Parametric Surface Representation Research Articles

A Posteriori Error Estimates for the Laplace--Beltrami Operator on Parametric $C^2$ Surfaces

We prove new a posteriori error estimates for surface finite element methods (SFEMs). SFEMs approximate solutions to PDEs posed on surfaces. Prototypical examples are elliptic PDEs involving the Laplace--Beltrami operator. Typically the surface is approximated by a polyhedral or higher-order polynomial approximation. The resulting finite element method exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEMs require practical access to geometric information about the surface in order to computably bound the geometric error. It is thus advantageous to allow for maximum flexibility in representing surfaces in practical codes when proving a posteriori error estimates for SFEMs. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on $C^2$ surfaces. Proofs of error estimates optimally reflecting the geometric error instead employ the closest point projection, which is defined using the signed distance function. Because the closest point projection is often unavailable or inconvenient to use computationally, a posteriori estimates using the signed distance function have notable practical limitations. We merge these two perspectives by assuming practical access only to a general parametric representation of the surface, but using the distance function as a theoretical tool. This allows us to derive sharper geometric estimators which exhibit improved experimentally observed decay rates when implemented in adaptive surface finite element algorithms.

Tools and techniques for building models for isogeometric analysis

Isogeometric Analysis (IGA) is a new computer method that has gained much attention in recent years. Its distinguishing feature is the use of parametric representation of curves, surfaces and solids, based on Non-Uniform Rational B-Splines, which are also used as shape functions in Galerkin approximation schemes. NURBS are commonly used in CAD modelling packages, thus among other advantages IGA facilitates use of CAD models as geometric domains for solving Boundary Value Problems. IGA is a relatively young method and despite its popularity there still is a small number of tools available, especially in comparison with Finite Element Method. This is notably nagging issue in case of tools for preparing geometric models for IGA. In this paper we discuss several popular tools and tricks that can be used in place of dedicated IGA preprocessor. The presented techniques can be particularly useful for researchers interested in starting work in IGA field, as they can speed up generation of comprehensive models and are incentive for data sharing, an important point from research reproducibility perspective.

Level set shape and topology optimization of finite strain bilateral contact problems

SummaryThis paper presents a method for the optimization of multicomponent structures comprised of 2 and 3 materials considering large motion sliding contact and separation along interfaces. The structural geometry is defined by an explicit level set method, which allows for both shape and topology changes. The mechanical model assumes finite strains, a nonlinear elastic material behavior, and a quasi‐static response. Identification of overlapping surface position is handled by a coupled parametric representation of contact surfaces. A stabilized Lagrange method and an active set strategy are used to model frictionless contact and separation. The mechanical model is discretized by the extended FEM, which maintains a clear definition of geometry. Face‐oriented ghost penalization and dynamic relaxation are implemented to improve the stability of the physical response prediction. A nonlinear programming scheme is used to solve the optimization problem, which is regularized by introducing a perimeter penalty into the objective function. Design sensitivities are determined by the adjoint method. The main characteristics of the proposed method are studied by numerical examples in 2 dimensions. The numerical results demonstrate improved design performance when compared to models optimized with a small strain assumption. Additionally, examples with load path dependent objectives display nonintuitive designs.

Parametric Surface Representation with Bump Image for Dense 3D Modeling Using an RBG-D Camera

When constructing a dense 3D model of an indoor static scene from a sequence of RGB-D images, the choice of the 3D representation (e.g. 3D mesh, cloud of points or implicit function) is of crucial ...

$${\mathcal {H}}$$ H -matrix Accelerated Second Moment Analysis for Potentials with Rough Correlation

We consider the efficient solution of partial differential equations for strongly elliptic operators with constant coefficients and stochastic Dirichlet data by the boundary integral equation method. The computation of the solution's two-point correlation is well understood if the two-point correlation of the Dirichlet data is known and sufficiently smooth. Unfortunately, the problem becomes much more involved in case of roughly correlated data. We will show that the concept of the $${\mathcal {H}}$$H-matrix arithmetic provides a powerful tool to cope with this problem. By employing a parametric surface representation, we end up with an $${\mathcal {H}}$$H-matrix arithmetic based on balanced cluster trees. This considerably simplifies the implementation and improves the performance of the $${\mathcal {H}}$$H-matrix arithmetic. Numerical experiments are provided to validate and quantify the presented methods and algorithms.

Parametric Surfaces Competition: Using Technology to Foster Creativity

Although most calculus students are comfortable with the Cartesian equations of curves and surfaces, they struggle with the concept of parameters. A multivariable calculus course is really the time to nail this concept down, once and for all, since it provides an easy way to represent many beautiful and useful surfaces, and graph them using a Computer Algebra System. Once the students grasp the concept of parametric representations of curves and surfaces, they can actually get quite creative, and even draw some very interesting computer art! In this paper we describe how we lead our students to combine mathematics and technology with creativity in an interesting way.

Gouge-free voxel-based machining for parallel processors

Manufacturers produce complex parts by utilizing computer-aided manufacturing (CAM) software to generate tool paths for a machine tool to follow. CAM systems traditionally rely on parametric surface representations of the parts and complex algorithms to produce the tool paths. This paper presents a new path generation framework that is based on parallel algorithms and hardware that utilizes voxel models. The use of new parallel algorithms allows rapid calculations of tool paths automatically from the voxel model. The core components of this framework are a method of generating digital voxel models from tessellated surface models, a method to obtain digital surface information by a parallel ray-casting approach, and a new approach to calculate gouge-free tool paths in parallel from surface information and generalized cutting geometry. The performance of utilizing digital models within this framework is then discussed with respect to timing and actual cutting results.

Comparison of fast boundary element methods on parametric surfaces

We compare fast black-box boundary element methods on parametric surfaces in R3. These are the adaptive cross approximation, the multipole method based on interpolation, and the wavelet Galerkin scheme. The surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels. Nonetheless, parametric surface representations are easily accessible from computer aided design (CAD) and are recently topic of the studies in isogeometric analysis. Especially, we can apply two-dimensional interpolation in the multipole method. A main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry. This results in a superior compression of the far-field compared to other cluster methods.

Semiautomatic bladder segmentation on CBCT using a population-based model for multiple-plan ART of bladder cancer

The aim of this study is to develop a novel semiautomatic bladder segmentation approach for selecting the appropriate plan from the library of plans for a multiple-plan adaptive radiotherapy (ART) procedure. A population-based statistical bladder model was first built from a training data set (95 bladder contours from 8 patients). This model was then used as constraint to segment the bladder in an independent validation data set (233 CBCT scans from the remaining 22 patients). All 3D bladder contours were converted into parametric surface representations using spherical harmonic expansion. Principal component analysis (PCA) was applied in the spherical harmonic-based shape parameter space to calculate the major variation of bladder shapes. The number of dominating PCA modes was chosen such that 95% of the total shape variation of the training data set was described. The automatic segmentation started from the bladder contour of the planning CT of each patient, which was modified by changing the weight of each PCA mode. As a result, the segmentation contour was deformed consistently with the training set to best fit the bladder boundary in the localization CBCT image. A cost function was defined to measure the goodness of fit of the segmentation on the localization CBCT image. The segmentation was obtained by minimizing this cost function using a simplex optimizer. After automatic segmentation, a fast manual correction method was provided to correct those bladders (parts) that were poorly segmented. Volume- and distance-based metrics and the accuracy of plan selection from multiple plans were evaluated to quantify the performance of the automatic and semiautomatic segmentation methods. For the training data set, only seven PCA modes were needed to represent 95% of the bladder shape variation. The mean CI overlap and residual error (SD) of automatic bladder segmentation over all of the validation data were 70.5% and 0.39 cm, respectively. The agreement of plan selection between automatic bladder segmentation and manual delineation was 56.7%. The automatic segmentation and visual assessment took on average 7.8 and 9.7 s, respectively. In 53.4% of the cases, manual correction was performed after automatic segmentation. The manual correction improved the mean CI overlap, mean residual error and plan selection agreement to 77.7%, 0.30 cm and 80.7%, respectively. Manual correction required on average 8.4 markers and took on average 35.5 s. The statistical shape-based segmentation approach allows automatic segmentation of the bladder on CBCT with moderate accuracy. Limited user intervention can quickly and reliably improve the bladder contours. This segmentation method is suitable to select the appropriate plan for multiple-plan ART of bladder cancer.

Parametric curves and surfaces: Mathematica demonstrations as a tool in exploration of architectural form

Through their work in the field of generic explorations, authors of this paper have developed a series of demonstration projects, based on the software package MATHEMATICA, related to the parametric representation of curves and surfaces in space. The main purpose of the demonstrations is exploration of generic potential of curves and surfaces in order to apply it to the process of generating architectural form. The paper presents the developed demonstration projects and their application within the generic explorations of architectural form.

Generating Triangulated Macromolecular Surfaces by Euclidean Distance Transform

Macromolecular surfaces are fundamental representations of their three-dimensional geometric shape. Accurate calculation of protein surfaces is of critical importance in the protein structural and functional studies including ligand-protein docking and virtual screening. In contrast to analytical or parametric representation of macromolecular surfaces, triangulated mesh surfaces have been proved to be easy to describe, visualize and manipulate by computer programs. Here, we develop a new algorithm of EDTSurf for generating three major macromolecular surfaces of van der Waals surface, solvent-accessible surface and molecular surface, using the technique of fast Euclidean Distance Transform (EDT). The triangulated surfaces are constructed directly from volumetric solids by a Vertex-Connected Marching Cube algorithm that forms triangles from grid points. Compared to the analytical result, the relative error of the surface calculations by EDTSurf is <2–4% depending on the grid resolution, which is 1.5–4 times lower than the methods in the literature; and yet, the algorithm is faster and costs less computer memory than the comparative methods. The improvements in both accuracy and speed of the macromolecular surface determination should make EDTSurf a useful tool for the detailed study of protein docking and structure predictions. Both source code and the executable program of EDTSurf are freely available at http://zhang.bioinformatics.ku.edu/EDTSurf.

A semi‐analytic approach for direct slicing of free form surfaces for layered manufacturing

PurposeThis paper aims to develop an efficient surface‐plane intersection (SPI) algorithm for direct slicing of free‐form surfaces to be produced by layered manufacturing.Design/methodology/approachA semi‐analytical method for direct slicing has been formulated and tested on Bezier and B‐spline surfaces commonly used in CAD modeling. This method solves for the intersection points by a “root” finding procedure and establishes their connectivity, unlike the conventional “marching” procedures.FindingsThe proposed algorithm solves intersection contours between free form surfaces and planes. The solution procedure is efficient with respect to computational time and accuracy (feature detection) over some of the conventional SPI strategies. The method involves a global solution procedure in contention with the traditional methodologies which are generally spatially distinctive in approach.Research limitations/implicationsUse of higher order terms in the representation of parametric surfaces makes the algorithm computationally intensive and time‐expensive.Practical implicationsThis algorithm would be of practical use in the direct slicing of free form surfaces used in CAD modeling. Direct slicing methods solve for the actual intersection of surface and plane without resorting to “tessellation.” Reducing the computation time and detection of features within a given resolution is of primary importance for developing commercial rapid prototyping software, which is achieved in the present paper.Originality/valueA novel method has been developed for SPI for use in direct slicing of CAD models. While a major proportion of the direct slicing strategies employ the “marching” procedure involving determination of “critical points,” the proposed method utilizes the evaluation of “roots” of a surface in a global manner to determine the intersection points with proper connectivity. Hence, it is effective in reducing the computation time and is simple but generic in approach. Although Bezier and B‐spline surfaces are used as the representative cases, the algorithm can be extended for any parametric surface for direct slicing.

Formulação geométrica do princípio de d'Alembert

Por meio de uma extensão natural da noção de representação paramétrica de superfícies no espaço euclidiano tridimensional, fazemos uma formulação geométrica do princípio de d'Alembert sem o uso de quantidades infinitesimais.

Single equation without inequalities to represent a composite curve

A scheme is proposed based on an extension of Analytic Geometry to represent a composite curve either with a single equation or with two equations in the case of parametric representation, or three equations in the case of parametric representation of 3D surfaces, without the use of inequalities and valid from minus infinity to plus infinity. The general equations of the regular polygon, the polygonal cylinder as well as the equation of the main view of the chain sprocket are established.

Surface parameterization in volumetric images for curvature-based feature classification

Curvature-based surface features are well suited for use in multimodal medical image registration. The accuracy of such feature-based registration techniques is dependent upon the reliability of the feature computation. The computation of curvature features requires second derivative information that is best obtained from a parametric surface representation. We present a method of explicitly parameterizing surfaces from volumetric data. Surfaces are extracted, without a global thresholding, using active contour models. A monge/spl acute/ basis for each surface patch is estimated and used to transform the patch into local, or parametric, coordinates. Surface patches are fit to a bicubic polynomial in local coordinates using least squares solved by singular value decomposition. We tested our method by reconstructing surfaces from the surface model and analytically computing Gaussian and mean curvatures. The model was tested on analytical and medical data.

Mesh generation in curvilinear domains using high‐order elements

AbstractThe ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high‐order algorithms in very complex geometries. Compact high‐order algorithms such as spectral element or p‐type finite element techniques offer the potential of high accuracy if the solution is smooth and a well‐behaved mapping exists between the local sub‐domains and a standard region. In this paper we address the problem of coupling high‐order algorithms with unstructured mesh generation and CAD representation techniques. We present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings. The first strategy investigates the influence of an anisometric parametric surface representation on the quality of high‐order unstructured meshes. We show that a straightforward splitting of a standard linear mesh for use in a high‐order method can lead to the generation of distorted, possibly invalid, meshes even for simple computational domains such as cubes. The second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries. In the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher‐order information about the surface into the mesh generation. Using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high‐order discretization of the computational domain. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Models for character animation

GORDON COLLINS and ADRIAN HILTON present a review of methods for the construction and deformation of character models. They consider both state of the art research and common practice. In particular they review applications, data capture methods, manual model construction, polygonal, parametric and implicit surface representations, basic geometric deformations, free form deformations, subdivision surfaces, displacement map schemes and physical deformation. Copyright © 2001 John Wiley & Sons, Ltd.

Methods to obtain quantitative parametric descriptions of the optical surfaces of the human crystalline lens from Scheimpflug slit-lamp images. I. Image processing methods.

Of the methods developed (e.g., phakometry, magnetic resonance imaging, etc.) for noninvasive measurement of the geometry of the anterior segment of the human eye, Scheimpflug photography offers the best resolution and the highest precision. The primary obstacle encountered with this or any other image-based method has been in obtaining quantitative measurements directly from the images. Image enhancement (gray-scale gradient analysis) and pattern recognition methods (Hough transform and recursive least-squares algorithms) are developed so that parametric representations of lens surfaces and zone boundaries can be obtained directly from the images. Methods to correct for nonlinear Scheimpflug camera reproduction ratios and provide error estimates for geometrical parameters are also developed and will be presented separately. Combined, these techniques yield representations of lens geometry having sufficient precision, to which paraxial ray tracing can be applied to determine lens optical properties by using well-posed optical models with one unknown.

Wavelet Multiresolution Representation of Curves and Surfaces

We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used filters. We also give examples of the applications of the wavelet approach: these include the derivation of compact hierarchical curve and surface representations using modified wavelet compression, the identification of smooth sections of surfaces, and a subdivision-like intersection algorithm for discrete plane curves.

Surface Curve Design by Orthogonal Projection of Space Curves Onto Free-Form Surfaces

A novel technique for designing curves on surfaces is presented. The design specifications for this technique derive from other works on curvature continuous surface fairing. Briefly stated, the technique must provide a computationally efficient method for the design of surface curves that is applicable to a very general class of surface formulations. It must also provide means to define a smooth natural map relating two or more surface curves. The resulting technique is formulated as a geometric construction that maps a space curve onto a surface curve. It is designed to be coordinate independent and provides isoparametric maps for multiple surface curves. Generality of the formulation is attained by solving a tensorial differential equation formulated in terms of local differential properties of the surfaces. For an implicit surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This technique has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations.