Quadratic Precision Research Articles

Quadratic serendipity element shape functions on general planar polygons

This paper proposes a method for the construction of quadratic serendipity element (QSE) shape functions on planar convex and concave polygons. Existing approaches for constructing QSE shape functions are linear combinations of the pair-wise products of generalized barycentric coordinates with linear precision, restricted to the convex polygonal domain or resort to numerical optimization. We extend the construction to general polygons with no more than three collinear consecutive vertices. This is done by defining coefficients of the linear combination as the oriented area of triangles with vertices from the polygonal domain, which can be either convex or concave. The proposed shape functions possess linear to quadratic precision. We prove the interpolation error estimates for mean value coordinate-based QSE shape functions on convex and concave polygonal domains satisfying a set of geometric constraints for standard finite element analysis. We also tailor a polygonal mesh generation scheme that improves the uniformity and avoids short edges of Voronoi diagrams for their use in the QSE-based polygonal finite element computation. Numerical tests for the 2D Poisson equations on various domains are presented, demonstrating the optimal convergence rates in both the L2-norm and the H1-seminorm.

Relationship between time-instant number and precision of ZeaD formulas with proofs

Recently, zeroing dynamics (ZD) method has shown wonderful effect in solving time-varying problems. Generally, the continuous-time ZD model should be discretized as a discrete-time algorithm for numerical computation. A good choice is to apply Zhang et al. discretization (ZeaD) formulas, which are effective time-discretization formulas. Actually, ZeaD formulas can also be applied to obtaining various discrete-time algorithms, not only the ZD algorithms. In the previous work, some piecemeal ZeaD formulas have been proposed and investigated. However, the relationship between the time-instant number and precision of ZeaD formulas is not found, which is emphatically investigated in this paper. Specifically, the ZeaD formulas from two to nine instants are investigated, and the general ZeaD formula groups are studied. Two-instant and three-instant ZeaD formula groups have linear precision at most. Four-instant ZeaD formula group has quadratic precision at most. Five-instant and six-instant ZeaD formula groups have cubic precision at most. Seven-instant and eight-instant ZeaD formula groups have quartic precision at most. Nine-instant ZeaD formula group has quintic precision at most. Theoretical analyses are presented to substantiate the relationship. Moreover, the ZeaD formulas as well as ZD method are applied to solving time-varying quadratic optimization problem, and the numerical results verify the effectiveness of ZeaD formulas.

Computing knots by quadratic and cubic polynomial curves

A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

On the hexagonal Shepard method

The problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit.

Quadratic serendipity finite elements over convex polyhedra

SummaryThe conventional approach to construct quadratic elements for an n‐sided polygon will yield n(n+1)/2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher‐order approximations over arbitrary polygons using the generalized barycentric and triangular coordinates. This approach ensures 2n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker delta property and satisfy constant, linear, and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate, and can tolerate extreme mesh distortion.

Extending linear finite elements to quadratic precision on arbitrary meshes

The method of finite elements can provide discrete versions of differential operators in general arrangements of nodes. Linear finite elements defined on a triangulation, like Delaunay’s, constitute a typical route towards this aim, as well as being a tool for interpolation. We discuss a procedure to build interpolating functions with quadratic precision from this functional set. The idea is to incorporate (extend) the products of the original functions in order to go from linear precision (the property that linear functions are reconstructed exactly) to quadratic precision (quadratic functions are reconstructed exactly). This procedure is applied to the standard Galerkin approach, in order to study the Poisson problem and the direct evaluation of the Laplacian. We discuss convergence of these problems in 1D and 2D. The extended method is shown to be superior in 2D, featuring convergence for the direct evaluation of the Laplacian in distorted lattices.

High-resolution images based on directional fusion of gradient

This paper proposes a novel method for image magnification by exploiting the property that the intensity of an image varies along the direction of the gradient very quickly. It aims to maintain sharp edges and clear details. The proposed method first calculates the gradient of the low-resolution image by fitting a surface with quadratic polynomial precision. Then, bicubic interpolation is used to obtain initial gradients of the high-resolution (HR) image. The initial gradients are readjusted to find the constrained gradients of the HR image, according to spatial correlations between gradients within a local window. To generate an HR image with high precision, a linear surface weighted by the projection length in the gradient direction is constructed. Each pixel in the HR image is determined by the linear surface. Experimental results demonstrate that our method visually improves the quality of the magnified image. It particularly avoids making jagged edges and bluring during magnification.

Enlarging Image by Constrained Least Square Approach with Shape Preserving

From a visual point of view, the shape of an image is mainly determined by the edges. Conventional polynomial interpolation of image enlarging methods would produce blurred edges, while edge-directed interpolation based methods would cause distortion in the non-edge areas. A new method for image enlarging is presented. The image is enlarged in two steps. In the first step, a fitting surface is constructed to interpolate the image data. To remove the zigzagging artifact, for each pixel, a fitting patch is constructed using edge information as constraints. The combination of all the patches forms the fitting surface which has the shape suggested by image data. Each point on the fitting surface can be regarded as a sampling point taken from a unit square domain, which means that when the fitting surface is used to enlarge the image, each sampling domain of the enlarged pixels is also a unit square, causing the enlarged image to lose some details. To make the enlarged image keep the details as many as possible, the sampling domain of the enlarged pixels should be less than a unit square. Then, in the second step, using the points taken from the fitting surface, new pixels are computed using constrained optimization technique to form the enlarged image, and the size of the sampling domain of the enlarged pixels is inversely proportional to the size of the enlarged image. The image enlarged by the new method has a quadratic polynomial precision. Comparison results show that the new method produces resized image with better quality.

Hierarchical Bases for Polygonal Finite Elements

Finite elements that are the shape of an arbitrary polygon offer several advantages over traditional triangles and quadrilaterals. Elements with linear and quadratic precision have been reported, but they have interpolatory bases. Hierarchical bases are described here. Elements with up to cubic precision are tested and it is confirmed that they give the expected convergence as the mesh density is increased, even when the elements are reentrant polygons. A magnetostatic problem is solved with elements of different orders.

Image Resizing by Piecewise Quadratic Polynomial with Edge Constraints

In the fleld of image resizing, general polynomial fltting surfaces which are based on the methods such as Lagrange, spline and least squared methods usually produce enlarged images with blurred edges. Our experiments showed, however, that when the edge information is used as a constraint to construct polynomial surfaces to resize the image, the enlarged image could preserve good edge characteristics and have better visual quality. A new method for constructing fltting patches to resize the image is presented. Associated with each pixel, a fltting patch is constructed using edge information as a constraint to make it have the shape suggested by the pixel and the adjacent pixels. The enlarged image pixels are computed by the combination of the points which are sampled from the fltting patches. The fltting patch has the quadratic polynomial precision. The new method has the advantages that it is easy to compute and produces the enlarged image with better visual quality and higher PSNR value. Comparison results of the new method with other methods are included.

Local computation of curve interpolation knots with quadratic precision

There are several prevailing methods for selecting knots for curve interpolation. A desirable criterion for knot selection is whether the knots can assist an interpolation scheme to achieve the reproduction of polynomial curves of certain degree if the data points to be interpolated are taken from such a curve. For example, if the data points are sampled from an underlying quadratic polynomial curve, one would wish to have the knots selected such that the resulting interpolation curve reproduces the underlying quadratic curve; in this case, the knot selection scheme is said to have quadratic precision. In this paper, we propose a local method for determining knots with quadratic precision. This method improves on our previous method that entails the solution of a global equation to produce a knot sequence with quadratic precision. We show that this new knot selection scheme results in better interpolation error than other existing methods, including the chord-length method, the centripetal method and Foley’s method, which do not possess quadratic precision.

A Quadratic precision generalized nonlinear global optimization migration velocity inversion method

An important research topic for prospecting seismology is to provide a fast accurate velocity model from pre-stack depth migration. Aiming at such a problem, we propose a quadratic precision generalized nonlinear global optimization migration velocity inversion. First we discard the assumption that there is a linear relationship between residual depth and residual velocity and propose a velocity model correction equation with quadratic precision which enables the velocity model from each iteration to approach the real model as quickly as possible. Second, we use a generalized nonlinear inversion to get the global optimal velocity perturbation model to all traces. This method can expedite the convergence speed and also can decrease the probability of falling into a local minimum during inversion. The synthetic data and Marmousi data examples show that our method has a higher precision and needs only a few iterations and consequently enhances the practicability and accuracy of migration velocity analysis (MVA) in complex areas.

Formula omitted] modeling with A-patches from rational trivariate functions

We approximate a manifold triangulation in R 3 using smooth implicit algebraic surface patches, which we call A-patches. Here each A-patch is a real iso-contour of a trivariate rational function defined within a tetrahedron. The rational trivariate function provides increased degrees of freedom so that the number of surface patches needed for free-form shape modeling is significantly reduced compared to earlier similar approaches. Furthermore, the surface patches have quadratic precision, that is they exactly recover quadratic surfaces. We give conditions under which a C 1 smooth and single sheeted surface patch is isolated from the multiple sheets.

Surface design using hand motion with smoothing

We describe elements used in a system to manipulate a polygon-based surface object using a data glove (made by Fifth Dimension Inc.). A direct-manipulation method is used that allows for placing new displacements on previous ones. Good results are obtained for manipulation at any incidence angle to the surface by using a new linear transformation on the selected vertices, in order to properly calculate their displacement magnitudes. The design of the manipulation system emphasizes intuitiveness and speed. A method for smoothing selected portions of the object is also presented. The smoothing tool does not change regions of the surface where vertex coordinates may be expressed as a second-order implicit function i.e. the method has quadratic precision. It operates by finding a least-squares fit, followed by a relaxation phase where vertices are moved near the surface approximation. The smoother operates on a set of disjoint points in space, no knowledge of connectivity is required by the surface smoother.

Nonnegative Interpolation with Clough‐Tocher Splines of Cubic Precision

AbstractIn ]4[ the Clough‐Tocher splines of quadratic precision are shown to allow always nonnegative interpolation of scattered data. The aim of the present note is to point out that Clough‐Tocher splines of cubic precision are suitable, too. This result extends to range restricted interpolation if the obstacles are piecewise constant. For piecewise linear and continuous obstacles the Clough‐Tocher splines may fail. However, for this type of restrictions it seems to be possible to derive an always working interpolation algorithm by means of the parametric Clough‐Tocher splines recently introduced in ]1[.

Cubic precision Clough-Tocher interpolation

The standard Clough-Tocher split-domain scheme constructs a surface element with quadratic precision. In this paper, I will look at methods for improving the degrees of freedom in Clough-Tocher schemes. In particular, I will discuss modifications to the cross-boundary construction that improve the interpolant from quadratic precision to cubic precision.

Using Subdivision on Hierarchical Data to Reconstruct Radiosity Distribution

Computing global illumination by finite element techniques usually generates a piecewise constant approximation of the radiosity distribution on surfaces. Directly displaying such scenes generates artefacts due to discretization errors. We propose to remedy this drawback by considering the piecewise constant output to be samples of a (piecewise) smooth function in object space and reconstruct this function by applying a binary subdivision scheme. We design custom taylored subdivision schemes with quadratic precision for the efficient refinement of cell‐ or pixel‐type data. The technique naturally allows to reconstruct functions from non‐uniform samples which result from adaptive binary splitting of the original domain (quadtree). This type of output is produced, e.g., by hierarchical radiosity algorithms. The result of the subdivision process can be mapped as a texture on the respective surface patch which allows to exploit graphics hardware for considerably accelerating the display.

Using Subdivision on Hierarchical Data to Reconstruct Radiosity Distribution

Computing global illumination by finite element techniques usually generates a piecewise constant approximation of the radiosity distribution on surfaces. Directly displaying such scenes generates artefacts due to discretization errors. We propose to remedy this drawback by considering the piecewise constant output to be samples of a (piecewise) smooth function in object space and reconstruct this function by applying a binary subdivision scheme. We design custom taylored subdivision schemes with quadratic precision for the efficient refinement of cell‐ or pixel‐type data. The technique naturally allows to reconstruct functions from non‐uniform samples which result from adaptive binary splitting of the original domain (quadtree). This type of output is produced, e.g., by hierarchical radiosity algorithms. The result of the subdivision process can be mapped as a texture on the respective surface patch which allows to exploit graphics hardware for considerably accelerating the display.

A discrete $C^1$ interpolant for tetrahedral data

We assume an underlying three-dimensional domain that has been tesselated into tetrahedra. A piecwise rational scheme is described that interpolates to function and gradient values at the vortices of the tetrahedra. No other data are required. The scheme is local and has quadratic precision. Its construction is described and numerical examples are given.

A symbolic system for computer‐aided development of surface interpolants

AbstractThe design and implementation of a symbolic input end computation package and its application to the development of several new surface interpolation schemes are described. Capabilities such as the composition of operators and symbolic differentiation have been incorporated into the system. This allows, in particular, the specification of Boolean sum projectors. The new schemes which have been implemented include an interpolant to randomly spaced data and a 'shape operator1 which has quadratic precision.