Implicit Polynomial Research Articles

Increasing the Fitting Precision of Implicit Polynomial Curves in 2D Modeling through Signomial Powers

Increasing the Fitting Precision of Implicit Polynomial Curves in 2D Modeling through Signomial Powers

Mathematical representation of 2D image boundary contour using fractional implicit polynomial

Mathematical representation of 2D image boundary contour using fractional implicit polynomial

Fitting objects with implicit polynomials by deep neural network

Fitting objects with implicit polynomials by deep neural network

Dynamic construction of a family of octic curves as geometric loci

<abstract><p>We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.</p></abstract>

Encoder-X: Solving Unknown Coefficients Automatically in Polynomial Fitting by Using an Autoencoder

Modeling, prediction, and recognition tasks depend on the proper representation of the objective curves and surfaces. Polynomial functions have been proved to be a powerful tool for representing curves and surfaces. Until now, various methods have been used for polynomial fitting. With a recent boom in neural networks, researchers have attempted to solve polynomial fitting by using this end-to-end model, which has a powerful fitting ability. However, the current neural network-based methods are poor in stability and slow in convergence speed. In this article, we develop a novel neural network-based method, called Encoder-X, for polynomial fitting, which can solve not only the explicit polynomial fitting but also the implicit polynomial fitting. The method regards polynomial coefficients as the feature value of raw data in a polynomial space expression and therefore polynomial fitting can be achieved by a special autoencoder. The entire model consists of an encoder defined by a neural network and a decoder defined by a polynomial mathematical expression. We input sampling points into an encoder to obtain polynomial coefficients and then input them into a decoder to output the predicted function value. The error between the predicted function value and the true function value can update parameters in the encoder. The results prove that this method is better than the compared methods in terms of stability, convergence, and accuracy. In addition, Encoder-X can be used for solving other mathematical modeling tasks.

Implicitization of the Vegter Yield Criterion

An advanced yield criterion plays a central role in rate-independent plasticity modeling of sheet metals in automotive and other industrial applications. Different from many non-quadratic yield criteria based on implicit polynomials, Vegter yield criterion is piecewise and quadratic as it is originally constructed by interpolation of several biaxial stress states using multiple second-order Bézier curves. Additional efforts in recent years have made Vetger yield criterion more user-friendly, flexible and robust for sheet metal forming applications. Nevertheless, Vetger yield criterion has only been presented in its parametric formulation in the literature. This work describes a method of reformulating the existing Vetger Lite yield criterion into the implicit form that is more commonly used in the conventional sheet metal plasticity modeling. The potential benefits of using both the original parametric and the new implicit forms of Vegter yield criterion for material parameter calibration, convexity certification, and computational simulations are discussed.

Investigation on multiple periodic solution branches of a system with piecewise linear nonlinear energy sink

The multiple solution problem of nonlinear energy sinks (NESs) is of great interest in academic research and engineering applications; however, there exists only limited work on the multiple solution branches of piecewise linear NESs. Piecewise linear NESs can give rise to the appearance of isolated solution branches (ISBs), which are harmful to the performance of dynamical systems. It is difficult to detect the ISBs of a dynamical system with a piecewise linear NES. Hence, in this study, the multiple periodic solution branch distribution of a four-degree-of-freedom dynamical system with a piecewise linear NES is investigated. A method based on polynomial homotopy that can capture ISBs is developed to search multiple solution branches. Spurious periodic solutions can exist owing to the use of the implicit polynomial approach; therefore, the harmonic balance method with high harmonic orders and alternating frequency/time-domain technique are employed to distinguish true periodic solutions. Numerical experiments demonstrate that the proposed method can search for multiple periodic solution branches. A parameter analysis is conducted on the multiple solution branches, and the inner and outer ISBs are observed.

Determining implicit equation of conic section from quadratic rational Bézier curve using Gröbner basis

The Gröbner Basis is a subset of finite generating polynomials in the ideal of the polynomial ring k[x 1,…,xn ]. The Gröbner basis has a wide range of applications in various areas of mathematics, including determining implicit polynomial equations. The quadratic rational Bézier curve is a rational parametric curve that is generated by three control points P 0(x 0,y 0), P 1(xi ,yi ), P 2(x 2,y 2) in ℝ2 and weights ω 0, ω 1, ω 2, where the weights ω i are corresponding to control points Pi (xi, yi ), for i = 0,1, 2. According to Cox et al (2007), the quadratic rational Bézier curve can represent conic sections, such as parabola, hyperbola, ellipse, and circle, by defining the weights ω 0 = ω 2 = 1 and ω 1 = ω for any control points P 0(x 0, y 0), P 1(x 1, y 1), and P 2(x 2, y 2). This research is aimed to obtain an implicit polynomial equation of the quadratic rational Bézier curve using the Gröbner basis. The polynomial coefficients of the conic section can be expressed in the term of control points P 0(x 0, y 0), P 1(x 1, y 1), P 2(x 2, y 2) and weight ω, using Wolfram Mathematica. This research also analyzes the effect of changes in weight ω on the shape of the conic section. It shows that parabola, hyperbola, and ellipse can be formed by defining ω = 1, ω > 1, and 0 < ω < 1, respectively.

Polynomial Fitting Algorithm Based on Neural Network

As a method of function approximation, polynomial fitting has always been the main research hotspot in mathematical modeling. In many disciplines such as computer, physics, biology, neural networks have been widely used, and most of the applications have been transformed into fitting problems using neural networks. One of the main reasons that neural networks can be widely used is that it has a certain sense of universal approximation. In order to fit the polynomial, this paper constructs a three-layer feedforward neural network, uses Taylor series as the activation function, and determines the number of hidden layer neurons according to the order of the polynomial and the dimensions of the input variables. For explicit polynomial fitting, this paper uses non-linear functions as the objective function, and compares the fitting effects under different orders of polynomials. For the fitting of implicit polynomial curves, the current popular polynomial fitting algorithms are compared and analyzed. Experiments have proved that the algorithm used in this paper is suitable for both explicit polynomial fitting and implicit polynomial fitting. The algorithm is relatively simple, practical, easy to calculate, and can efficiently achieve the fitting goal. At the same time, the computational complexity is relatively low, which has certain application value.

HUMAN-INSPIRED ALGEBRAIC CURVES FOR WEARABLE ROBOT CONTROL.

Having unified representations of human walking gait data is of paramount importance for wearable robot control. In the rehabilitation robotics literature, control approaches that unify the gait cycle of wearable robots are more appealing than the conventional approaches that rely on dividing the gait cycle into several periods, each with their own distinct controllers. In this article we propose employing algebraic curves to represent human walking data for wearable robot controller design. In order to generate algebraic curves from human walking data, we employ the 3L fitting algorithm, a tool developed in the pattern recognition literature for fitting implicit polynomial curves to given datasets. For an impedance model of the knee joint motion driven by the hip angle signal, we provide conditions by which the generated algebraic curves satisfy a robust relative degree condition throughout the entire walking gait cycle. The robust relative degree property makes the algebraic curve representation of walking gaits amenable to various nonlinear output tracking controller design techniques.

Curve and surface fitting by implicit polynomials: Optimum degree finding and heuristic refinement

Finding an implicit polynomial that fits a set of observations X is the goal of many researches in recent years. However, most existing algorithms assume the knowledge of the degree of the implicit polynomial that best represents the points. This paper presents two main contributions. First, a new distance measure between X and the implicit polynomial is defined. Second, this distance is used to define an algorithm able to find the degree of the polynomial needed for the representation of the data set. The proposed algorithm is based on the idea of gradually increase the degree, while there is an improvement in the smoothness of the solutions. The experiments confirm the validity of the approach for the selected 2D and 3D datasets.

Detecting When an Implicit Equation or a Rational Parametrization Defines a Conical or Cylindrical Surface, or a Surface of Revolution.

Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e., the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations.

Sampling and Reconstruction of Shapes With Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of $\mathds{1}_{\{p\leq0\}}$ to the image plane, where $\mathds{1}$ denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.

A Novel Fractional Implicit Polynomial Approach for Stable Representation of Complex Shapes

Implicit polynomials (IPs) are applied to represent 2D object shapes in image processing and computer vision. However, it is difficult for IPs to represent complex object shapes due to high computational cost and high instability. In this work, we present a new representation model based on IPs, which is called fractional implicit polynomial (FIP). Firstly, the general formula for FIP and a definition of base are given; secondly, we investigate the properties of FIPs and conclude that the FIP representation exhibits higher stability and higher power than IP representation due to the presence of the base. Thirdly, we develop an algorithm for determination of a moderate degree for an FIP to represent a given shape, which can be obtained by only computing the number of stationary points on the shapes. We compare FIPs with IPs by test on various object shapes and the results show that the FIP is indeed sufficiently powerful to represent the complex object shapes.

Almost-toric hypersurfaces

An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit polynomial equation.

An improved star test for implicit polynomial objects

For a given point set, a particular point is called a star if it can see all the boundary points of the set. The star test determines whether a candidate point is a star for a given set. It is a key component of some topology computing algorithms such as Connected components via Interval Analysis (CIA), Homotopy type via Interval Analysis (HIA), etc. Those algorithms decompose the input object using axis-aligned boxes, so that each box is either not intersecting or intersecting with the object and in this later case its center is a star point of the intersection. Graphs or simplicial complexes describing the topology of the objects can be obtained by connecting these star points following different rules. The star test is performed for simple primitive geometric objects, because complex objects can be constructed using Constructive Solid Geometry (CSG), and the star property is preserved via union and intersection. In this paper, we improve the method to perform the test for implicit objects. For a primitive set defined by an implicit polynomial equation, the polynomial is made homogeneous with the introduction of an auxiliary variable, thus the degree of the star condition is reduced. A linear programming optimization is introduced to further improve the performance. Several examples are given to show the experimental results of our method.

Geometric operations using sparse interpolation matrices

Based on the computation of a superset of the implicit support, implicitization of a parametrically given hypersurface is reduced to computing the nullspace of a numeric matrix. Our approach predicts the Newton polytope of the implicit equation by exploiting the sparseness of the given parametric equations and of the implicit polynomial, without being affected by the presence of any base points. In this work, we study how this interpolation matrix expresses the implicit equation as a matrix determinant, which is useful for certain operations such as ray shooting, and how it can be used to reduce some key geometric predicates on the hypersurface, namely membership and sidedness for given query points, to simple numerical operations on the matrix, without need to develop the implicit equation. We illustrate our results with examples based on our Maple implementation.

The Richer Representation the Better Registration

In this paper, the registration problem is formulated as a point to model distance minimization. Unlike most of the existing works, which are based on minimizing a point-wise correspondence term, this formulation avoids the correspondence search that is time-consuming. In the first stage, the target set is described through an implicit function by employing a linear least squares fitting. This function can be either an implicit polynomial or an implicit B-spline from a coarse to fine representation. In the second stage, we show how the obtained implicit representation is used as an interface to convert point-to-point registration into point-to-implicit problem. Furthermore, we show that this registration distance is smooth and can be minimized through the Levengberg–Marquardt algorithm. All the formulations presented for both stages are compact and easy to implement. In addition, we show that our registration method can be handled using any implicit representation though some are coarse and others provide finer representations; hence, a tradeoff between speed and accuracy can be set by employing the right implicit function. Experimental results and comparisons in 2D and 3D show the robustness and the speed of convergence of the proposed approach.

Phase transitions of thep-spin model on pure Husimi lattices

We consider the p-spin model with spin 1/2 on all pure Husimi lattices. Using an effective representation of the recursion relations, the phase transitions of the model on all pure Husimi lattices are investigated. First, the nonexistence of the second order phase transitions in the model on all pure Husimi lattices is proven exactly. Then the existence and properties of the first order phase transitions in a zero external magnetic field are studied in detail. An implicit polynomial equation for determining temperatures below which the paramagnetic and ferromagnetic phases coexist in the model on all pure Husimi lattices is found. In addition, an implicit equation for exactly determining the transition temperatures of the first order phase transitions in a zero external magnetic field on all pure Husimi lattices is derived and discussed.

A coarse-to-fine IP-driven registration for pose estimation from single ultrasound image

A fast registration making use of implicit polynomial (IP) models is helpful for the real-time pose estimation from single clinical free-hand Ultrasound (US) image, because it is superior in the areas such as robustness against image noise, fast registration without enquiring correspondences, and fast IP coefficient transformation. However it might lead to the lack of accuracy or failure registration.In this paper, we present a novel registration method based on a coarse-to-fine IP representation. The approach starts from a high-speed and reliable registration with a coarse (of low degree) IP model and stops when the desired accuracy is achieved by a fine (of high degree) IP model. Over the previous IP-to-point based methods our contributions are: (i) keeping the efficiency without requiring pair-wised correspondences, (ii) enhancing the robustness, and (iii) improving the accuracy. The experimental result demonstrates the good performance of our registration method and its capabilities of overcoming the limitations of unconstrained freehand ultrasound data, resulting in fast, robust and accurate registration.