Non-uniform B-spline Surfaces Research Articles

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.

Efficient EM Scattering Analysis of Uncertain Inhomogeneous Medium

An efficient solver based on volume-surface integral equation is proposed for the electromagnetic (EM) scattering of inhomogeneous medium targets with uncertain geometrical shapes and dielectric constants. On the one hand, the nonuniform ration B-spline surfaces can be used to constructed the uncertain geometrical shapes with a number of geometrical variables. On the other hand, the uncertain dielectric constant can also be derived as the permittivity variables. Then these random variables can be expanded by the first-order Taylor series, which represent the uncertain geometrical shapes and dielectric constant. At last, the mean and variance of the radar cross section (RCS) are acquired for the uncertain inhomogeneous medium targets. The numerical results can be compared with the traditional Monte Carlo (MC) method, which can indicate the accuracy and effectiveness of the method in this letter.

Shape transformer nets: Generating viewpoint-invariant 3D shapes from a single image

Single-view 3D shapes generation has achieved great success in recent years. However, current methods always blind the learning of shapes and viewpoints. The generated shape only fit the observed viewpoints and would not be optimal from unknown viewpoints. In this paper, we propose a novel encoder–decoder based network which contains a disentangled transformer to generate the viewpoint-invariant 3D shapes. The differentiable and parametric Non-uniform B-spline (NURBS) surface generation and 3D-to-3D viewpoint transformation are incorporated to learn the viewpoint-invariant shape and the camera viewpoint, respectively. Our new framework allows us to learn the latent geometric parameters of shapes and viewpoints without knowing the ground truth viewpoint. That can simultaneously generate camera-viewpoint and viewpoint-invariant 3D shapes of the object. We analyze the effects of disentanglement and show both quantitative and qualitative results of shapes generated at various unknown viewpoints.

3D Interactive Segmentation With Semi-Implicit Representation and Active Learning.

Segmenting complex 3D geometry is a challenging task due to rich structural details and complex appearance variations of target object. Shape representation and foreground-background delineation are two of the core components of segmentation. Explicit shape models, such as mesh based representations, suffer from poor handling of topological changes. On the other hand, implicit shape models, such as level-set based representations, have limited capacity for interactive manipulation. Fully automatic segmentation for separating foreground objects from background generally utilizes non-interoperable machine learning methods, which heavily rely on the off-line training dataset and are limited to the discrimination power of the chosen model. To address these issues, we propose a novel semi-implicit representation method, namely Non-Uniform Implicit B-spline Surface (NU-IBS), which adaptively distributes parametrically blended patches according to geometrical complexity. Then, a two-stage cascade classifier is introduced to carry out efficient foreground and background delineation, where a simplistic Naïve-Bayesian model is trained for fast background elimination, followed by a stronger pseudo-3D Convolutional Neural Network (CNN) multi-scale classifier to precisely identify the foreground objects. A localized interactive and adaptive segmentation scheme is incorporated to boost the delineation accuracy by utilizing the information iteratively gained from user intervention. The segmentation result is obtained via deforming an NU-IBS according to the probabilistic interpretation of delineated regions, which also imposes a homogeneity constrain for individual segments. The proposed method is evaluated on a 3D cardiovascular Computed Tomography Angiography (CTA) image dataset and Brain Tumor Image Segmentation Benchmark 2015 (BraTS2015) 3D Magnetic Resonance Imaging (MRI) dataset.

Optimization of freeform surfaces using intelligent deformation techniques for LED applications

Abstract For many years, optical designers have great interests in designing efficient optimization algorithms to bring significant improvement to their initial design. However, the optimization is limited due to a large number of parameters present in the Non-uniform Rationaly b-Spline Surfaces. This limitation was overcome by an indirect technique known as optimization using freeform deformation (FFD). In this approach, the optical surface is placed inside a cubical grid. The vertices of this grid are modified, which deforms the underlying optical surface during the optimization. One of the challenges in this technique is the selection of appropriate vertices of the cubical grid. This is because these vertices share no relationship with the optical performance. When irrelevant vertices are selected, the computational complexity increases. Moreover, the surfaces created by them are not always feasible to manufacture, which is the same problem faced in any optimization technique while creating freeform surfaces. Therefore, this research addresses these two important issues and provides feasible design techniques to solve them. Finally, the proposed techniques are validated using two different illumination examples: street lighting lens and stop lamp for automobiles.

A PIA optimization algorithm for non-uniform B-spline curves and surfaces

This paper aims to address an important issue in B-spline interpolation. In order to reduce shape distortion due to parameterization in the iterative interpolation of B-spline curves and surfaces, we consider both the initial iteration direction of the control vertices and the parameter values corresponding to the nearest points on the initial iterative curves or surfaces. Genetic algorithm with a carefully designed objective function is employed in finding the optimal parameter solutions for high order B-spline curves and surfaces. Experiments show that the shape distortion due to parameterization has been significantly reduced.

A risk index for pediatric patients undergoing diagnostic imaging with 99mTc-dimercaptosuccinic acid that accounts for body habitus**This work was supported by:R01 EB013558 with the National Institute for Biomedical Imaging and Bioengineering (NIBIB).

Published guidelines for administered activity to pediatric patients undergoing diagnostic nuclear medicine imaging are currently obtained through expert consensus of the minimum values as a function of body weight as required to yield diagnostic quality images. We have previously shown that consideration of body habitus is also important in obtaining diagnostic quality images at the lowest administered activity. The objective of this study was to create a series of computational phantoms that realistically portray the anatomy of the pediatric patient population which can be used to develop and validate techniques to minimize radiation dose while maintaining adequate image quality. To achieve this objective, we have defined an imaging risk index that may be used in future studies to develop pediatric patient dosing guidelines. A population of 48 hybrid phantoms consisting of non-uniform B-spline surfaces and polygon meshes was generated. The representative ages included the newborn, 1 year, 5 year, 10 year and 15 year male and female. For each age, the phantoms were modeled at their 10th, 50th, and 90th height percentile each at a constant 50th weight percentile. To test the impact of kidney size, the newborn phantoms were modeled with the following three kidney volumes: −15%, average, and +15%. To illustrate the impact of different morphologies on dose optimization, we calculated the effective dose for each phantom using weight-based 99mTc-DMSA activity administration. For a given patient weight, body habitus had a considerable effect on effective dose. Substantial variations were observed in the risk index between the 10th and 90th percentile height phantoms from the 50th percentile phantoms for a given age, with the greatest difference being 18%. There was a dependence found between kidney size and risk of radiation induced kidney cancer, with the highest risk indices observed in newborns with the smallest kidneys. Overall, the phantoms and techniques in this study can be used to provide data to refine dosing guidelines for pediatric nuclear imaging studies while taking into account the effects on both radiation dose and image quality.

Non‐Uniform B‐Spline Surface Fitting from Unordered 3D Point Clouds for As‐Built Modeling

AbstractThe three‐dimensional mapping of the built environment is of particular importance for engineering applications such as monitoring work‐in‐progress and energy performance simulation. The state‐of‐the‐art methods for fitting primitives, non‐uniform B‐Spline surface (NURBS) and solid geometry to point clouds still fail to account for all the topological variations or struggle with mapping of physical space to parameter space given unordered, incomplete, and noisy point clouds. Assuming an input of points that can be described by a single non‐self‐intersecting NURBS, this article presents a new method that leverages segmented point clouds and outputs NURBS surfaces. It starts by successively fitting uniform B‐Spline curves in two‐dimensional as planar cross‐sectional cuts on each surface. An intermediate B‐Spline surface is then computed by globally optimizing and lofting over the cross‐sections. This surface is used to parameterize the points and perform final refinement to a NURBS. For cylindrical segments such as pipes, a new supervised method is also introduced to string the fitted segments, identify connection types, standardize the connections, and then refine them using NURBS optimization. Experimental results show the applicability of the proposed methods for as‐built modeling purposes.

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic-trigonometric T-spline surfaces (NUAT T-splines for short) of odd bi-degree. The NUAT T-spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic-trigonometric B-spline surfaces (NUAT B-spline surfaces). Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd bi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.

Vibration and Buckling of Thick Plates using Isogeometric Approach

A study on the free vibration and linear buckling analyses of thick plates is described in this article. In order to determine the natural frequencies and buckling loads of plates, a plate element is developed by using isogeometric approach. The Non-uniform B-spline surface (NURBS) is used to represent both plate geometry and the unknown displacement field of plate. All terms required in isogeometric formulation are consistently derived by NURBS definition. The capability of the present plate element is demonstrated by using several numerical examples. From numerical results, it is found to be that the present isogeometric element can predict accurate natural frequencies and buckling loads of plates.

Non-uniform recursive Doo–Sabin surfaces

This paper presents a generalization of Catmull–Clark-variant Doo–Sabin surfaces and non-uniform biquadratic B-spline surfaces called Non-Uniform Recursive Doo–Sabin Surfaces (NURDSes). One step of NURDS refinement can be factored into one non-uniform linear subdivision step plus one dual step. Compared to the prior non-uniform Doo–Sabin surfaces (i.e., quadratic NURSSes), NURDSes are convergent for arbitrary n-sided faces. Closed form limit point rules, which are important for applications in adaptive rendering and NC machining, are given as well.

New Digital Method of Reverse Engineering about Free from Surface

In this paper, it is proposed a new method of free-from surface’s reverse engineering, making data acquisition and surface reconstruction form closed loop system, solving no feedback problems in the measuring and modeling process, shortening the time of the whole reverse engineering, improving the quality of reconstructed models. The core of this paper is used the CMM adaptive measuring method and non-uniform b-spline surface reconstruction method, integrating the free-from surface measuring and modeling in a closed loop system, realizing the CMM real-time online measurement and reconstructed surface real-time update.

Application of a real-coded genetic algorithm for the fitting of a ship hull surface through a single non-uniform B-spline surface

In digital ship-design processes, surface modeling needs to be as accurate as possible for effectiveness in ship production as well as numerical analysis of the performance. Traditionally, the form of a ship hull is constructed from a set of cross-sectional data. This approach entails difficulties in the cross-sectional spacing and accuracy of the characteristic curves, such as the stern and bow profiles, deck side line, bottom tangential line, and unconnected curves. Genetic algorithms (GAs) have attracted increasing attention as a multimodal optimization solution for surface reconstruction that enable construction of a single non-uniform B-spline (NUB) surface at the initial stage of ship design with constraints such as knuckles, discontinuity conditions, and bulbous bows with high curvatures, . The first, simultaneous multi-fitting GA determines the boundary curves, such as the stem and stern profiles, and finds the common knot values for both curves. Similarly, the same GA technique is applied for other boundary curves at the bottom and the deck. The second GA is employed to fit the interior data points after the boundary curves are fitted. The encoded design variables for surface construction are the locations of the vertices and the knot values. Those variables are modified for improving the surface quality until a predefined degree of precision is attained. In four instances of application, the GA technique developed in this research has been shown to provide good, single, NUB surfaces with high efficiency. In the early design stage, a single NUB surface is more convenient for performance visualization and finite-element methods. It can be readily translated into many CAD/CAM packages, which facilitate the smooth transition of data across the different design stages.

Solution of Poisson Equation using Isogeometric Formulation

Isogeometric solution of Poisson equation is provided. NURBS (NonUniform B-spline Surface) is introduced to express both geometry of structure and unknown field of governing equation. The terms of stiffness matrix and load vector are consistently derived with very accurate geometric definition. The validity of the isogeometric formulation is demonstrated by using two numerical examples such as square plate and L-shape plate. From numerical results, the present solutions have a good agreement with analytical and finite element (FE) solutions with the use of a few cells in isogeometric analysis.

BLIND WATERMARKING OF NON-UNIFORM B-SPLINE SURFACES

In this paper, we propose a watermarking scheme for non-uniform B-spline (NUBS) surface. Firstly, we first do sampling on a NUBS surface and get the sample points, then the watermark is embedded into the DCT coefficients of the sample points and the watermarked sample points are transformed back, finally the watermarked surface is reconstructed from watermarked sample points using global interpolation. A sign correlation detector is used to test for the presence of the watermark, and the original surface is not required at this stage. Experimental results show that our algorithm can preserve the shape of the original surface within a specified error, and that it is robust against attacks including knot insertion, order elevation, addition of white noise, rotation, scaling, translation and further watermarking.

Fragment-based Evaluation of Non-Uniform B-spline Surfaces on GPUs

AbstractIn this paper, we propose a fragment-based evaluation method for non-uniform B-spline surfaces using recent programmable graphics hardware (GPU). A position on a non-uniform B-spline surface is evaluated by the linear combination of both control points and B-spline basis functions. Hence the computational costs can be reduced by pre-computing a knot interval of a parameter from a knot vector. We show that efficient computation of positions and normal vectors for B-spline surfaces can be done on GPUs by applying these ideas. Our algorithm computes an exact position, a derivative and a curvature per fragment. We demonstrate that this achieves high-quality surface rendering. We also discuss about the extension to NURBS and trimmed surfaces.

Single-knot wavelets for non-uniform B-splines

We propose a flexible and efficient wavelet construction for non-uniform B-spline curves and surfaces. The method allows to remove knots in arbitrary order minimizing the displacement of control points when a knot is re-inserted. Geometric detail subtracted from a shape by knot removal is represented by an associated wavelet coefficient replacing one of the control points at a coarser level of detail. From the hierarchy of wavelet coefficients, perfect reconstruction of the original shape is obtained. Both knot removal and insertion have local impact. Wavelet synthesis and analysis are both computed in linear time, based on the lifting scheme for biorthogonal wavelets. The method is perfectly suited for multiresolution surface editing, progressive transmission, and compression of spline curves and surfaces.

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we present the iterative non-uniform B-spline curve and surface to settle a key problem in computer aided geometric design and reverse engineering, that is, constructing the curve (surface) fitting (interpolating) a given ordered point set without solving a linear system. We start with a piece of initial non-uniform B-spline curve (surface) which takes the given point set as its control point set. Then by adjusting its control points gradually with iterative formula, we can get a group of non-uniform B-spline curves (surfaces) with gradually higher precision. In this paper, using modern matrix theory, we strictly prove that the limit curve (surface) of the iteration interpolates the given point set. The non-uniform B-spline curves (surfaces) generated with the iteration have many advantages, such as satisfying the NURBS standard, having explicit expression, gaining locality, and convexity preserving, etc.

T-splines and T-NURCCs

This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline control grids permit T-junctions, so lines of control points need not traverse the entire control grid. T-splines support many valuable operations within a consistent framework, such as local refinement, and the merging of several B-spline surfaces that have different knot vectors into a single gap-free model. The paper focuses on T-splines of degree three, which are C 2 (in the absence of multiple knots). T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull-Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. T-NURCCs enable true local refinement of a Catmull-Clark-type control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit surface. T-NURCCs use stationary refinement rules and are C 2 except at extraordinary points and features.

A unified architecture for the computation of B-spline curves and surfaces

B-Splines, in general, and Non-Uniform Rational B-Splines (NURBS), in particular, have become indispensable modeling primitives in computer graphics and geometric modeling applications. In this paper, a novel high-performance architecture for the computation of uniform, nonuniform, rational, and nonrational B-Spline curves and surfaces is presented. This architecture has been derived through a sequence of steps. First, a systolic architecture for the computation of the basis function values, the basis function evaluation array (the BFEA), is developed. Using the BFEA as its core, an architecture for the computation of NURBS curves is constructed. This architecture is then extended to compute NURBS surfaces. Finally, this architecture is augmented to compute the surface normals, so that the output from this architecture can be directly used for rendering the NURBS surface.