Can local NURBS refinement be achieved by modifying only the user interface?

© 2015 The Authors. NURBS patches have a serious restriction: they are constrained to a strict rectangular topology. This means that a request to insert a single new control point will cause a row of control points to appear across the NURBS patch, a global refinement of control. We investigate a method that can hide unwanted control points from the user so that the user's interaction is with local, rather than global, refinement. Our method requires only straightforward modification of the user interface and the data structures that represent the control mesh, making it simpler than alternatives that use hierarchical or T-constructions. Our results show that our method is effective in many cases but has limitations where inserting a single new control point in certain cases will still cause a cascade of new control points to appear across the NURBS patch.

An effective algorithm for planning a collision-free path based on linear parametric curve is developed. A collision-free path is viewed as a series of segmented polynomial curves in a space that does not interfere with any object in a given work space. It is assumed that the path connecting start and target points has no width. The algorithm presented here uses a linear parametric curve as a base curve and maps objects in Euclidean Space (ES) into objects in Control Point Space (CPS) through intersection checks between path and obstacles. A path having a single control point is investigated here. The Free Space (FS) of CPS identifies a collision-free path in ES, hence any point in the FS of CPS is a candidate for a collision-free path. A CPS completely occupied by obstacle images indicates no collision-free path is available with a single control point and a search based on a multiple control point is required. The shortest path with minimum search time is obtained by setting CPS in elliptic coordinate. Considerations are given to get a path with the smallest maximum curvature by selecting the control point toward the bisection line of ST and smoothing the path. ©1997 John Wiley & Sons, Inc.

The paper presents the first optimal conjugation design methodology based on free-form conjugation modeling theory. The methodology is implemented for planar circular gearing and general for any planar gearing. According to the previous research of free-form conjugation, conjugate profiles are modeled by contact path geometries or cutter geometries which are represented by NURBS (non-uniform rational B-splines). The interchangeability between control points and interpolation points of NURBS are introduced in general to offer reasonable constraints of special interpolation conditions in conjugation design. To adapt to the flexibility brought by free-form techniques, the determination of an important conjugate property—contact ratio is carried out through geometric relationship. To make use of NURBS for optimal design, conjugate properties and their differentiations are represented by important parameters and then by control points and interpolation points of NURBS. The interested properties are relative curvature, specific sliding ratio and nominal Hertz contact stress, which are the main factors of gear efficiency and wear. The properties are well-known for their difficulties in optimization. The paper shows that with appropriate manipulations in mathematics and programming, it is feasible for gradient based optimization methods, which are accurate and fast in convergence. The methodology is consistent with the regular optimization of geometry design, and can be integrated into geometry design systems. The examples show the effectiveness of the proposed optimization framework.

This paper describes a synthesis method of designing flexible cam profiles by using quintic non-uniform rational B spline (NURBS) curves. The cam profile curve can be optimized by adjusting knot vector, control points and weight factors of NURBS curve. The multiobjective optimization for a cam curve was described to satisfy both kinematic and dynamic requirements. To solve the optimization model for NURBS cam curve, an improved genetic algorithm (GA) called INSGA in the paper was proposed. Application examples on the kinematical and dynamical optimization of cam curves are presented together with detailed discussions. These examples, especial the dynamic optimization to control the residual vibration for a cam curve used in an indexing cam mechanism, sufficiently illustrate the effectiveness of the proposed approach.

To guarantee the temporal correctness of a digital circuit a set of multiple path delay faults called primitive faults need to be tested. Primitive faults can contain one or more faulty paths. Existing techniques can identify and test primitive faults containing up to two or three paths. Identifying and testing primitive faults that consist of a larger number of paths is impractical for large designs. We propose a design for testability method that assures the temporal correctness of the circuit without the need to test all primitive faults in the circuit. In the test mode, only primitive faults that contain up to two paths can affect the circuit performance. Our methodology efficiently identifies a small set of potential locations for inserting control points to eliminate primitive faults with more than two paths. Addition of a single control point can lower the cardinality of several primitive faults. Our approach re-evaluates primitive delay fault testability of the circuit after insertion of every control point. After a few iterations only primitive faults with at most two paths can exist in the circuit in the test mode. Experimental results on several circuits are included to demonstrate our method.

We present current research in which both left and right ventricular deformation is estimated from tagged cardiac magnetic resonance imaging using volumetric deformable models constructed from nonuniform rational B-splines (NURBS). The four model types considered include Cartesian-based NURBS models with both a cylindrical and prolate-spheroidal parameterization, prolate spheroidal-based NURBS models with a prolate-spheroidal parameterization, and cylindrical-based NURBS models with a cylindrical parameterization. For each frame subsequent to end-diastole, a NURBS model is constructed by fitting two surfaces with the same parameterization to the corresponding set of epicardial and endocardial contours from which a volumetric model is created. Using normal displacements of the three sets of orthogonal tag planes as well as displacements of contour/tag line intersection points and tag plane intersection points, one can solve for the optimal homogeneous coordinates, in a weighted least squares sense, of the control points of the deformed NURBS model at end-diastole using quadratic programming. This allows for subsequent forward displacement fitting from end-diastole to all later time frames. After fitting to all time points of data, lofting the NURBS model at each time point creates a comprehensive 4-D NURBS model. From this model, we can extract 3-D myocardial deformation fields and corresponding strain maps which are local measures of non-rigid deformation. The results show that, in the case of simulated data, the quadratic Cartesian-based NURBS model outperformed its counterparts in predicting normal strain. This model was used to then calculate normal Lagrangian and Eulerian strains in canine data.

Soil particle-size distribution or soil texture presents one of the most important physical properties. There are various systems of the classification systems for soil particle-size fractions with different boundaries. Our effort was concentrated on the mathematical approach to evaluate the existing data and convert it to the form of a reconstructed cumulative particle-size curve which will allow reading concentration of any desired particle size. Non-Uniform Rational B-Splines (NURBS) curves therefore represent a generalization of B-splines and BĂŠzier curves by extending the definition by an element of rationality, which is represented by the weights of the control points, and a nodal vector of parametrization, which represents the element of uniformity. The NURBS curve was used for smooth (depending on the degree of the curve used) and as tight as possible approximation of the arranged control points, the connecting lines of which forms a convex envelope for its individual parts. The NURBS approximation curve is therefore determined by the ordered control points and their connecting lines, the weights of these points, the degree of the curve and the nodal vector of parametrization. However, the construction of the approximation curve is primarily dependent on a limited number of points of the experimentally determined particle-size distribution curves, and for curves with significant breaks in the course, one must consider either a lower accuracy of the approximation or the necessity of “improving” the approximation using the weights of individual points, inserting additional points or working with a nodal vector of parametrization. For basic approximation, the PUGIS system (Czech soil information system) offers automatic approximation using all variants mentioned in the article as well as the possibility of individual changes in the weights of control points, in their number and position, and in the nodal vector of parametrization.

The receiver operating characteristic (ROC) curve is a commonly used statistical method to assess the efficacy of a diagnostic test or biomarker measured on a continuous scale. This work presents a versatile approach using a non-uniform rational B-spline (NURBS) for estimating the ROC curve. This approach uses control points, weights, and the knot sequence to more accurately estimate the true ROC curve. The new method applies linear constraints to the NURBS basis function coefficients to smooth the empirical ROC curve and guarantee a non-decreasing function. Moreover, as a specific case, a NURBS curve devoid of interior knots simplifies to the Bernstein polynomial when all weight values are equal. We conduct Monte Carlo simulation studies to evaluate how well the NURBS-based estimator works in different scenarios. We compare our estimator to the empirical ROC, the kernel-based ROC, and Bernstein polynomial estimators in terms of the averaged squared errors. We also apply our method to two real medical datasets, such as metastatic kidney cancer and diffuse large B-cell lymphoma datasets. According to the findings from both the real and simulated data, the NURBS method is a powerful alternative for estimating the ROC curve.

Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves

We present current research in which left ventricular deformation is estimated from tagged cardiac magnetic resonance imaging using volumetric deformable models constructed from nonuniform rational B-splines (NURBS). From a set of short and long axis images at end-diastole, the initial NURBS model is constructed by fitting two surfaces with the same parameterization to the set of epicardial and endocardial contours from which a volumetric model is created. Using normal displacements of the three sets of orthogonal tag planes as well as displacements of both tag line and contour/tag line intersection points, one can solve for the optimal homogeneous coordinates, in a least squares sense, of the control points of the NURBS model at a later time point using quadratic programming. After fitting to all time points of data, lofting the NURBS model at each time point creates a comprehensive 4-D NURBS model. From this model, we can extract 3-D myocardial displacement fields and corresponding strain maps, which are local measures of non-rigid deformation.

In this paper, structural shape of stent has been optimized using NURBS for parameterization of stent structure and target those objectives which are critical for vascular injury. NURBS modeling is done using python coding in RHINO 3D software. For later part of the design, Solidworks is used. The objectives considered in our study are dogboning, foreshortening and arterial wall stresses, all of which are strongly linked to vascular injury leading to restenosis. We use control point weights, strut thickness and strut width as design variables for Latin Hypercube sampling (LHS) in order to generate dataset for Stent deployment simulations. In our study, we generate 80 design data points using LHS in Matlab R2014a. Finite element analysis of stent deployment process is then carried out using ANSYS for all 80 designs of stent generated using LHS. Thereafter, we use Kriging for surrogate modeling and non-dominated sorting genetic algorithm (NSGA-II) in MATLAB for multi-objective design optimization so as to minimize dogboning, foreshortening and arterial wall stresses. As a result, we obtain a range of pareto optimal design parameter values which can be used in clinical design guides so as to accommodate variations observed across different patients.

Isogeometric size optimization of bi-directional functionally graded beams under static loads

A surrogate model based on Non-Uniform Rational B-Splines hypersurfaces

K-surfaces are an interactive modeling technique for Bézier-spline surfaces. Inspired by k -curves by [Yan et al. 2017], each patch provides a single control point that is being interpolated at a local extremum of Gaussian curvature. The challenge is to solve the inverse problem of finding the center control point of a Bézier patch given the boundary control points and the handle. Unlike the situation in 2D, bi-quadratic Bézier patches may exhibit none, one, or several extrema, and finding them is non-trivial. We solve the difficult inverse problem, including the possible selection among several extrema, by learning the desired function from samples, generated by computing Gaussian curvature of random patches. This approximation provides a stable solution to the ill-defined inverse problem and is much more efficient than direct numerical optimization, facilitating the interactive modeling framework. The local solution is used in an iterative optimization incorporating continuity constraints across patches. We demonstrate that the surface varies smoothly with the handle location and that the resulting modeling system provides local and generally intuitive control. The idea of learning the inverse mapping from handles to patches may be applicable to other parametric surfaces.

Purpose. This study explored a novel technique to improve the MV imaging based fiducial visibility for a cohort of prostate radiotherapy patients, without compromising the original treatment plan. The study also compared these results to visibility using single MLC control points, as well as short arcs. Methods. Geometric data from 68 prostate radiotherapy treatments, each with implanted gold fiducials, was retrospectively analysed. Fiducials were contoured for each patient, and conventional and SBRT treatment plans were generated using a VMAT technique. Using an in-house script, fiducial contours were projected onto the VMAT MLC control points. Resulting data was assessed to determine whether the fiducial contours were theoretically visible for single MLC control points and groups of MLC control points (short arcs), both being surrogates for intra-fraction MV imaging. Using this data, a theoretical quadrant technique was investigated that assessed the region surrounding each fiducial to determine if visualisation would theoretically improve. Results. Using a conventional treatment type, mean fiducial visibility for single MLC control points across the patient cohort ranged from 2.5% up to 17.8%. For SBRT, fiducial visibility ranged from 1.8% up to 19.7%. For short arcs, fiducial visibility for conventional treatment types ranged from 5.9% up to 20.7%. For SBRT, fiducial visibility ranged from 4.6% up to 23.1%. When the novel fiducial quadrant technique was used, theoretical visibility improved two-fold, from 22.7% up to 52.5% and from 24.7% up to 55.3% for conventional and SBRT treatment types respectively. Conclusions. Fiducial visibility was assessed for a cohort of VMAT prostate patients. Using the novel quadrant technique, it was demonstrated that theoretical visualisation and localisation of the implanted fiducials could be improved two-fold, without sacrificing treatment plan quality.