Geometric Interpolation Research Articles

Hermite G 1 rational spline motion of degree six

Applying geometric interpolation techniques to motion construction has many advantages, e.g., the parameterization is chosen automatically and the obtained rational motion is of the lowest possible degree. In this paper a G 1 Hermite rational spline motion of degree six is presented. An explicit solution of nonlinear equations that determine the spherical part of the motion is derived. Particular emphasis is placed on the construction of the translational part of the motion. Since the center trajectory is a G 1 continuous for an arbitrary choice of lengths of tangent vectors, additional free parameters are obtained, which are used to minimize particular energy functionals. Thenumerical examples provide an evidence that the obtained motions have nice shapes.

Avaliação do programa de controle da tuberculose em cenário Amazônico: desafios em Coari

Pesquisa avaliativa, tipo estudo de caso, do contexto e grau de implantação do Programa de Controle da Tuberculose (PCT) no município de Coari, Amazonas, 2001-2008. As técnicas de pesquisa conjugaram abordagens quantitativa e qualitativa, com uso de modelo lógico e matriz de julgamento. O componente quantitativo utilizou dados do Sistema de Informação de Agravos de Notificação e dados populacionais do Departamento de Informática do Sistema Único de Saúde. As variáveis foram, idade, sexo, forma clínica, procedência, tipo de entrada, meios diagnósticos e situação de encerramento. Para a população foram feitos cálculos de interpolação geométrica, por faixa etária, sexo e procedência (urbana e rural). O componente qualitativo combinou técnicas e fontes: pesquisa documental; observação e entrevista. O modelo lógico propiciou o entendimento da estrutura, processo e resultado do PCT; e a matriz de julgamento permitiu identificar o grau de implantação. A taxa média de incidência da tuberculose na área urbana foi 27 casos por 100.000 habitantes e de 9,4 por 100.000 habitantes na área rural, com razão masculino/feminino de 1,6. A avaliação do contexto externo, vigilância epidemiológica e operacionalização do PCT mostrou grau de implantação intermediário. No contexto organizacional, estrutura física das Unidades Saúde, recursos humanos e efetividade, o resultado foi incipiente. No geral, o grau de implantação foi intermediário. A flutuação nas notificações e taxas anuais de incidência, associada ao não cumprimento das metas do PCT, à desarticulação entre o PCT e o restante da atenção básica e à não interiorização da assistência indicam sérias dificuldades na operacionalização das ações de controle da tuberculose.

GEOMETRIC HERMITE INTERPOLATION FOR PLANAR PYTHAGOREAN-HODOGRAPH CUBICS

We solve the geometric Hermite interpolation problem with planar Pythagorean-hodograph cubics. For every Hermite data, we determine the exact number of the geometric Hermite interpolants and represent the interpolants explicitly. We also present a simple criterion for determining whether the interpolants have a loop or not.

Application of morphing technique with mesh-merging in rapid hull form generation

Morphing is a geometric interpolation technique that is often used by the animation industry to transform one form into another seemingly seamlessly. It does this by producing a large number of 'intermediate' forms between the two 'extreme' or 'parent' forms. It has already been shown that morphing technique can be a powerful tool for form design and as such can be a useful addition to the armoury of product designers. Morphing procedure itself is simple and consists of straightforward linear interpolation. However, establishing the correspondence between vertices of the parent models is one of the most difficult and important tasks during a morphing process. This paper discusses the mesh-merging method employed for this process as against the already established mesh-regularising method. It has been found that the merging method minimises the need for manual manipulation, allowing automation to a large extent.

Hermite interpolation by rational [formula omitted] motions of low degree

Interpolation by rational spline motions is an important issue in robotics and related fields. In this paper a new approach to rational spline motion design is described by using techniques of geometric interpolation. This enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion. A general approach to geometric interpolation by rational spline motions is presented and two particularly important cases are analyzed, i.e., geometrically continuous quartic rational motions and second order geometrically continuous rational spline motions of degree six. In both cases sufficient conditions on the given Hermite data are found which guarantee the uniqueness of the solution. If the given data do not fulfill the solvability conditions, a method to perturb them slightly is described. Numerical examples are presented which confirm the theoretical results and provide evidence that the obtained motions have nice shapes.

Application of morphing technique with mesh-merging in rapid hull form generation

Morphing is a geometric interpolation technique that is often used by the animation industry to transform one form into another seemingly seamlessly. It does this by producing a large number of ‘intermediate’ forms between the two ‘extreme’ or ‘parent’ forms. It has already been shown that morphing technique can be a powerful tool for form design and as such can be a useful addition to the armoury of product designers. Morphing procedure itself is simple and consists of straightforward linear interpolation. However, establishing the correspondence between vertices of the parent models is one of the most difficult and important tasks during a morphing process. This paper discusses the mesh-merging method employed for this process as against the already established mesh-regularising method. It has been found that the merging method minimises the need for manual manipulation, allowing automation to a large extent.

Convergence analysis for B-spline geometric interpolation

In this paper, we propose a sufficient condition for the convergence of a geometric algorithm for interpolating a given polygon using non-uniform cubic B-splines. Geometric interpolation uses the given polygon as the initial shape of the control polygon of the B-spline and reduces the approximate error by iteratively updating the control points with the deviations from the corresponding interpolated vertices to their nearest footpoints on the current B-spline curve. The convergence condition is derived by employing a spectral radius estimation technique. The primary goal is to find for each control point a parametric interval within which the nearest footpoint should be confined such that the spectral radius of the error iteration matrix is smaller than 1. A convergent condition for the geometric interpolation of uniform B-splines can be derived as a special case of the new scheme.

Geometric point interpolation method in [formula omitted] space with tangent directional constraint

This paper discusses a cubic B-spline interpolation problem with tangent directional constraint in R3 space. Given m points and their tangent directional vectors as well, the interpolation problem is to find a cubic B-spline curve which interpolates both the positions of the points and their tangent directional vectors. Given the knot vector of the resulting B-spline curve and parameter values to all of the data points, the corresponding control points can often be obtained by solving a system of linear equations. This paper presents a piecewise geometric interpolation method combining a unclamping technique with a knot extension technique, with which there is no need to solve a system of linear equations. It firstly uses geometric methods to construct a seed curve segment, which interpolates several data point pairs, i.e., positions and tangent directional vectors of the points. The seed segment is then extended to interpolate the remaining data point pairs one by one in a piecewise fashion. We show that a B-spline curve segment can always be extended to interpolate a new data point pair by adding two more control points. Methods for a curve segment extending to interpolate one more data point pair by adding one more control point are also provided, which are utilized to construct an interpolation B-spline curve with as small a number of control points as possible. Numerical examples show the effectiveness and the efficiency of the new method.

A simple and conservative operator-splitting semi-Lagrangian volume-of-fluid advection scheme

A semi-Lagrangian, operator-splitting, high-performance volume-of-fluid advection scheme (SLOSH-VOF) is developed. In the backward time integration a directional splitting strategy is adopted, which greatly simplifies the definition and locating of the departure volume and reduces it to a grid cell expanded or compressed in one grid direction. The VOF value in the departure cell is estimated using a geometrical interpolation algorithm with a piecewise linear interface calculation (PLIC) scheme for the interfacial cells. The proposed SLOSH-VOF method is unconditionally stable and very large time steps can be used to significantly speed up the overall computations. It is conceptually simple and can be very easily implemented due to the direction-split advection. Its performance is evaluated through several benchmark advection tests. The SLOSH-VOF results are comparable to those from an Eulerian VOF method in terms of interface position errors, and improved with regard to mass conservation, even with CFL numbers increased up to one order of magnitude.

Quantitative CT with finite element analysis: towards a predictive tool for bone remodelling around an uncemented tapered stem

We used quantitative CT in conjunction with finite element analysis to provide a new tool for assessment of bone quality after total hip arthroplasty in vivo. The hypothesis of this prospective five-year study is that the combination of the two modalities allows 3D patient-specific imaging of cortical and cancellous bone changes and stress shielding. We tested quantitative CT in conjunction with finite elements on a cohort of 29 patients (31 hips) who have been scanned postoperatively and at one year, two years and five years follow-up. The method uses cubic Hermite finite element interpolation for efficient mesh generation directly from qCT datasets. The element Gauss points that are used for the geometric interpolation functions are also used for interpolation of osteodensitometry data. The study showed changes of bone density suggestive of proximal femur diaphysis load transfer with osteointegration and moderate metaphyseal stress shielding. Our model revealed that cortical bone initially became porous in the greater trochanter, but this phenomenon progressed to the cortex of the lesser trochanter and the posterior aspect of the metaphysis. The diaphyseal area did not experience major change in bone density for either cortical or cancellous bone. The combination of quantitative CT with finite element analysis allows visualization of changes to bone density and architecture. It also provides correlation of bone density/architectural changes with stress patterns enabling the study of the effects of stress shielding on bone remodelling in vivo. This technology can be useful in predicting bone remodeling and the quality of implant fixation using prostheses with different design and/or biomaterials.

기하학적 보간과 최적화를 이용한 고속 에어포일 형상 설계 연구

In this paper, a study on the design of high speed airfoil is described. Various airfoils are investigated and existing airfoils are geometrically interpolated to generate new airfoils. An optimization method is applied to theses new airfoils and their aerodynamic performances are optimized. Through this study, it is demonstrated that the airfoil can be designed using the geometrical interpolation and the optimization method to exhibit good aerodynamic performances.

Formula omitted]-spline surface fitting by iterative geometric interpolation/approximation algorithms

Recently, the use of B-spline curves/surfaces to fit point clouds by iteratively repositioning the B-spline’s control points on the basis of geometrical rules has gained in popularity because of its simplicity, scalability, and generality. We distinguish between two types of fitting, interpolation and approximation. Interpolation generates a B-spline surface that passes through the data points, whereas approximation generates a B-spline surface that passes near the data points, minimizing the deviation of the surface from the data points. For surface interpolation, the data points are assumed to be in grids, whereas for surface approximation the data points are assumed to be randomly distributed. In this paper, an iterative geometric interpolation method, as well as an approximation method, which is based on the framework of the iterative geometric interpolation algorithm, is discussed. These two iterative methods are compared with standard fitting methods using some complex examples, and the advantages and shortcomings of our algorithms are discussed. Furthermore, we introduce two methods to accelerate the iterative geometric interpolation algorithm, as well as a method to impose geometric constraints, such as reflectional symmetry, on the iterative geometric interpolation process, and a novel fairing method for non-uniform complex data points. Complex examples are provided to demonstrate the effectiveness of the proposed algorithms.

Cortical surface registration using spherical thin-plate spline with sulcal lines and mean curvature as features

Analysis of cortical patterns requires accurate cortical surface registration. Many researchers map the cortical surface onto a unit sphere and perform registration of two images defined on the unit sphere. Here we have developed a novel registration framework for the cortical surface based on spherical thin-plate splines. Small-scale composition of spherical thin-plate splines was used as the geometric interpolant to avoid folding in the geometric transform. Using an automatic algorithm based on anisotropic skeletons, we extracted seven sulcal lines, which we then incorporated as landmark information. Mean curvature was chosen as an additional feature for matching between spherical maps. We employed a two-term cost function to encourage matching of both sulcal lines and the mean curvature between the spherical maps. Application of our registration framework to fifty pairwise registrations of T1-weighted MRI scans resulted in improved registration accuracy, which was computed from sulcal lines. Our registration approach was tested as an additional procedure to improve an existing surface registration algorithm. Our registration framework maintained an accurate registration over the sulcal lines while significantly increasing the cross-correlation of mean curvature between the spherical maps being registered.

PYTHAGOREAN-HODOGRAPH CUBICS AND GEOMETRIC HERMITE INTERPOLATION

In this paper, we present the geometric Hermite interpolation for planar Pythagorean-hodograph cubics for some general Hermite data. curve ro(t) = r(t) + dn(t) at a xed distance d from a given polynomial curve r(t), in the direction of its unit normal vector n(t), is an important problem in computer aided geometric design. The oset ro(t) is not, in general, a rational curve. Thus we need some approximation schemes to deal with osets of a polynomial curves. However, if the speed function of a given polynomial curve is a polynomial, then an oset curve can be expressed in a rational parametrization. For this reason, the Pythagorean-hodograph (PH) curves, whose speed function is a polynomial, were introduced by Farouki and Sakkalis ((6)). Since then, there have been vast researches on this class of curves by themselves and others ((1), (2), (3), (4), (5), and (11)). A PH curve r(t) = (x(t);y(t)) means a special polyno- mial curve which is characterized by the algebraic property that its hodograph r 0 (t) = (x 0 (t);y 0 (t)) satises the Pythagorean condition, that is, x 02 (t) +y 02 (t) = 2 (t) for some polynomial (t). So, the arc-length function and the osets

Lagrange geometric interpolation by rational spatial cubic Bézier curves

In the paper, the Lagrange geometric interpolation by spatial rational cubic Bézier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, i.e., order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme.

Hermite Geometric Interpolation by Rational Bézier Spatial Curves

Polynomial geometric interpolation by parametric curves has become one of the standard techniques for interpolation of geometric data. An obvious generalization leads to rational geometric interpolation schemes, which are a much less investigated research topic. The aim of this paper is to present a general framework for Hermite geometric interpolation by rational Bezier spatial curves. In particular, cubic $G^2$ and quartic $G^3$ interpolations are analyzed in detail. Systems of nonlinear equations are derived in a simplified form, and the existence of admissible solutions is studied. For the cubic case, geometric conditions implying solvability of the nonlinear system are also stated. The asymptotic analysis is done in both cases, and optimal approximation orders are proved. Numerical examples are given, which confirm the theoretical results.

Least eccentric ellipses for geometric Hermite interpolation

We present a rational Bézier solution to the geometric Hermite interpolation problem. Given two points and respective unit tangent vectors, we provide an interpolant that can reproduce a circle if possible. When the tangents permit an ellipse, we produce one that deviates least from a circle. We cast the problem as a theorem and provide its proof, and a method for determining the weights of the control points of a rational curve. Our approach targets ellipses, but we also present a cubic interpolant that can find curves with inflection points and space curves when an ellipse cannot satisfy the tangent constraints.

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G 1 cubic interpolatory spline.

An approach to geometric interpolation by Pythagorean-hodograph curves

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d???2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

Automated statistical method to align 2D chemical maps with 3D X-ray computed micro-tomographic images of soils

Recent 2-dimensional measurements reveal that soils are chemically very heterogeneous at nanometric and micrometric scales. Direct measurement techniques are still lacking to extend these observations to 3 dimensions. Sequential sectioning of soils, followed by 2-dimensional mapping of chemical elements and geometric interpolation to 3D, appears to be the only available alternative. Unfortunately, sectioning of soil samples suffers from geometric distortions that are difficult to avoid in practise. In this regard, the objective of the research described in this article was to develop a procedure enabling one to locate, in a 3D X-ray microtomographic image of a soil sample, a physical surface that is obtained by sectioning and for which a number of chemical maps are available. This procedure involves three steps: (1) the reconstitution of the physical structure of the soil layer surface, (2) the alignment of the chemical maps with the reconstituted soil surface image, and (3) the 3D alignment of the 2D chemical maps with the internal structure of the soil cube. Visual comparison of the C and Si maps and of the reconstituted CT images of the layer surfaces suggests a good correspondence between them, which is supported by Pearson correlation coefficients of − 0.57, − 0.58, 0.45, and 0.43 for the different surfaces and elements considered. Relative to the original 3D X-ray CT image of the soil sample, the planes associated with the C and Si maps, respectively, are nearly superposed, which further confirms the validity of the alignment procedure.