Coordinate Interpolation Research Articles

SU-E-T-734: Evaluation of a New, Grid-Based Linear Boltzmann Transport Equation Solver for Dose Calculation in the Presence of High-Density Materials

Purpose: To evaluate the accuracy of a new, deterministic linear Boltzmann transport equation solver, AcurosXB, for radiotherapydose calculations in the presence of high‐density materials by comparison with Monte Carlo.Methods: AcurosXB, EGSnrc‐based Monte Carlo and Analytic Anisotropic Algorithm (AAA) were used to model 6MV and 18MV photon beams incident on phantom and patient geometries containing stainless steel and titanium high‐density objects. Monte Carlo was used as the evaluation benchmark while AAA was included as a representative of the convolution‐based clinical modeling algorithms currently used for radiotherapy treatment planning. Calculation geometries include a virtual cubic phantom created in MatLab, a cubic water phantom containing stainless steel rods and prosthetic hips and patient CT images with associated clinical treatment plans. Dose distributions were processed and analyzed in MatLab.Results: Agreement between AcurosXB and Monte Carlo is within 5% and 2mm. There is a backscatter peak position offset of up to one pixel (2mm) between AcurosXB and Monte Carlo which can be accounted for by differences in voxel coordinate definition, contour definition and interpolation effects. Additional differences between AcurosXB and Monte Carlo fall within statistical variance associated with Monte Carlo. Disagreement between AAA and Monte Carlo was far more dramatic; AAA was unable to model the sharp backscatter peak associated with photon transport across low‐ to high‐density boundaries resulting in inaccurate dose distribution through and past high density regions Conclusions: AcurosXB has shown to be an improvement on existing algorithms for calculating dose distribution in the presence of high‐density regions, nearly matching the current gold standard Monte Carlo method. Discrepancies between AcurosXB and Monte Carlo do not translate into clinically significant dose differences, while AAA has shown to under‐ or over‐estimate dose by as much as 20%. Varian Medical Systems has provided AcurosXB hardware and software for research activities.

The third form of the quadrilateral area coordinate method (QACM-III): Theory, application, and scheme of composite coordinate interpolation

Two quadrilateral area coordinate methods (hereinafter referred to as QACM-I and QACM-II) have already been proposed in 1999 and 2008, respectively. Many successful applications demonstrate that these new coordinate systems are efficient tools for developing robust quadrilateral finite element models. In this paper, another new quadrilateral area coordinate method (hereinafter referred to as QACM-III), containing two components T 1 and T 2, is systematically established. This new coordinate system (QACM-III) not only keeps the most important advantages of the QACM-I and QACM-II, but also possesses other distinguishing characters. Then, the QACM-III is applied to formulate two new 4-node membrane elements with internal parameters. The first new element, completely formulated by the QACM-III and denoted as QACIII6, exhibits much better performance than that of Wilson's Q6 element in benchmark problems, and has the simplest formulations when compared with the similar models formulated by the QACM-I and -II. The second element, denoted as CQAC6, is developed by a new scheme of composite coordinate interpolation, in which the QACM-I and -III are both used. It can strictly pass Iron's patch test and exhibits better accuracy than the Taylor's QM6 element. Finally, the new scheme of composite coordinate interpolation is employed again. Three different quadrilateral area coordinate methods, QACM-I, -II and -III, are used simultaneously to develop a new 8-node membrane element CQAC-Q8. On the one hand, element CQAC-Q8 is insensitive to mesh distortion, which the traditional isoparametric element Q8 cannot achieve; on the other hand, its construction procedure and formulations are quite simpler than the models formulated only by any single area coordinate method. The new coordinate method, the element models, and the scheme of composite coordinate interpolation presented in this paper provide new tools and ideas for high-performance finite element analysis.

3D visualization method of radiotherapy dose distribution in medical TPS

According to the requirements of GyroRTPS,a 3D visualization method used to display the distribution of radiotherapy dose was proposed to assist doctors to look over the dose distribution from different angles.Firstly,some preprocesses of coordinate transformation and interpolation on volume data and dosage data were discussed.Secondly,after these processing,the data for visualization was normalized according to VTK specifications.Finally,the method based on VTK to joint display dose data and human body data in 3D way was realized.The test shows that the reliability of this method and this system can better help the doctor carry on the radiotherapy.

Cage-based deformation transfer

We present a cage-based method for transferring animation from a mesh sequence or motion capture data to geometric models in variant representations. To reach the aimed generality, the target model is first embedded into a cage by Green Coordinates interpolation, which preserves the geometric details inherently. The deformation gradient sequences of some user-selected points on the source are then extracted and used as the gradient constraints in the cage to guide the target deformation. The variation of deformation gradients in the cage is minimized to avoid degeneration and ensure smoothness of the result. Position and length constraints can also be applied for more controls. The optimal positions of target cage vertices can be efficiently evaluated by solving a non-linear minimization problem in terms of a few variables. The main advantage of our algorithm is that both source and target can be in a wide range of shapes and deformation representations. Moreover, we can transfer the deformation from multiple sequences onto a single target model.

Structure of a Type III Restriction Endonuclease by Single-Particle Electron Microscopy

Type III restriction endonucleases recognize DNA sequences and cleave at a specific distance from the recognition sequence. These enzymes are composed of two protein subunits: Mod, necessary for specific binding and methylation, and Res, necessary for ATP-dependent restriction. A Res2Mod2 subunit assembly has been determined for the best-known examples of Type III restriction endonucleases, EcoP1I and EcoP15I, however no more structural information is available yet about type III restriction endonucleases. We have used electron microscopy (EM) and single-particle image processing to start deciphering the 3D structure of EcoP15I.Advances in single-particle reconstruction methods have made it possible to elucidate the 3D structure of many macromolecules from EM images. However, accurate 3D structural determination of non-polymeric proteins smaller than 500 kDa such as EcoP15I is still a challenge. To optimize the extraction of information from our EM dataset, we review systematically different aspects of the reconstruction algorithms such as image de-noising, alignment, and classification. De-noising is an important factor in the quality of reconstruction because images of small macromolecules have low signal-to-noise ratio. We are comparing various techniques of de-noising based on statistical and geometrical characteristics of images. We are also investigating possible improvement of alignment based on coordinate transformation, interpolation, and statistical information to reduce alignment errors. Finally, we are evaluating sophisticated classification algorithms based on statistical information of the images.Our preliminary 2D averages of EcoP15I show an oval structure of ∼160x100 A major and minor axes, and at least four distinguishable domains. The different classes correspond to different various projection angles, readily providing 3D structural information. Our preliminary 3D reconstruction confirms the above finding. The enhancement prospects of the image processing developments will be discussed in the context of the determination of the 3D structure of EcoP15I.

Area-efficient pixel rasterization and texture coordinate interpolation

In this paper, new pixel rasterization and texture coordinate interpolation algorithms are presented to reduce silicon area. The proposed pixel rasterization based on the characteristics of edge function saves silicon area in terms of gate count by 38.9% and 35.3% compared to the previous centerline and scanline algorithms, respectively. The proposed texture coordinate interpolation combines the benefits of division and midpoint iteration in order to reduce silicon area without performance loss in computing the fraction part of texture coordinates, which is required for texture filtering. The proposed texture coordinate interpolation architecture uses less silicon gates than the architecture using dividers, and the gate count reduction ratios are 25.2% and 37.0% for 16- and 32-bit texture coordinates, respectively. The hardware feasibility of the proposed architecture is proved by implementation into a three-dimensional (3D) graphics SoC.

Optimization of elastic systems using absolute nodal coordinate finite element formulation

An approach to a shape optimization of elastic dynamic multibody systems is presented. The proposed method combines an appropriate shape parameterization concept and recently introduced finite element type using absolute nodal coordinate formulation (ANCF). In ANCF, slopes and displacements are used as the nodal coordinates instead of infinitesimal or finite rotations. This way one avoids interpolation of rotational coordinates and problems with finite rotations. ANCF elements are able to describe nonlinear deformation accurately; therefore, this method is very useful for simulations of lightweight multibody structures, where large deformations have to be taken into account. The optimization problem is formulated as a nonlinear programming problem and a gradient-based optimization procedure is implemented. The introduced optimization design variables are related to the cross-sectional parameters of the element and to the shape of the whole structure. The shape parameterization is based on the design element techniqueand a rational B ezier body is used as a design element. A body-like design element makes possible to unify the shape optimization of both simple beams and beam-like (skeletal) structures.

Projected tetrahedra revisited: a barycentric formulation applied to digital radiograph reconstruction using higher-order attenuation functions

This paper presents a novel method for volume rendering of unstructured grids. Previously, we introduced an algorithm for perspective-correct interpolation of barycentric coordinates and computing polynomial attenuation integrals for a projected tetrahedron using graphics hardware. Here, we enhance the algorithm by providing a simple and efficient method to compute the projected shape (silhouette) and tessellation of a tetrahedron, in perspective and orthographic projection models. Our tessellation algorithm is published here for the first time. Compared with works of other groups on rendering unstructured grids, the main contributions of this work are: 1) A new algorithm for finding the silhouette of a projected tetrahedron. 2) A method for interpolating barycentric coordinates and thickness on the faces of the tetrahedron. 3) Visualizing higher-order attenuation functions using GPU without preintegration. 4) Capability of applying shape deformations to a rendered tetrahedral mesh without significant performance loss. Our visualization model is independent of depth-sorting of the cells. We present imaging and timing results of our implementation, and an application in time-critical "2D-3D" deformable registration of anatomical models. We discuss the impact of using higher-order functions on quality and performance.

Polynomial interpolation of GPS satellite coordinates

This article describes an algorithm for polynomial interpolation of GPS satellite coordinates and its implementation in MATLAB. The algorithm is intended for realtime processing software and comput ...

MovieMaker: a web server for rapid rendering of protein motions and interactions

MovieMaker is a web server that allows short (∼10 s), downloadable movies of protein motions to be generated. It accepts PDB files or PDB accession numbers as input and automatically calculates, renders and merges the necessary image files to create colourful animations covering a wide range of protein motions and other dynamic processes. Users have the option of animating (i) simple rotation, (ii) morphing between two end-state conformers, (iii) short-scale, picosecond vibrations, (iv) ligand docking, (v) protein oligomerization, (vi) mid-scale nanosecond (ensemble) motions and (vii) protein folding/unfolding. MovieMaker does not perform molecular dynamics calculations. Instead it is an animation tool that uses a sophisticated superpositioning algorithm in conjunction with Cartesian coordinate interpolation to rapidly and automatically calculate the intermediate structures needed for many of its animations. Users have extensive control over the rendering style, structure colour, animation quality, background and other image features. MovieMaker is intended to be a general-purpose server that allows both experts and non-experts to easily generate useful, informative protein animations for educational and illustrative purposes. MovieMaker is accessible at .

A new method for constructing multidimensional potential energy surfaces by a polar coordinate interpolation technique

An interpolation technique was developed for constructing ab initio potential energy surfaces (PES) around the equilibrium structure efficiently and accurately, using polar coordinates coupled with systematic generation of sample points on the hypersphere. Test calculations were performed for H2O and HCHO molecules, and interpolated PES and direct ab initio data were compared for randomly chosen 1000 points. Root mean square errors were found to be 4.8×10−5 (2.4 cm−1) for H2O and 1.7×10−4 (8.4 cm−1) for HCHO with respect to the maximum energy of 50000 cm−1. The present approach was found to provide considerably more accurate PES than conventional methods.

The Interpolation Problem in Gps‐Supported Aerial Triangulation

This paper gives the results of an investigation into the use of different sampling rates and models for the interpolation of perspective‐centre coordinates in GPS‐assisted aerial triangulation. Industry‐standard models and rigorous signal reconstruction filtering have been implemented for two largescale photogrammetric blocks. GPS sampling rate was found to have the most impact on bundle adjustment results in terms of the perspective‐centre residuals. However, the accuracy of object‐space point coordinates was found to be invariant to both GPS sampling rate and interpolation method

Finite Element Incremental Approach and Exact Rigid Body Inertia

In the dynamics of multibody systems that consist of interconnected rigid and deformable bodies, it is desirable to have a formulation that preserves the exactness of the rigid body inertia. As demonstrated in this paper, the incremental finite element approach, which is often used to solve large rotation problems, does not lead to the exact inertia of simple structures when they rotate as rigid bodies. Nonetheless, the exact inertia properties, such as the mass moments of inertia and the moments of mass, of the rigid bodies can be obtained using the finite element shape functions that describe large rigid body translations by introducing an intermediate element coordinate system. The results of application of the parallel axis theorem can be obtained using the finite element shape functions by simply changing the element nodal coordinates. As demonstrated in this investigation, the exact rigid body inertia properties in case of rigid body rotations can be obtained using the shape function if the nodal coordinates are defined using trigonometric functions. The analysis presented in this paper also demonstrates that a simple expression for the kinetic energy can be obtained for flexible bodies that undergo large displacements without the need for interpolation of large rotation coordinates.

Bidimensional Regression

Since its invention by Francis Galton in 1877 regression analysis has been found useful in almost all disciplines. Comparison of geographic phenomena requires a two‐dimensional extension of this technique. In this manner geographic maps can be compared with each other. Possible applications include geometric comparison of ancient and modern maps, or of “mental” maps, or for rubber‐sheeting as used in Geographic Information Systems and in remote sensing. Other applications, for example, in biology for the comparison of shapes of leaves, fish, faces, or skulls after the manner of D'Arcy W. Thompson are also possible, as are higher‐dimensional and multivariate cases. The method implements, and puts into this new context, existing models from the field of cartography. The linear case yields an easy definition of a Pearsonian‐like correlation coefficient. The bidimensional case is richer in mathematical options than is the usual unidimensional version. The curvilinear case is of even greater utility. Here the regression coefficients constitute a spatially varying, but coordinate invariant, second‐order tensor field defined by the matrix of partial derivatives of the transformation. This can be shown to be essentially equivalent to Tissot's Indicatrix, used in cartography to determine the properties of a map projection. In a computer implementation a nonparametric approach allows visualization of the regression by automatically plotting the pair of scatter diagrams, drawing of the displacement field, differentiable smooth interpolation of the warped coordinates and predicted image, by a diagram of the principal strains, and with contour maps of the estimated local angular, areal, and total distortion.

A generalized patched-grid algorithm with application to the F-18 forebody with actuated control strake

Analysis of complex configurations using structured grids virtually demands multiple blocks of grids. To facilitate calculations involving multiple blocks, a general grid-block patching algorithm based on generalized coordinate interpolation has been developed. The computational grid may contain as many arbitrarily shaped blocks as required to make the grid generation problem tractable and to accurately model the flow features. Results are presented for several test cases as well as for the F-18 forebody control strake. The methodology developed has application to overlapped or embedded grids.

Hybrid quadrilateral element based on mindlin/reissner plate theory

Two quadrilateral hybrid Mindlin plate elements Q PL4 and Q PL6 based on a new functional [1] are proposed here. In this formulation the displacements are decomposed into two parts, and the internal displacements are able to impose two different types of constraints on the shear strains of the element. For the first type, locking is eliminated indirectly through increasing the constraints on the shear forces and bending moments, while for the second type the constraints are imposed on the shear strains directly, and corresponds to the technique employed for the displacement type models. It will be shown that the element Q PL6 , which belongs to the first type, is a superior element and can achieve high accuracy and convergence without oscillation over a wide range of plate thickness/span ratios. By adopting the natural coordinate interpolation system, the elements are coordinate invariant. Furthermore, the separation of stress field variables enables the element to yield more accurate stresses. A number of examples is used to demonstrate the accuracy and reliability of the proposed elements.

A new thin plate finite element by basis transformation

A simple nine-freedom thin plate finite element is obtained by using cubic Hermitian interpolation to define displacements at the third points on each side. A tenth “local” freedom at the centroid is then approximated using an interpolation devised by Bazeley et al. together with Hermitian interpolation on the sides. The result is a ten freedom “local” element to which the standard cubic Lagrangian areal coordinate interpolation can be applied. The element is of similar accuracy to a number of other well-known elements of comparable complexity and it provides another useful example of the “method of nested interpolations” which, along with penalty factors and Lagrange multipliers, brings us to the state of the art of the finite element method[1].

Elastostatic Infinite Elements for Layered Half Spaces

The far field behavior of a homogeneous half space and a layered half space is investigated under torsional, vertical, horizontal and moment loadings. Based on the derived far field behavior, three different finite‐element based algorithms, i.e., Ordinary Infinite Elements (OIE), Finite Elements by Singular Contraction (FESC) and Exactly Integrable Infinite Elements (EIIE) are developed to model the far field of multilayered half spaces. The coordinate mapping functions and displacement interpolation functions are selected in accordance with the derived far field model. All three schemes satisfy compatibility and completeness. Excellent results are obtained at a very low computational cost by modeling the near field using a small finite‐element mesh, together with any of the present schemes modeling the domain exterior to the finite elements. Considering the computational efficiency and ease in incorporating FESC into an existing finite element package, FESC is superior to the other two algorithms.

Interpolation and approximation properties of rational coordinates over quadrilaterals

This is a study of the properties of rational coordinate functions for the purposes of interpolation and approximation of functions over an arbitrary convex quadrilateral in the plane. In particular, we investigate the properties of the space of all real homogeneous polynomials in the rational coordinate functions. We show that the class B n of all real homogeneous polynomials of degree n has the dimension ( n + 1) 2 and contains the set of all real polynomials of degree n or less in the Cartesian coordinates. We construct a monomial basis for B n , and a canonical basis for Lagrange interpolation which can be used in finite element approximations. Finally, we define an approximation procedure in Sobolev spaces and derive estimates for the norms of the error function.