Algebraic Spline Research Articles

C2 Cubic Algebraic Hyperbolic Spline Interpolating Scheme by Means of Integral Values

In this paper, a cubic Hermite spline interpolating scheme reproducing both linear polynomials and hyperbolic functions is considered. The interpolating scheme is mainly defined by means of integral values over the subintervals of a partition of the function to be approximated, rather than the function and its first derivative values. The scheme provided is C2 everywhere and yields optimal order. We provide some numerical tests to illustrate the good performance of the novel approximation scheme.

Implicit surface reconstruction with total variation regularization

Implicit representations have been widely used for surface reconstruction on account of their capability to describe shapes that exhibit complicated geometry and topology. However, extra zero-level sets or spurious sheets usually emerge in implicit algorithms and damage the reconstruction results. In this paper, we propose a reconstruction approach that involves the total variation (TV) of the implicit representation to minimize the occurrence of spurious sheets. Proof is given to show that the recovered shape has the simplest topology with respect to the input data. By using algebraic spline functions as the implicit representation, an efficient discretization is presented together with effective algorithms to solve it. Hierarchical structures with uniform subdivisions can be applied in the framework for fitting fine details. Numerical experiments demonstrate that our algorithm achieves high quality reconstruction results while reducing the existence of extra sheets.

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.

An Algebraic Spline Model of Molecular Surfaces for Energetic Computations

In this paper, we describe a new method to generate a smooth algebraic spline (AS) approximation of the molecular surface (MS) based on an initial coarse triangulation derived from the atomic coordinate information of the biomolecule, resident in the Protein data bank (PDB). Our method first constructs a triangular prism scaffold covering the PDB structure, and then generates a piecewise polynomial F on the Bernstein-Bezier (BB) basis within the scaffold. An ASMS model of the molecular surface is extracted as the zero contours of F, which is nearly C1 and has dual implicit and parametric representations. The dual representations allow us easily do the point sampling on the ASMS model and apply it to the accurate estimation of the integrals involved in the electrostatic solvation energy computations. Meanwhile comparing with the trivial piecewise linear surface model, fewer number of sampling points are needed for the ASMS, which effectively reduces the complexity of the energy estimation.

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G 2 continuous contact with the control polygon at two endpoints and is G 2 continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.

AN EFFICIENT HIGHER-ORDER FAST MULTIPOLE BOUNDARY ELEMENT SOLUTION FOR POISSON-BOLTZMANN BASED MOLECULAR ELECTROSTATICS.

In order to compute polarization energy of biomolecules, we describe a boundary element approach to solving the linearized Poisson-Boltzmann equation. Our approach combines several important features including the derivative boundary formulation of the problem and a smooth approximation of the molecular surface based on the algebraic spline molecular surface. State of the art software for numerical linear algebra and the kernel independent fast multipole method is used for both simplicity and efficiency of our implementation. We perform a variety of computational experiments, testing our method on a number of actual proteins involved in molecular docking and demonstrating the effectiveness of our solver for computing molecular polarization energy.

FAST MOLECULAR SOLVATION ENERGETICS AND FORCE COMPUTATION.

The total free energy of a molecule includes the classical molecular mechanical energy (which is understood as the free energy in vacuum) and the solvation energy which is caused by the change of the environment of the molecule (solute) from vacuum to solvent. The solvation energy is important to the study of the inter-molecular interactions. In this paper we develop a fast surface-based generalized Born method to compute the electrostatic solvation energy along with the energy derivatives for the solvation forces. The most time-consuming computation is the evaluation of the surface integrals over an algebraic spline molecular surface (ASMS) and the fast computation is achieved by the use of the nonequispaced fast Fourier transform (NFFT) algorithm. The main results of this paper involve (a) an efficient sampling of quadrature points over the molecular surface by using nonlinear patches, (b) fast linear time estimation of energy and inter-molecular forces, (c) error analysis, and (d) efficient implementation combining fast pairwise summation and the continuum integration using nonlinear patches.

Paths of algebraic hyperbolic curves

Cubic algebraic hyperbolic (AH) Bezier curves and AH spline curves are defined with a positive parameter α in the space spanned by {1, t, sinht, cosht}. Modifying the value of α yields a family of AH Bezier or spline curves with the family parameter α. For a fixed point on the original curve, it will move on a defined curve called “path of AH curve” (AH Bezier and AH spline curves) when α changes. We describe the geometric effects of the paths and give a method to specify a curve passing through a given point.

Simultaneous blending of convex polyhedra by [formula omitted] algebraic splines

In this paper, we present a method of simultaneous blending of implicitly defined convex polyhedral angles and solid models. This algorithm starts with a new suitable partition of the n -simplex over which the algebraic splines are defined. Then a cubic algebraic spline is constructed which meets the initial surfaces with G 2 continuity. Examples are provided to demonstrate the blending process. The results show some advantages of the new blending method.

Distance bounds of [formula omitted]-points on hypersurfaces

ε -Points were introduced by the authors (see [S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2–3) (2004) 627–650 (Special issue); S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147–181; S. Pérez-Díaz, J.R. Sendra, J. Sendra, Distance properties of ε -points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45–61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of ε -point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an ε -point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the ε -point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an ε -point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.

Data Smoothing and Interpolation Using Eighth-order Algebraic Splines

A new type of algebraic spline is used to derive a filter for smoothing or interpolating discrete data points. The spline is dependent on control parameters that specify the relative importance of data fitting and the derivatives of the spline. A general spline of arbitrary order is first formulated using matrix equations. We then focus on eighth-order splines because of the continuity of their first three derivatives (desirable for motor and robotics applications). The spline's matrix equations are rewritten to give a recursive filter that can be implemented in real time for lengthy data sequences. The filter is lowpass with a bandwidth that is dependent on the spline's control parameters. Numerical results, including a simple image processing application, show the tradeoffs that can be achieved using the algebraic splines.

Least-squares fitting of algebraic spline surfaces ∗

We present an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. By simultaneously approximating points and associated normal vectors, we obtain a method which is computationally simple, as the result is obtained by solving a system of linear equations. In addition, the result is geometrically invariant, as no artificial normalization is introduced. The potential applications of the algorithm include the reconstruction of free-form surfaces in reverse engineering. The paper also addresses the generation of exact error bounds, directly from the coefficients of the implicit representation.

Regular algebraic curve segments (III)—applications in interactive design and data fitting

In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves defined within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properties and possess a minimal number of inflection points. The regular algebraic spline curves considered here have many attractive features: They are easy to construct. There exist convenient geometric control handles to locally modify the shape of the curve. The error of the approximation is controllable. Since the spline curve is always inside the parallelogram, the error of the fit is bounded by the size of the parallelogram. The spline curve can be rapidly displayed, even though the algebraic curve segments are implicitly defined.

A-splines: local interpolation and approximation using Gk-continuous piecewise real algebraic curves

We provide sufficient conditions for the Bernstein–Bézier (BB) form of an implicitly defined bivariate polynomial over a triangle, such that the zero contour of the polynomial defines a smooth and single sheeted real algebraic curve segment. We call a piecewise G k -continuous chain of such real algebraic curve segments in BB-form as an A-spline (short for algebraic spline). We prove that the degree n A-splines can achieve in general G 2 n−3 continuity by local fitting and still have degrees of freedom to achieve local data approximation. As examples, we show how to construct locally convex cubic A-splines to interpolate and/or approximate the vertices of an arbitrary planar polygon with up to G 4 continuity, to fit discrete points and derivatives data, and approximate high degree parametric and implicitly defined curves. Additionally, we provide computable error bounds.

An Evaluation of Spline Functions for use in Cam Design

This paper shows how spline functions can be employed for kinematic motion specification in cam design. The polynomial spline is introduced as a special case of a continuous piecewise function. Cubic and quintic splines are derived and their properties are discussed in the cam design context. It is shown how standard cam laws can be approximated extremely accurately with a small number of points and appropriate boundary conditions. The modified sinusoidal acceleration cam law is used as an example. The application of quintic splines to non-standard and special motions is discussed. The algebraic and B-spline representations of spline functions are compared. The former is considered preferable in this context and a list of useful algorithms is given. The real power of the spline function, in particular the algebraic quintic spline, is its simplicity, ease of computation and adaptability to non-standard design problems. The use of parametrized, deficient and exponential splines is proposed for specific applications.