Quadratic Patches Research Articles

The fast methods of high-frequency wave scattering in complex media and structures

In this review work, we give a review of recent works of high-frequency electromagnetic (EM) wave scattering in complex media and structures. The EM wave scattering from electrically large scatterers remains an important and challenging problem. We introduce the quadratic patches to discretize the electrically large scatterers and present the fast physical optics (FPO) method to solve the PO radiation integral. The computational workload and accuracies of the physical optics (PO) scattered fields from the quadratic patches are error-controllable and frequency-independent, respectively. To remedy the limitations of the accuracies of the PO scattered fields in the lit region of the electrically large scatterers, we consider the Fock current from the convex scatterers to cover the contributions of creeping wave fields in the shadow region. For the coated material from the perfectly conducting structure, we propose the multi-level technique together with the PO method to calculate the scattered field in an efficient way. In the process of mesh discretization for electrically large scatterers, we adopt the quadratic patches and further extend them to adaptive patches. For the multi-scale scatterers, we use the hybrid method of moment and FPO (MoM-FPO) method to calculate the scattered field from the multi-scale scatterers.

Survey of Point Cloud Registration Methods and New Statistical Approach

The use of a 3D range scanning device for autonomous object description or unknown environment mapping leads to the necessity of improving computer methods based on identical point pairs from different point clouds (so-called registration problem). The registration problem and three-dimensional transformation of coordinates still require further research. The paper attempts to guide the reader through the vast field of existing registration methods so that he can choose the appropriate approach for his particular problem. Furthermore, the article contains a regression method that enables the estimation of the covariance matrix of the transformation parameters and the calculation of the uncertainty of the estimated points. This makes it possible to extend existing registration methods with uncertainty estimates and to improve knowledge about the performed registration. The paper’s primary purpose is to present a survey of known methods and basic estimation theory concepts for the point cloud registration problem. The focus will be on the guiding principles of the estimation theory: ICP algorithm; Normal Distribution Transform; Feature-based registration; Iterative dual correspondences; Probabilistic iterative correspondence method; Point-based registration; Quadratic patches; Likelihood-field matching; Conditional random fields; Branch-and-bound registration; PointReg. The secondary purpose of this article is to show an innovative statistical model for this transformation problem. The new theory needs known covariance matrices of identical point coordinates. An unknown rotation matrix and shift vector have been estimated using a nonlinear regression model with nonlinear constraints. The paper ends with a relevant numerical example.

Polyhedrons the Faces of which are Special Quadric Patches

We seize an idea of Oswald Giering (see [1] and [2]), who replaced pairs of faces of a polyhedron by patches of hyperbolic paraboloids and link up edge-quadrilaterals of a polyhedron with the pencil of quadrics determined by that quadrilateral. Obviously only ruled quadrics can occur. There is a simple criterion for the existence of a ruled hyperboloid of revolution through an arbitrarily given quadrilateral. Especially, if a (not planar) quadrilateral allows one symmetry, there exist two such hyperboloids of revolution through it, and if the quadrilateral happens to be equilateral, the pencil of quadrics through it contains even three hyperboloids of revolution with pairwise orthogonal axes. To mention an example, for right double pyramids, as for example the octahedron, the axes of the hyperboloids of revolution are, on one hand, located in the plane of the regular guiding polygon, and on the other, they are parallel to the symmetry axis of the double pyramid. Not only for platonic solids, but for all polyhedrons, where one can define edge-quadrilaterals, their pairs of face-triangles can be replaced by quadric patches, and by this one could generate roofing of architectural relevance. Especially patches of hyperbolic paraboloids or, as we present here, patches of hyperboloids of revolution deliver versions of such roofing, which are also practically simple to realize.

The Fast Physical Optics Method on Calculating the Scattered Fields From Electrically Large Scatterers

In this article, we propose the fast physical optics (FPO) method to calculate the scattered fields from the electrically large scatterers. The quadratic patches are adopted to discretize the surfaces of the scatterers. Invoking the quadratic amplitude and phase functions approximations, the closed-form formulas could be obtained with the complementary error function. Hence, the FPO method is time-saving compared with the conventional methods. Moreover, the accuracy and the physical insight of the closed-form formulas are expounded. Numerical examples demonstrate the efficiency of the FPO method with respect to both the accuracy and CPU time.

The Two-Dimensional Numerical Steepest Descent Path Method for Calculating the Physical Optics Scattered Fields From Different Quadratic Patches

In this article, the Two-Dimensional numerical steepest descent path (TD-NSDP) method is proposed to calculate the physical optics (PO) scattered fields from the quadratic patches. Compared with the NSDP method, the TD-NSDP method avoids adopting the complementary error function, and hence, the TD-NSDP method could significantly improve the accuracy of the PO scattered fields by around four to five digits. Moreover, the computational workload is also frequency independent. To demonstrate the efficiencies of the TD-NSDP method, numerical examples for both quadratic convex and concave surface patches are conducted. Finally, the radar cross section (RCS) simulation from the realistic and complex scatterer is conducted to verify the practicability of this method in electromagnetic (EM) scattering application.

Accelerating smooth molecular surface calculation.

This study proposes a novel approach, namely, skin flow complex algorithm (SFCA), to decompose the molecular skin surface into topological disks. The main contributions of SFCA include providing a simple decomposition and fast calculation of the molecular skin surface. Unlike most existing works which partition the molecular skin surface into sphere and hyperboloid patches, SFCA partitions the molecular skin surface into triangular quadratic patches and rectangular quadratic patches. Each quadratic patch is proven to be a topological disk and rendered by a rational Bézierpatch. The skin surface is constructed by assembling all rational Bézierpatches. Experimental results show that the SFCA is more efficient than most existing algorithms, and produces a triangulation of molecular skin surface which is decomposable, deformable, smooth, watertight and feature-preserved.

Geometric elements and classification of quadrics in rational Bézier form

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational Bézier triangular form (such as the center, the conic at infinity, the vertex and the axis of paraboloids and the principal planes), using just the control vertices and the weights for the quadric patch. The results are extended also to quadric tensor product patches. Our results rely on using techniques from projective algebraic geometry to find suitable bilinear forms for the quadric in a coordinate-free fashion, considering a pencil of quadrics that are tangent to the given quadric along a conic. Most of the information about the quadric is encoded in one coefficient, involving the weights of the patch, which allows us to tell apart oval from ruled quadrics. This coefficient is also relevant to determine the affine type of the quadric. Spheres and quadrics of revolution are characterized within this framework.

An intuitive way for constructing parametric quadric triangles

We present an intuitive algorithm for providing quadric surface design elements with shape parameters. To this end, we construct rational parametric triangular quadratic patches which lie on quadrics. The input of the algorithm is three vertex data points in 3D and normals at these points. It emanates from a thorough analysis of two existing methods for the construction of rational parametric Bezier triangles on quadrics, that allows to establish an interesting geometric relation between them. The sufficient condition for a configuration of vertex and normal data to allow for the existence of a rational triangular quadratic patch lying on a quadric whose tangent planes at the vertices are those prescribed by the given normals is the concurrence of certain cevians. When these conditions are not met we offer an optimization procedure to tweak the normals, without varying the vertex data, so that for the new normals there is a rational triangular quadratic patch that lies on a quadric. The resulting quadric design element offers three free shape parameters.

Shape analysis based on feature-preserving Elastic Quadratic Patch Modeling

Abstract An explicit shape representation, which is robust to noise and feature-preserving, is important to various tasks such as shape analysis and image processing. Elastic Quadratic Wire (EQW) is a recently proposed spline based shape representation model. It is particularly highlighted for good capacity in eliminating shape noises and preserving salient shape features, but is limited in representing only 2D planar shapes. In this study, we extend the EQW model into an Elastic Quadratic Patch (EQP) model to represent 3D parametric surfaces. In our model, we construct a planar overlapping parameterization space and represent all the surface points by quadratic patches. We then form the 0th and 1st order discontinuities between the neighboring patches into a quadratic energy function with an analytic minimal value. Therefore, the shape can be gradually fitted using an efficient iterative style. In experiments, we validate the EQP model in terms of efficient computation, effects on the adjustable parameter selection and performance in de-noising and preserving features. Experimental results on 3D facial surfaces demonstrate that our EQP model inherits all the strong points of the seminal EQW model, and is comparable or better than other frequently-used models in the shape smoothing task.

Continuous collision detection for composite quadric models

A composite quadric model (CQM) is an object modeled by piecewise linear or quadric patches. We study the continuous collision detection (CCD) problem of a special type of CQM objects which are commonly used in CAD/CAM, with their boundary surfaces intersect only in straight line segments or conic curve segments. We derive algebraic formulations and compute numerically the first contact time instants and the contact points of two moving CQMs in R3. Since it is difficult to process CCD of two CQMs in a direct manner because they are composed of semi-algebraic varieties, we break down the problem into subproblems of solving CCD of pairs of boundary elements of the CQMs. We present procedures to solve CCD of different types of boundary element pairs in different dimensions. Some CCD problems are reduced to their equivalents in a lower dimensional setting, where they can be solved more efficiently.

Exact tunneling solutions in Minkowski spacetime and a candidate for dark energy

We study exact tunneling solutions in scalar field theory for potential barriers composed of linear or quadratic patches. We analytically continue our solutions to imaginary Euclidean radius in order to study the profile of the scalar field inside the growing bubble. We find that generally there is a non-trivial profile of the scalar field, generating a stress-energy tensor, that depending on the form of the potential, can be a candidate for dark energy.

Design Quadratic Patch and Cubic Patch of the Surface

This paper describes a new method for designing quadric patch and cubic patch of the surfaces and show how to use Mathematica programs for computations the equations of the patch of the surfaces. The surface with quadratic patches is defined through a given set of six points and the surface with cubic patches is defined through a given set of ten points.

Surface Reconstruction from Unorganized Point Data with Quadrics

Abstract We present a reverse engineering method for constructing a surface approximation scheme whose input is a set of unorganized noisy points in space and whose output is a set of quadric patches. The local surface properties, necessary for the subsequent segmentation, are estimated directly from the data using a simple and efficient data structure—the neighborhood graph. Our segmentation scheme, based on principal curvatures, constructs initial point subsets, which may be enlarged or further subdivided based on associated approximation error estimates obtained through approximation of the initial segments by quadric surfaces. Our method is highly efficient and produces a high‐quality piecewise quadric surface approximation of engineering objects, which we demonstrate for several simple and complex example data sets.

Curvature-based algorithms for nonrigid motion and correspondence estimation

We present a novel technique for utilizing the Gaussian curvature information in 3D nonrigid motion estimation in the absence of known correspondence. Differential-geometric constraints derived in the paper allow one to estimate parameters of the local affine motion model given the values of Gaussian curvature before and after motion. These constraints can be further combined with the previously known constraints based on the unit normals before and after motion. Our experiments demonstrate that the resulting hybrid algorithm is more accurate than each of its constituents and more accurate than the classical ICP algorithm. We also present a technique for curvilinear orthogonalization of quadratic Monge patches that is essential in our derivation and useful in other applications.

Free-Form Modeling in Bilateral Brep and CSG Representation Schemes

This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.

Determination and classification of triangular quadric patches

A method for determining, if a given rational triangular Bézier patch of degree 2 lies on a quadric surface, and if so, for establishing the quadric's affine type, is presented. First, the question whether the patch is a quadric patch is solved by means of the related Veronese surface in five-dimensional projective space. Once established that the patch lies on a quadric the Gaussian curvature in one of the corner points of the patch is used for a rough classification yielding the projective type of the quadric. Then, the quadric's affine type is obtained by means of the quadric's intersection with the plane at infinity. An easy algorithm for the method is finally presented, together with several examples.

Transfinite interpolation of conic triangles using quadric patches

The use of quadric surface patches to form a continuous approximating surface to given curved surfaces was discussed in a paper by Baart and McLeod. A method was proposed for the construction of continuous patches on a covering mesh of conic segments, where each patch consists of three quadric subpatches, and each conic boundary segment interpolates three points on the given surface. This construction is not unique, and in this paper we investigate the effect of various combinations of the parameters on the resulting quadric patches. We also propose a method that will ensure that, were the given sculptured patch itself quadric, the approximation will be exact.

A graphics system based on quadrics

Today, hardware graphics systems use three and four-sided polygons as display primitives. Polygons are very easy to scan-convert and have largely influenced software and hardware developments. Yet, as triangles have well-known limitations, the purpose of this paper is to explore a new way of displaying scenes in real-time with other primitives: quadrics. Quadrics are usually used in CSG as natural quadrics. Moreover, recent works on low degree algebraic surfaces use them as quadric patches. After having presented the modeling aspects, we define a new basic object made of one quadric and planes. The main advantage of this object is genericity, i.e., it includes polygons, natural quadrics, quadric patches, and polyhedra. Specific computations and algorithms are necessary to scan-convert our basic object. They are described precisely, with hardware constraints in mind. Finally, a processor is proposed and its cost is estimated. So, we conclude that the processor is one step toward the design of a complete graphics system based on quadrics.

Quadric and cubic bitetrahedral patches

Quadric and cubic bitetrahedral patches

A Framework for Constructing Probability Distributions on the Space of Image Segmentations

The goal of traditional probabilistic approaches to image segmentation has been to derive a single, optimal segmentation, given statistical models for the image formation process. In this paper, we describe a new probabilistic approach to segmentation, in which the goal is to derive a set of plausible segmentation hypotheses and their corresponding probabilities. Because the space of possible image segmentations is too large to represent explicitly, we present a representation scheme that allows the implicit representation of large sets of segmentation hypotheses that have low probability. We then derive a probabilistic mechanism for applying Bayesian, model-based evidence to guide the construction of this representation. One key to our approach is a general Bayesian method for determining the posterior probability that the union of regions is homogeneous, given that the individual regions are homogeneous. This method does not rely on estimation and properly treats the issues involved when sample sets are small and estimation performance degrades. We present experimental results for both real and synthetic range data, obtained from objects composed of piecewise planar and implicit quadric patches.