CVGIP: Graphical Models and Image Processing
Regular ArticleAn Extension of Chaiken′s Algorithm to B-Spline Curves with Knots in Geometric Progression
Abstract
Chaiken′s algorithm is a procedure for inserting new knots into uniform quadratic B-spline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken′s algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken′s algorithm to B-spline curves of arbitrary degree for knot sequences in geometric and affine progression.
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Subdivision algorithms with modular arithmetic
2024, Computer Aided Geometric DesignWe study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle atGiven a sequence of integers mod m, we connect consecutive values on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that . Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.
Bivariate non-uniform subdivision schemes based on L-systems
2023, Applied Mathematics and ComputationL–systems have been used to describe non-uniform, univariate, subdivision schemes, which offer more flexible refinement processes than the uniform schemes, while at the same time are easier to analyse than the geometry driven non-uniform schemes. In this paper, we extend L–system based non-uniform subdivision to the bivariate setting. We study the properties that an L–system should have to be the suitable descriptor of a subdivision refinement process. We derive subdivision masks to construct the regular parts of the subdivision surface as cubic B-spline patches. Finally, we describe stencils for the extraordinary vertices, which after a few steps become stationary, so that the scheme can be studied through simple eigenanalysis. The proposed method is illustrated through two new subdivision schemes, a Binary-Ternary, and a Fibonacci scheme with average refinement rate below two.
Non-uniform interpolatory subdivision schemes with improved smoothness
2022, Computer Aided Geometric DesignSubdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate and limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one.
L-system specification of knot-insertion rules for non-uniform B-spline subdivision
2012, Computer Aided Geometric DesignSubdivision schemes are based on a hierarchy of knot grids in parameter space. A univariate grid hierarchy is regular if all knots are equidistant on each level, and irregular otherwise. We use L-systems to design a wide class of systematically described irregular grid hierarchies. Furthermore, we give sufficient conditions on the L-system which guarantee that the subdivision scheme, based on the non-uniform B-spline of degree d defined on the initial knot grid, is uniformly convergent. If n is the number of symbols in the alphabet of the L-system, this subdivision scheme is defined with a finite set of masks (at most ) which does not depend on the subdivision step. We provide an implementation of such schemes which is available as a worksheet for Sage software.
Adjustable speed surface subdivision
2009, Computer Aided Geometric DesignWe introduce a non-uniform subdivision algorithm that partitions the neighborhood of an extraordinary point in the ratio , where . We call σ the speed of the non-uniform subdivision and verify continuity of the limit surface. For , the Catmull–Clark limit surface is recovered. Other speeds are useful to vary the relative width of the polynomial spline rings generated from extraordinary nodes.
Selective knot insertion for symmetric, non-uniform refine and smooth B-spline subdivision
2009, Computer Aided Geometric DesignNURBS surfaces can be non-uniform and defined for any degree, but existing subdivision surfaces are either uniform or of fixed degree. The resulting incompatibility forms a barrier to the adoption of subdivision for CAD applications. Motivated by the search for NURBS-compatible subdivision schemes, we present a non-uniform subdivision algorithm for B-splines in the spirit of the uniform Lane–Riesenfeld ‘refine and smooth’ algorithm. In contrast to previous approaches, our algorithm is independent of index direction (symmetric), and also allows a selection of knot intervals to remain unaltered by the subdivision process. B-splines containing multiple knots, an important non-uniform design tool, can therefore be subdivided without increasing knot multiplicity.