Projection Equations Research Articles

Instantaneous solitons and fractal solitons for a (2+1)-dimensional nonlinear system

By an improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Broek-Kaup system is derived. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as instantaneous solitons and fractal solitons are investigated.

Fusion and fission solitons for the (2+1)-dimensional generalized Breor–Kaup system

With a projective equation and a linear variable separation method, this paper derives new families of variable separation solutions (including solitory wave solutions, periodic wave solutions, and rational function solutions) with arbitrary functions for (2+1)-dimensional generalized Breor–Kaup (GBK) system. Based on the derived solitary wave excitation, it obtains fusion and fission solitons.

Perspective Reconstruction of a Spheroid from an Image Plane Ellipse

The perspective reconstruction of a spheroid's position and orientation from a single image plane ellipse is derived, by direct inversion of the projection equations, assuming the semi-major and semi-minor axes are known. Attention is paid to the geometric interpretation of the reconstruction. The reconstruction is formulated and reduced to an eigenvalue problem, to yield 2 solutions for the spheroid's position and orientation. The symmetry of the polar planes for these solutions are then deduced.

Multi-Soliton Excitations and Chaotic Patterns for the (2+1)-Dimensional Breaking-Soliton System

Starting from a projective equation and a linear variable separation approach, some solitary wave solutions with arbitrary functions for the (2+1)-dimensional breaking soliton system are derived. Based on the derived solution and by selecting appropriate functions, some novel localized excitations such as multi-solitons and chaotic-solitons are investigated.

Fractal Structures and Chaotic Behaviors in a (2 + 1)-Dimensional Nonlinear System

With a new projective equation, a series of solutions of the (2 + 1)-dimensional dispersive long-water wave system (LWW) is derived. Based on the derived solitary wave solution, we obtain some special fractal localized structures and chaotic patterns.

A series of exact solutions for (2 + 1)-dimensional Wick-type stochastic generalized Broer–Kaup system via a modified variable-coefficient projective Riccati equation mapping method

A modified variable-coefficient projective Riccati equation mapping method is applied to (2 + 1)-dimensional Wick-type stochastic generalized Broer–Kaup system. With the help of Hermit transformation, we obtain a series of new exact stochastic solutions to the stochastic Broer–Kaup system in the white noise environment.

A Study of Trajectory Models for Satellite Image Triangulation

Many Spaceborne imagery products are provided with metadata or support data having diverse types, representations, frequencies, and conventions. According to the variability of metadata, a compatible physical sensor model approach must be constructed. Among the three components of the sensor model, i.e., trajectory model, projection equations, and parameter subset selection, the construction of the position and attitude trajectory is closely linked with the availability and type of support data. In this paper, we show how trajectory models can be implemented based on support data from six satellite image types: QuickBird, Hyperion, SPOT-3, ASTER, PRISM, and EROS-A. Triangulation for each image is implemented to investigate the feasibility and suitability of the different trajectory models. The results show the effectiveness of some of the simple models while indicating that careful use of dense ephemeris information is necessary. These results are based on having a number of high quality ground control points.

New traveling waves solutions to generalized Kaup–Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup–Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.

Propagating solitons and chaotic behaviour of (2+1)-dimensional Korteweg-de Vries system

With a projective equation and a linear variable separation method, new exact solutions of the (2+1)-dimensional Korteweg-de Vries system(KdV) is derived. Based on the derived solitary wave excitation, we obtain the propagating solitons and study the chaotic behaviours with a new chaotic system.

Exact Solutions to KdV6 Equation by Using a New Approach of the Projective Riccati Equation Method

We study a new integrable KdV6 equation from the point of view of its exact solutions by using an improved computational method. A new approach to the projective Riccati equations method is implemented and used to construct traveling wave solutions for a new integrable system, which is equivalent to KdV6 equation. Periodic and soliton solutions are formally derived. Finally, some conclusions are given.

Instantaneous embed soliton and instantaneous taper-like soliton for the (3+1)-dimensional Burgers system

With a projective equation method and a variable separation method, new families of variable separation solutions (including solitory wave solutions, periodic wave solutions, and rational function solutions) for (3+1)-dimensional Burgers system is derived. Based on a derived solitory wave solution and a rational function solution, we obtain some novel localized structures such as instantaneous embed soliton and instantaneous taper-like soliton.

Multi-soliton solutions and fractal structures in a (2+1)-dimensional soliton system

By the projective equation approach and the variable separation method, a series of excitations of the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system is derived. Based on the derived solitary wave excitation, we obtain multi solitons and fractal solitons of the system.

New Exact Solutions and Novel Localized Excitations for the (2+1)-Dimensional Broek-Kaup System with Variable Coefficients

In this paper, a new projective equation (Φ' = σΦ +Φ2) is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Broek-Kaup system with variable coefficients (VCBK). Based on the derived solutions and by selecting appropriate functions, some novel folded solitary wave evolutional behaviours are investigated

Multi Dromion-Solitoff and Fractal Excitations for (2+1)-Dimensional Boiti–Leon–Manna–Pempinelli System

In this short note, a new projective equation (φ′ = σφ + φ2) is used to obtain the variable separation solutions with two arbitrary functions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli system (BLMP). Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as multi dromion-solitoffs and fractal-solitons are investigated.

A globally convergent non-interior point algorithm with full Newton step for second-order cone programming

A non-interior point algorithm based on projection for second-order cone programming problems is proposed and analyzed. The main idea of the algorithm is that we cast the complementary equation in the primal-dual optimality conditions as a projection equation. By using this reformulation, we only need to solve a system of linear equations with the same coefficient matrix and compute two simple projections at each iteration, without performing any line search. This algorithm can start from an arbitrary point, and does not require the row vectors of A to be linearly independent. We prove that our algorithm is globally convergent under weak conditions. Preliminary numerical results demonstrate the effectiveness of our algorithm.

Estimation of orientation of a textured planar surface using projective equations and separable analysis with M-channel wavelet decomposition

Estimation of the orientation of a textured planar surface is one of the basic tasks in the area of “shape from texture”. For the solution of this task, many successful approaches were proposed. In this paper, we have examined a few unaddressed questions: First, is there a mathematical formulation that relates the spectral characteristics of the texture pattern and the orientation of an inclined planar surface in a polar-coordinate system? Second, is there a good wavelet-based approach that produces an accurate estimate of the orientation angle of the textured planar surface by analyzing the spectral behavior of one single uncalibrated image? To answer these questions at first we present the formulation of a “texture projective equation”, which relates the depth and orientation of an inclined planar surface in a polar coordinate system with the spectral properties of its image texture. A suitable imaging geometry has been considered to enable separable analysis of the effect of inclination of the texture surface. Next, a method for shape from texture is presented based on discrete wavelet analysis to estimate the orientation of the planar surface. This approach although designed mainly for M-channel wavelets, is also applicable for dyadic wavelet analysis. Texture characteristics in the subbands of wavelet decomposition are analyzed using scalograms, and quantitatively evaluated based on texture projective equations. The proposed method of estimation of the orientation of a planar texture surface is evaluated using a set of simulated and real world textured images.

Camera calibration using two concentric circles: linear approach

We present a new algorithm for camera calibration using two concentric circles, which is a linear approach. In the calibration, a pinhole camera model is used. Different from previous methods, we take the projective equations of 3-D circles, which include the intrinsic parameter matrix of the camera, as the basis of our calibration approach. According to the special structure of the projective equations in algebra, we can get a linear equation system about the intrinsic parameters. After enough equations are constructed, the calibration can be easily finished. With at least three images of the two concentric circles, all five intrinsic parameters can be recovered. Experiments using computer simulated data and real data demonstrate the robustness and accuracy of our method.

Towards multi-perspective rasterization

We present a novel framework for real-time multi-perspective rendering. While most existing approaches are based on ray-tracing, we present an alternative approach by emulating multi-perspective rasterization on the classical perspective graphics pipeline. To render a general multi-perspective camera, we first decompose the camera into piecewise linear primitive cameras called the general linear cameras or GLCs. We derive the closed-form projection equations for GLCs and show how to rasterize triangles onto GLCs via a two-pass rendering algorithm. In the first pass, we compute the GLC projection coefficients of each scene triangle using a vertex shader. The linear raster on the graphics hardware then interpolates these coefficients at each pixel. Finally, we use these interpolated coefficients to compute the projected pixel coordinates using a fragment shader. In the second pass, we move the pixels to their actual projected positions. To avoid holes, we treat neighboring pixels as triangles and re-render them onto the GLC image plane. We demonstrate our real-time multi-perspective rendering framework in a wide range of applications including synthesizing panoramic and omnidirectional views, rendering reflections on curved mirrors, and creating multi-perspective faux animations. Compared with the GPU-based ray tracing methods, our rasterization approach scales better with scene complexity and it can render scenes with a large number of triangles at interactive frame rates.

Magnetothermoelectric response near quantum critical points

Following on from our previous work [M. J. Bhaseen , Phys. Rev. Lett. 98, 166801 (2007)] we examine the finite temperature magnetothermoelectric response in the vicinity of a quantum critical point (QCP). We begin with general scaling considerations relevant to an arbitrary QCP, either with or without Lorentz invariance, and in arbitrary dimension. In view of the broad connections to high-temperature superconductivity and cold atomic gases, we focus on the quantum critical fluctuations of the relativistic Landau-Ginzburg theory. This paradigmatic model arises in many contexts and describes the (particle-hole symmetric) superfluid-Mott insulator quantum phase transition in the Bose-Hubbard model. The application of a magnetic field opens up a wide range of physical observables, and we present a detailed overview of the charge and thermal transport and thermodynamic response. We combine several different approaches including the epsilon expansion and associated quantum Boltzmann equation, entropy drift, and arguments based on Lorentz invariance. The results differ markedly from the zero-field case, and we include an extended discussion of the finite thermal conductivity which emerges in the presence of a magnetic field. We derive an integral equation that governs its response and explore the crossover upon changing the magnetic field. This equation may be interpreted as a projection equation in the low-field limit, and clearly highlights the important role of collision invariants (or zero modes) in the hydrodynamic regime. Using an epsilon expansion around three dimensions, our analytic and numerical results interpolate between our previously published value and the exact limit of two-dimensional relativistic magnetohydrodynamics.

A New Projection-Based Neural Network for Constrained Variational Inequalities

This paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations. Five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of Lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function. The proposed neural network includes an existing model, and can be applied to solve some nonmonotone and nonsmooth problems. The validity and transient behavior of the proposed neural network are demonstrated by some numerical examples.