A multi-level filtering approach for fairing planar cubic B-spline curves

doi:10.1016/j.cagd.2006.09.004

Computer Aided Geometric Design

Volume 24, Issue 1, January 2007, Pages 53-66

https://doi.org/10.1016/j.cagd.2006.09.004 Get rights and content

Abstract

In this paper a new approach to the problem of fairing planar B-spline curves is introduced. We propose an algorithm based on a multi-level representation of cubic B-spline curves, which enables the identification of bad control points that need to be faired. The multi-level representation allows splitting a curve into its low resolution and details function parts. The details function permits the formulation of a different approach to the selection of bad control points, differing from others methods that are based on the evaluation of curve and curvature derivatives. Moreover, this new technique leads to an increased interaction with designers that can identify faster the set of bad control points and then operate on them through their level-of-detail (LOD) representation in manner to obtain the expected shape.

Hence, designers have more control over the entire slope by thresholding details in several manners fully described in the following chapters. Several numerical examples are presented to validate the effectiveness of this algorithm compared with another technique in (Farin, G., Sapidis, N., 1989. Curvature and the fairness of curves and surfaces. IEEE Computer Graphics and Applications 9 (2), 52–57).

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Besides unified approaches for fitting and fairing, B-spline curves can also be faired independently after fitting or construction. Typical algorithms for B-spline curve fairing include knot removal [10,11], local or global energy minimization [12,13] and multi-scale filtering [14,15], etc. Compared with B-spline curves, NURBS curves are more flexible in shape representation [1,2,16].
This paper proposes techniques to fit and fair sequences of points together with normals or tangents at the points by matrix weighted NURBS curves. Given a set of Hermite-type data, a matrix weighted NURBS curve is constructed by choosing the input points as control points and computing the weight matrices using the normals or tangents. Unlike traditional B-spline or NURBS curves that have only linear precision, matrix weighted NURBS curves with point-normal or point-tangent control pairs have almost circular or helical precision. Matrix weighted NURBS curves constructed from Hermite-type data can be fair and fit the input points closely when the original data were regularly sampled from curves with smoothly varying tangents and curvatures. If the original data are non-uniformly spaced or noisy, fair fitting curves can still be obtained by repeatedly sampling points from previously constructed curves and constructing new matrix weighted NURBS curves using the resampled data.
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The early wavelet fairing algorithms were generally for the uniform B-splines [13,14]. In 2007, Amati proposed a wavelet fairing algorithm for quasi-uniform B-spline curves [16]. However, it has two disadvantages: (a) its efficiency is low; (b) the quasi-uniform wavelet transform limits the amount of control points.
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Cho et al. introduced a fairing method with boundary continuity based on the uniform B-spline wavelets [7]. In 2006, Amati proposed a multi-level filtering approach for fairing planar cubic B-spline curves [1]. This method works for local and global fairing, but it is not extended to surface fairing.
This paper describes a novel simplification and fairing method for NURBS (Non-Uniform Rational B-Spline) curves and surfaces using orthogonal non-uniform spline wavelets. The compression nature of wavelets is utilized for NURBS simplification. Sufficient conditions for $C^{0}$ - and $G^{1}$ -continuity at the shared boundary of multiple NURBS patches are proved and utilized to preserve the boundary continuity. Our fairing method filters out unwanted imperfections and noise on the NURBS curves and surfaces while preserving geometric details via wavelet decomposition and selective reconstruction. The continuity between neighboring patches is preserved automatically during fairing. Our algorithm is accurate, automatic, robust, efficient, and can be extensively applied to any NURBS models in various research fields such as Computer Aided Design (CAD), Reverse Engineering (RE), finite element analysis and isogeometric analysis. Several examples are presented in this paper, including a propeller CAD model, a Navy boat model and a $G^{1}$ -continuous NURBS model of the thoracic aorta constructed from medical CT (Computed Tomography) data.
Geometric constraint modelling: Boundary planar B-spline curves and control polyhedra for 5-axes response surface graph
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This paper introduces a geometric constraint modelling system for 5-axes response surface graph based on free-form boundary planar curves and cross-boundary curves. Both are developed on the basis of nonperiodic B-spline curves. It is evident that generating the boundary planar curves is better done symmetrically on the main planes and the 45°-angular planes. While, generating the cross-boundary curves require to additionally identify and involve control points on the 30°- and 60°-angular planes. As a result, a mesh $3 \times 4$ size on parametric $R^{2}$ Euclidean space is obtained that symmetry and equally spaced interval from the design center. Advancement of the mesh onto $R^{3}$ Euclidean space forms the control polyhedra. Then, the B-spline surface will shape uniquely to follow the polyhedra. The emerging B-spline surfaces are estimated in partial on four different $R^{3}$ Euclidean spaces that, ultimately, converge as a composite 5-axes response surface graph.
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They preserve the volume constraint using a quadratic minimization process when the mesh is deformed through multiresolution decomposition. Amanti [11] proposes a wavelet based multi-level analysis approach to fair planar cubic B-spline curves. This approach is useful to find the curve segments that need to be smoothed.
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This paper utilizes a recursive formula for transformation matrices of B-spline bases to construct biorthogonal nonuniform B-spline wavelets based on discrete norm $l_{2}$ and proposes the reconstruction and decomposition algorithms for the MRA of the wavelets. The computation of reconstruction matrices of the proposed wavelets is based on simple recursion and integral operation is avoided. The reconstruction and decomposition algorithms are both simple, efficient and with linear time complexity. Moreover, the analysis for the performances of the proposed wavelets reveals that the wavelets possess smaller supports in general cases compared with that of semiorthogonal nonuniform B-spline wavelets, with good approximation property and suitable for any uniform and nonuniform nested sequences of knot vectors. Several examples are given for the applications of the proposed wavelets in the MRA of nonuniform B-spline curves. The experimental results show that as any other nonuniform B-spline wavelets, the proposed wavelets provide greater flexibility than uniform B-spline wavelets do for the applications of the MRA of B-spline curves and surfaces. With the proposed wavelets we can easily construct rich multiresolution levels and maintain the specified characteristic points as well as local details of the decomposed curves to satisfy various requirements in the applications.

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A multi-level filtering approach for fairing planar cubic B-spline curves

Abstract

Computer-Aided Design

Computer Aided-Design

Computer Aided Geometric Design

An Introduction to Wavelets

Local energy fairing of b-spline curves

Computing Supplementum

Curves and Surfaces for CAGD: A Practical Guide

Curvature and the fairness of curves and surfaces

IEEE Computer Graphics and Applications