatitude Pixelization (HEALPix) is described and shown to be one of a hybrid class that combines the cylindrical equal-area and Collignon projections, not previously documented in the cartographic literature. Projection equations are derived for the class in general and are used to investigate its properties. It is shown that the HEALPix projection suggests a simple method of (i) storing and (ii) visualizing data sampled on the grid of the HEALPix pixelization, and also suggests an extension of the pixelization that is better suited for these purposes. Potentially useful properties of other members of the class are described, and new triangular and hexagonal pixelizations are constructed from them. Finally, the standard formalism is defined for representing the celestial coordinate system for any member of the class in the FITS data format.

Periodic structures of atmospheric internal gravity wave under the influence of air–sea interaction

A simple incompressible non-hydrostatic Boussinesq equation with the influence of air–sea interaction is applied to investigate the non-linear internal gravity waves in the atmosphere. By the phase plane analysis method and power series expansion method, we derived the KdV equation and Gardner equation. We obtain various periodic structures for the internal gravity waves with the help of direct integration method and the transformation method based on the projective Riccati equation analysis. It is shown that the air–sea interaction is very important for these periodic structures of the waves.