Optimal arc spline approximation

Maier, Georg

doi:10.1016/j.cagd.2014.02.011

Cited by 14 publications

(18 citation statements)

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“…This opens the door to using recent approximation algorithms for 0 -minimization for optimal spline approximation. For arc splines-curves consisting of a finite sequence of circular arcs and line segments-a geometric approach for approximating a sequence of points by a G 1 -continuous arc spline with a minimum number of segments has been recently proposed by Maier [Mai14]. Contemporaneous to our research, Kang et al [KCL * 15] proposed an equivalent reformulation of the optimal spline problem for the case of B-splines in one dimension.…”

Section: Related Workmentioning

confidence: 98%

“…Optimal approximation by polylines has a more local character: changing the position of knots will have no global effect on the solution, as is the case for splines, where differentiability conditions introduce global dependencies. This local property of polylines (and other types of curve which are only demanded to be continuous) allows the use of Dynamic Programming algorithms, which can give optimal approximations (given either a maximum number of segments, a cost per segment or a maximum approximation error in the 2 or ∞ norm) by polylines [PV94,GB04], line segments and circular arcs [Kol12,Mai14], or piecewise polynomials [MJEM02]. 0 -Minimization.…”

Section: Related Workmentioning

confidence: 99%

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Optimal Spline Approximation via ℓ₀‐Minimization

Brandt

Seidel

Hildebrandt

2015

Computer Graphics Forum

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Splines are part of the standard toolbox for the approximation of functions and curves in R d . Still, the problem of finding the spline that best approximates an input function or curve is ill-posed, since in general this yields a "spline" with an infinite number of segments. The problem can be regularized by adding a penalty term for the number of spline segments. We show how this idea can be formulated as an 0 -regularized quadratic problem. This gives us a notion of optimal approximating splines that depend on one parameter, which weights the approximation error against the number of segments. We detail this concept for different types of splines including B-splines and composite Bézier curves. Based on the latest development in the field of sparse approximation, we devise a solver for the resulting minimization problems and show applications to spline approximation of planar and space curves and to spline conversion of motion capture data.

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Section: Related Workmentioning

confidence: 98%

Section: Related Workmentioning

confidence: 99%

Optimal Spline Approximation via ℓ₀‐Minimization

Brandt

Seidel

Hildebrandt

2015

Computer Graphics Forum

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“…Biarcs have several interesting properties, first of all, they are easy to understand and to use: in fact the arclength computation is straightforward, the tangent vector field is continuous and defined everywhere, the curvature is defined almost everywhere and is piecewise constant. Moreover, they are very useful in several applications, for instance, they are effectively used in the approximation of higher degree curves [Mai14,DM14] machining and milling, where the cutting devices follow the so called G-code, i.e. path composed of straight lines and circles.…”

Section: Introductionmentioning

confidence: 99%

A Note on Robust Biarc Computation

Bertolazzi¹,

Frego²

2019

CAD&A

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A new robust algorithm for the numerical computation of biarcs, i.e. G 1 curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical configurations. The proposed algorithm uses an algebraic construction which is reduced to the solution of a single 2 by 2 linear system. Singular angles configurations are treated smoothly by using the pseudoinverse matrix when solving the linear system. The proposed algorithm is compared with the Matlab's routine rscvn that solves geometrically the same problem. Numerical experiments show that Matlab's routine sometimes fails near singular configurations and does not select the correct solution for large angles, whereas the proposed algorithm always returns the correct solution. The proposed solution smoothly depends on the geometrical parameters so that it can be easily included in more complex algorithms like splines of biarcs or least squares data fitting.

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“…We use this algorithm to improve the numerical stability of the SMAP (smooth minimum arc path) approach which computes an approximating smooth arc spline with the minimal number of segments within a specified maximal tolerance, cf. [6] or [8] for an application in vehicle self-localization. The basic task in the SMAP algorithm is closely related to the computation of the circular visibility set from a starting arc.…”

Section: Introductionmentioning

confidence: 99%