Tensor Product B-spline Surfaces Research Articles

Improving the approximation quality of tensor product B-spline surfaces by local parameterization

Abstract Freeform surfaces like tensor product B-spline surfaces have been proven to be a suitable tool to model laser scanner point clouds, especially those representing artificial objects. However, when it comes to the modelling of point clouds representing natural surfaces with a lot of local structures, tensor product B-spline surfaces reach their limits. Refinement strategies are usually used as an alternative, but their functional description is no longer nearly as compact as that of classical tensor product B-spline surfaces, making subsequent analysis steps considerably more cumbersome. In this publication, the approximation quality of classical tensor product B-spline surfaces is improved by means of local parameterization. By using base surfaces with a local character, relevant information about local structures of the surface to be estimated are stored in the surface parameters during the parameterization step. As a consequence, the resulting tensor product B-spline surface is able to represent these structures even with only a small number of control points. The developed locally parameterized B-spline surfaces are used to model four data sets with different characteristics. The results reveal a clear improvement compared to the classical tensor product B-spline surfaces in terms of correctness, goodness-of-fit and stability.

Linear-Time Algorithm for Computing the Bernstein–Bézier Coefficients of B-spline Basis Functions

A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. In other cases, the complexity is increased by at most O ( m 3 ) . When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. This algorithm scales well when evaluating the B-spline curve at multiple points, e.g., in order to render it, since one only needs to find the coefficients for each knot span once. When evaluating many B-spline curves at multiple points (as is the case when rendering tensor product B-spline surfaces), such approach has lower computational complexity than using the de Boor-Cox algorithm. The numerical tests show that the new method is efficient. The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly. • A new differential-recurrence relation for the B-spline functions of the same degree is proved. • From this relation, a recursive method of computing the coefficients of B-spline functions of degree m in the Bernstein-Bézier form is derived. • Its complexity is proportional to the number of coefficients in the case of coincident boundary knots. This means that, asymptotically, the algorithm is optimal. • When the Bernstein-Bézier coefficients of B-spline basis functions are known, it is possible to compute any B-spline function in linear time with respect to its degree by performing the geometric algorithm proposed recently by the authors. • This algorithm scales well when evaluating many B-spline curves with the same knot sequences at multiple points, e.g., when rendering tensor product B-spline surfaces. • When evaluating many B-spline curves at multiple points, such approach has lower computational complexity than using the de Boor-Cox algorithm. • The numerical tests show that the new method is efficient. • The problem of finding the coefficients of the B-spline functions in the power basis can be solved similarly.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

SURFACE APPROXIMATION OF COASTAL REGIONS: LR B-SPLINE FOR DETECTION OF DEFORMATION PATTERN

Abstract. Geospatial data acquisition of terrains produces huge, noisy and scattered point clouds. An efficient use of the acquired data requires structured and compact data representations. Working directly in a point cloud is often not appealing. To face this challenge, approximation with tensor product B-spline surfaces is attractive. It reduces the point cloud description to relatively few coefficients compared to the volume of the original point cloud. However, this representation lacks the ability to adapt the resolution of the shape to local variations in the point cloud. The result is frequently that noise is approximated and that surfaces have unwanted oscillations.Locally Refined (LR) B-spline surfaces were introduced to face this challenge and provide a tool for approximating Geographic Information System point clouds. In our LR B-spline based approximation algorithm, iterative least-squares approximation is combined with a Multilevel B-spline Approximation to reduce memory consumption. We apply the approach to data sets from coastal regions in Norway and the Netherlands, and compare the obtained approximation with a raster method. We further highlight the potential of LR B-spline volumes for spatio-temporal visualisation of deformation patterns.

Studies on deformation analysis of TLS point clouds using B-splines – A control point based approach (Part I)

Abstract This paper presents a method to overlay laser scan data from at least two different epochs using B-splines. Here, an approach is described in which the laser scan points are directly taken as control points for the B-spline curves or surfaces. In combination with a second part of the publication, the presented technique should be able to detect small deformations compared to the object size based on the curvature parameter. However, the method does not aim at detecting object movements, rotations or translations. Before the epoch-wise laser scan acquisition of an object, a sampling rate adjustment according to the object properties has to be performed. By means of a defining algorithm, control polygons with different numbers of points are defined from the point clouds, whose points are available in scanning order, however not in a uniform grid structure, but an irregular distributed form. As a basis for a bidirectional grid structure of tensor product B-spline surfaces a knot insertion algorithm (Boehm’s algorithm) is to be applied. While the previous process steps are implemented for both B-spline curves and surfaces, the following process steps are only available for B-spline curves so far. In a correlation analysis where the ranks of the curvatures of the B-spline curve points are used, one shift position of the curve can be generated in both epochs. This forms the foundation for the determination of identical curve points, which are iteratively improved to provide the basis for a Helmert transformation to epoch 0. The rotation and translation parameters calculated with the Helmert transformation are only used to map the B-spline curve of an epoch n to epoch 0. A determination of an object rotation is not possible, because the laser scans are not georeferenced. The transformation applied allows the representation of coordinate differences between the laser scan data of different epochs. An implementation of the process steps is available for B-spline curves so far and has to be completed for B-spline surfaces. The finalization of the implementation for B-spline surfaces will be done in a second part of the publication. The calculation process of the standard deviations of the laser scan data up to the standard deviations of the curve and surface points for detecting deformations by means of a significance test will also be listed in more detail in the second part of the publication.

Quadratic B-Spline Surfaces with Free Parameters for the Interpolation of Curve Networks

In this paper, we propose a method for constructing spline surfaces interpolating a B-spline curve network, allowing the presence of free parameters, in order to model the interpolating surface. We provide a constructive algorithm for its generation in the case of biquadratic tensor product B-spline surfaces and bivariate B-spline surfaces on criss-cross triangulations. Finally, we present graphical results.

Multi-sided B-spline surfaces over curved, multi-connected domains

We propose a new surface representation, the Generalized B-spline (GBS) patch, that combines ribbon interpolants given in B-spline form. A GBS patch can connect to tensor-product B-spline surfaces with arbitrary Gm continuity. It supports ribbons not only along the perimeter loop, but also around holes in the interior of the patches.This is a follow-up paper of a recent publication (Várady et al., 2020) that described multi-sided Bézier surfaces over curved multi-sided domains. While the fundamental concept is retained, several new details have been elaborated. The weighting functions are modified to be products of B-spline and Bernstein basis functions, multiplied by rational terms. A new local parameterization method is introduced using harmonic functions, that handles periodic hole loops, as well. Interior shape control is adapted to the B-spline representation of the ribbons. Several examples illustrate the capabilities of the proposed scheme. Our implementation is based on a computationally efficient discretization.

Untrimming: Precise conversion of trimmed-surfaces to tensor-product surfaces

Trimmed B-spline surfaces are common in the geometric computer aided design (CAD) community due to their capability to represent complex shapes that can not be modeled with ease using tensor product B-spline and NURBs surfaces. However, in many cases, handling trimmed-surfaces is far more complex than tensor-product (non-trimmed) surfaces. Many algorithms that operate on tensor-product surfaces, such as algorithms toward rendering, analysis and manufacturing, need to be specially adapted to consider the trimming domains. Frequently, these special adaptations result in lack of accuracy and elevated complexity. In this paper, we present an algorithm for converting general trimmed surfaces into a set of tensor-product (typically B-spline) surfaces. We focus on two algorithms to divide the parametric space of the trimmed surface into four-sided quadrilaterals with freeform curved boundaries, which is the first step of the algorithm. Then, the quadrilaterals are parameterized as planar parametric patches, only to be lifted to the Euclidean space using a surface-surface composition, resulting in tensor product surfaces that precisely tile the input trimmed surface in Euclidean space. The algorithm is robust and precise. We show that we can handle complex, industrial level, objects, with numerous high orders and rational surfaces and trimming curves. Finally, the algorithm provides user control on some properties of the generated tensor-product surfaces.

Progressive iterative approximation for regularized least square bivariate B-spline surface fitting

Recently, the use of progressive iterative approximation (PIA) to fit data points has received a deal of attention benefitting from its simplicity, flexibility, and generality. In this paper, we present a novel progressive iterative approximation for regularized least square bivariate B-spline surface fitting (RLSPIA). RLSPIA extends the PIA property of univariate NTP (normalized totally positive) bases to linear dependent non-tensor product bivariate B-spline bases, which leads to a lower order fitting result than common tensor product B-spline surface. During each iteration, the weights for generating fairing updating surface are obtained by solving an energy minimization problem with box constraints iteratively. Furthermore, an accelerating term is introduced to speed up the convergence rate of RLSPIA, which is comparable favourably with the theoretical optimal one. Several examples are provided to illustrate the efficiency and effectiveness of the proposed method. • RLSPIA extends the PIA property to a set of bivariate non-tensor blending bases. • RLSPIA generalizes the scope of PIA property to a set of linear dependent bases. • RSPIA has faster convergence rate by using the accelerating term. • RSPIA is flexible to impose additional operation, such as surface fairing.

Construction of B-spline surface with B-spline curves as boundary geodesic quadrilateral

This paper is concerned with construction of a B-spline surface that interpolates a B-spline curvilinear quadrilateral as boundary geodesics. The construction consists of two parts. First, from the corner data (i.e., position, unit tangent vector and curvature at the end points of each curve), a four-quartic B-spline curvilinear quadrilateral with minimum strain energy is constructed to satisfy the constraints of the crossing geodesics on a surface. Second, a tensor-product B-spline surface of degree (5, 5) is constructed to interpolate the quadrilateral as boundary geodesics. The control points of the interpolation surface are determined by two steps, and the surface which adheres to the NURBS standard and employs geometric shape handles meeting the requirements of the design scenario. The method is illustrated with several computational examples.

IGA-based point cloud fitting using B-spline surfaces for reverse engineering

Reverse engineering is a viable method to create a 3D virtual model of real physical parts. Usually, reverse engineering consists of two main steps: (1) measure the object and (2) reconstruct it as a 3D model. The measured data are usually represented as a point cloud without topological information and must therefore often be converted into a tensor product B-spline surface format, which has become an industry standard in computer graphics and in CAD systems. In this paper, a new immune genetic algorithm (IGA) for point cloud fitting that fits a noisy 3D point cloud using a B-spline surface with approximate G1 continuity is presented. The point cloud is first segmented into a set of quadrilateral patches. For every patch, a B-spline surface is reconstructed using a least-squares approximation method, and then the surface is optimized to increase the approximation precision using an IGA-based knots adjustment algorithm. Finally, the B-spline patches are stitched together with approximate G1 continuity with a numerical method and the particle swarm optimization (PSO) algorithm. A set of experimental results shows that the proposed method achieves better approximation accuracy than the Bezier-based method and the GA-based method.

Volumetric Boolean sum

Boolean sum is a well-known surface construction operation (Cohen et al., 2001). In the light of the growing interest in trivariate B-spline and NURBs, for example in Isogeometry analysis, in this work we extend this operator for trivariate volumetric elements. Consider six arbitrary tensor product B-spline and/or NURBs surfaces that share boundaries along a cube-like topology. The volume that is enclosed by these six surfaces is parameterized using a volumetric extension of the Boolean sum for surfaces, while the boundaries of the proposed volumetric extension interpolate the six input surfaces. Finally, a generalization of the Boolean sum idea is presented for the general multivariate case.

Polynomial splines over general T-meshes

The present authors have introduced polynomial splines over T-meshes (PHT-splines) and provided theories and applications for PHT-splines over hierarchical T-meshes. This paper generalizes PHT-splines to arbitrary topology over general T-meshes with any structures (GPT-splines). GPT-spline surfaces can be constructed through a unified scheme to interpolate the local geometric information at the basis vertices of the T-mesh. We also discuss general edge insertion and removal algorithms for GPT-splines. As applications, we present algorithms to construct a GPT-spline surface from a quadrilateral mesh and to simplify a tensor-product B-spline surface into a GPT-spline surface with superfluous edges removal.

Monotonicity preserving representations of non-polynomial surfaces

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bézier and tensor product B-spline surfaces.

Shape-based multi-spectral optical image reconstruction through genetic algorithm based optimization

Multi-spectral optical imaging of skin and skin-lesions has been of significant interest for various biomedical applications. A multi-spectral optical imaging instrument called Nevoscope has been developed to acquire images of skin and skin-lesion through trans-illumination. Nevoscope based multi-spectral trans-illumination imaging is aimed at characterization of skin-lesions for early diagnosis of skin-cancers by reconstructing distributions of melanin and blood. Conventional approaches usually involve dividing the field of view into a number of voxels and assuming constant optical properties in each voxel. The optical properties are reconstructed in terms of measurements at multiple wavelengths and the distribution of melanin and blood are subsequently calculated. However, since the inverse problem is generally an ill-posed and under-determined one, it is hard to get quantitatively accurate reconstruction. In this paper, a shape-based multi-constrained reconstruction algorithm is presented, which uses a genetic algorithm based optimization methods to find the best possible reconstruction solution. A skin-lesion such as melanoma is modeled as melanin and blood parts, which are delineated by two cubic tensor-product B-spline surfaces. This reduces the number of unknowns to a few control parameters of the surfaces. The parameters are then coded into a genetic algorithm to find a solution through global optimization. Reasonable constraints are incorporated into the genetic algorithm to stabilize the solution. Results of reconstructions of melanin and blood parts are presented for simulated lesions using multi-spectral wavelengths at 580 nm and 800 nm.

Composite [formula omitted] subdivision surfaces

This paper presents a new unified framework for subdivisions based on a 2 splitting operator, the so-called composite 2 subdivision. The composite subdivision scheme generalizes 4-direction box spline surfaces for processing irregular quadrilateral meshes and is realized through various atomic operators. Several well-known subdivisions based on 2 splitting operator and based on 1-4 splitting operator for quadrilateral meshes are properly included in the newly proposed unified scheme. Typical examples include the midedge and 4-8 subdivisions based on the 2 splitting operator that are now special cases of the unified scheme as the simplest dual and primal subdivisions, respectively. Variants of Catmull–Clark and Doo–Sabin subdivisions based on the 1-4 splitting operator also fall in the proposed unified framework. Furthermore, unified subdivisions as extension of tensor-product B-spline surfaces also become a subset of the proposed unified subdivision scheme. In addition, Kobbelt interpolatory subdivision can also be included into the unified framework using VV-type (vertex to vertex type) averaging operators.

Generation of Discrete Bicubic G1 B-Spline Ship Hullform Surfaces from a Given Curve Network Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G1 continuous B-spline surfaces that interpolate the curve network of a ship hullform. First, the curves in the network are classified into two types: boundary curves and “reference curves”. The boundary curves correspond to a set of rectangular (or triangular) topological type that can be represented with tensor-product (or degenerate) B-spline surface patches. Next, in the interior of the patches, surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing “virtual” iso-parametric curves. Finally, a discrete G1 continuous B-spline surface is generated by a surface fitting algorithm. Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.

Piecewise B-spline Surfaces Fitting to Arbitrary Triangle Meshes

Due to technical progress in shape acquiring from range data, triangle mesh representations of models can now be obtained conveniently. However, triangle meshes are not the final representation form in many applications, for reasons of compactness, appearance, modification, and manufacturability etc. In this paper, we present a solution for fitting smooth B-spline surfaces to triangle meshes of arbitrary topology. The procedure consists of two major phases: (1) partitioning triangle meshes of arbitrary topology into quadrilateral patches, (2) fitting tensor-product B-spline surfaces to each quadrilateral patch and keeping patches continuous across shared boundaries and corners. One merit of our solution is that it makes a reasonable tradeoff between automation and human control in complex model segmentation and surface fitting. Experiments in the paper show that our solution is practical and effective.

On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree

We introduce a new class of subdivision surfaces which generalize uniform tensor product B-spline surfaces of any bi-degree to meshes of arbitrary topology. Surprisingly, this can be done using subdivision rules that involve direct neighbors only. Consequently, our schemes are very easy to implement, regardless of degree. The famous Catmull–Clark scheme is a special case. Similarly we show that triangular box splines of total degree 3m+1 can be generalized to arbitrary triangulations. Loop subdivision surfaces are a special case when m=1. Our new schemes should be of interest to the high-end design market where surfaces of bi-degree up to 7 are common.

Extensions of the general polar value based control point specification method in constructing tensor product B-spline surfaces

Blossoming has been utilized as a powerful tool for understanding Bezier and B-spline curves. Recently, general polar values (GPV) have been employed as an aid to specify control points for polynomial curves. Based on this technique, it is possible for us to use a set of polar values to control a polynomial curve. In this paper, we first construct linear systems of tensor product B- spline surfaces (TPBS) under the GPV-based framework. We call this extension as the direct extension scheme. Subsequently, based on similar principles, we investigate the issue of extending the linear systems based on degree elevation and partial derivatives evaluation. For the direct extension and degree elevation schemes, high flexibility is provided though a large set of corresponding free parameters. For the partial derivative scheme, partial derivatives on a surface can be used as parameters. In all schemes, a restriction on parameter selection is imposed by the invertibility of the related matrices. Finally, based on the subtle relationships among the basis functions, a fast computation algorithm is developed for evaluating those related basis-function matrices.