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Research Papers

An Adaptive Curvature-Guided Approach for the Knot-Placement Problem in Fitted Splines

[+] Author and Article Information
Enrique Aguilar

School of Engineering and Science,
Tecnológico de Monterrey, Tlalpan,
Mexico City 14380, Mexico
e-mail: enriqueam1409@gmail.com

Hugo Elizalde

School of Engineering and Science,
Tecnológico de Monterrey, Tlalpan,
Mexico City 14380, Mexico
e-mail: hugo.elizalde@itesm.mx

Diego Cárdenas

School of Engineering and Science,
Tecnológico de Monterrey, Tlalpan,
Mexico City 14380, Mexico
e-mail: diego.cardenas@itesm.mx

Oliver Probst

School of Engineering and Science,
Tecnológico de Monterrey,
Monterrey 64849, Nuevo Leon, Mexico
e-mail: oprobst@itesm.mx

Pier Marzocca

Aerospace, Mech. & Manuf. Eng. Department,
RMIT University,
P.O. Box 71,
Bundoora 3083, Victoria, Australia
e-mail: pier.marzocca@rmit.edu.au

Ricardo A. Ramirez-Mendoza

School of Engineering and Science,
Tecnológico de Monterrey,
Monterrey 64849, Nuevo Leon, Mexico
e-mail: ricardo.ramirez@itesm.mx

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received November 27, 2015; final manuscript received July 18, 2018; published online September 5, 2018. Editor: Satyandra K. Gupta.

J. Comput. Inf. Sci. Eng 18(4), 041013 (Sep 05, 2018) (9 pages) Paper No: JCISE-15-1390; doi: 10.1115/1.4040981 History: Received November 27, 2015; Revised July 18, 2018

This paper presents an adaptive and computationally efficient curvature-guided algorithm for localizing optimum knot locations in fitted splines based on the local minimization of an objective error function. Curvature information is used to narrow the searching area down to a data subset where the local error function becomes one-dimensional, convex, and bounded, thus guaranteeing a fast numerical solution. Unlike standard curvature-guided methods, typically relying on heuristic rules, the novel method here presented is based on a phenomenological approach as the error function to be minimized represents geometrical properties of the curve to be fitted, consequently reducing case-sensitivity issues and the possibility of defining spurious knots. A knot-readjustment procedure performed in the vicinity of a newly created knot has the ability of dispersing knots from otherwise highly knot-populated regions, a feature known to generate undesired local oscillations. The performance of the introduced method is tested against three other methods described in the literature, each handling the knot-placement problem via a different paradigm. The quality of the fitted splines for several datasets is compared in terms of the overall accuracy, the number of knots, and the computing efficiency. It is demonstrated that the novel algorithm leads to a significantly smaller knot vector and a much lower computing time, while preserving or improving the overall accuracy.

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Copyright © 2018 by ASME
Topics: Splines , Algorithms , Errors
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Figures

Grahic Jump Location
Fig. 2

Flowchart of the main algorithm

Grahic Jump Location
Fig. 1

Generic illustrations for describing the knot-placement algorithm. The solid lines represent the original polygon segments, the dashed lines represent the new subsegments, and the dotted lines represent the dataset. (a) Representative curvature function and “parent knots”; (b) piecewise linear polygon (solid line), segments (and associated data subsets) t−1 and t; (c) a new knot Kt+γ is inserted at an optimum location within segment t; and (d) orthogonal deviations ei and ej.

Grahic Jump Location
Fig. 3

The nine sample datasets used in this work for comparing the quality of the splines generated by each of the selected methods

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Fig. 4

Splines and correspondent knots generated by methods A, C, and D for a zoomed view of the sample dataset 6. Method A is applied (a) without the knot-readjustment scheme and (b) with the knot-readjustment scheme. Insets: Zoom into the critical near-vertical regions.

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