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.

References

References
1.
Yoshimoto
,
F.
,
Moriyana
,
M.
, and
Harada
,
T.
,
1999
, “
Automatic Knots Placement by a Genetic Algorithm for Data Fitting With a Spline
,” SMI'99, Washington, DC, Mar. 1–4, p. 162.
2.
Yoshimoto
,
F.
,
Harada
,
T.
, and
Yoshimoto
,
Y.
,
2003
, “
Data Fitting With a Spline Using a Real Coded Genetic Algorithm
,”
Comput.-Aided Des.
,
35
(
8
), pp.
751
760
.
3.
Gálvez
,
A.
, and
Iglesias
,
A.
,
2011
, “
Efficient Particle Swarm Optimization Approach for Data Fitting With Free Knot B-Splines
,”
Comput. Aided Des.
,
43
(
12
), pp.
1683
1692
.
4.
Hughes
,
T. J. R.
,
Cottrell
,
J. A.
, and
Bazilevs
,
Y.
,
2005
, “
Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
39–41
), pp.
4135
4195
.
5.
Cottrell
,
J. A.
,
Hughes
,
T. J. R.
, and
Bazilevs
,
Y.
,
2009
,
Isogeometric Analysis: Toward Integration of CAD and FEA
,
Wiley
, Hoboken, NJ.
6.
Cárdenas
,
D.
,
Elizalde
,
H.
,
Marzocca
,
P.
,
Probst
,
O.
,
Ramirez
,
R.
, and
Toledo
,
J. P.
,
2014
, “
The Poly-SAFE Method: A Semi-Analytical Representation of Finite Element Models Via Nested Polynomial Reduction of Modal Data
,”
Compos. Struct.
,
111
, pp.
301
316
.
7.
Seo
,
Y. D.
,
Kim
,
H. J.
, and
Youn
,
S. K.
,
2010
, “
Shape Optimization and Its Extension to Topological Design Based on Isogeometric Analysis
,”
Int. J. Solids Struct.
,
47
(
11–12
), pp.
1618
1640
.
8.
Seo
,
Y. D.
,
Kim
,
H. J.
, and
Youn
,
S. K.
,
2010
, “
Isogeometric Topology Optimization Using Trimmed Spline Surfaces
,”
Comput. Methods Appl. Mech. Eng.
,
199
(
49–52
), pp.
3270
3296
.
9.
Li
,
W.
,
Xu
,
S.
,
Zhao
,
G.
, and
Goh
,
L. P.
,
2005
, “
Adaptive Knot Placement in B-Spline Curve Approximation
,”
Comput.-Aided Des.
,
37
(
8
), pp.
791
797
.
10.
Park
,
H.
,
Kim
,
K.
, and
Lee
,
S.-C.
,
2000
, “
A Method for Approximate NURBS Curve Compatibility Based on Multiple Curve Refitting
,”
Comput.-Aided Des.
,
32
(
4
), pp.
237
252
.
11.
Park
,
H.
,
2004
, “
An Error-Bounded Approximate Method for Representing Planar Curves Is B-Splines
,”
Comput.-Aided Geom. Des.
,
21
(
5
), pp.
479
497
.
12.
Park
,
H.
, and
Lee
,
J.-H. B.
,
2007
, “
Spline Curve Fitting Based on Adaptive Curve Refinement Using Dominant Points
,”
Comput.-Aided Des.
,
39
(
6
), pp.
439
451
.
13.
Robinson
,
E. C.
,
Jbabdi
,
S.
,
Andersson
,
J.
,
Smith
,
S.
,
Glasser
,
M. F.
,
Van Essen
,
D.
, and
Jenkinson
,
M.
,
2013
, “
Multimodal Surface Matching: Fast and Generalizable Cortical Registration Using Discrete Optimization
,”
Information Processing in Medical Imaging
,
Springer
,
Berlin
, pp.
475
486
.
14.
Zhao
,
X.
,
Zhang
,
C.
,
Yang
,
B.
, and
Li
,
P.
,
2011
, “
Adaptive Knot Placement Using a GMM-Based Continuous Optimization Algorithm in B-Spline Curve Approximation
,”
Comput.-Aided Des.
,
43
(
6
), pp.
598
604
.
15.
Ulker
,
E.
, and
Arslan
,
A.
,
2009
, “
Automatic Knot Adjustment Using an Artificial Immune System for B-Spline Curve Approximation
,”
Inf. Sci.
,
179
(
1
), pp.
1483
1494
.
16.
Jupp
,
A. L.
,
1978
, “
B Approximation to Data by Splines With Free Knots
,”
J. Numer. Anal.
,
15
(
2
), pp.
328
343
.
17.
Valenzuela
,
O.
,
Delgado-Marquez
,
B.
, and
Pasadas
,
M.
,
2013
, “
Evolutionary Computation for Optimal Knots Allocation in Smoothing Splines
,”
Appl. Math. Modell.
,
37
(
8
), pp.
5851
5863
.
18.
Jiang
,
S.
,
Wang
,
Y.
, and
Zhicheng
,
J.
,
2014
, “
Convergence Analysis and Performance of an Improved Gravitational Search Algorithm
,”
Appl. Soft Comput.
,
24
, pp.
363
384
.
19.
Razdan
,
A.
,
1999
,
Knot Placement for B-Spline Curve Approximation
,
Arizona State University
,
Tempe, AZ
.
20.
Asada
,
H.
, and
Brady
,
M.
,
1986
, “
The Curvature Primal Sketch
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
8
(1), pp. 2–14.
21.
Hamann
,
B.
, and
Chen
,
J.-L.
,
1994
, “
Data Point Selection for Piecewise Linear Curve Approximation
,”
Comput. Aided Geometric Des.
,
11
(
3
), pp.
289
301
.
22.
Pei
,
S.-C.
, and
Horng
,
J.-H.
,
1995
, “
Fitting Digital Curve Using Circular Arcs
,”
Pattern Recognit.
,
28
(
1
), pp.
107
116
.
23.
Rattarangsi
,
A.
, and
Chin
,
R. T.
,
1992
, “
Scaled-Detection of Corners of Planar Curves
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
14
(
4
), pp.
430
449
.
24.
Horn
,
J.-H.
,
2003
, “
An Adaptive Smoothing Approach for Fitting Digital Planar Curves With Line Segments and Circular Arcs
,”
Pattern Recognit. Lett.
,
24
(
1–3
), pp.
565
577
.
25.
Cao
,
M. R.
,
Xu
,
M.
, and
Ostachowicz
,
W.
,
2014
, “
W. Indentification of Multiple Damage in Beams Based on Robust Curvature Mode Shapes
,”
Mech. Syst. Signal Process.
,
46
(
2
), pp.
468
480
.
26.
Lin
,
W.-Y.
,
Chiu
,
Y.-L.
,
Widder
,
K. R.
,
Hu
,
Y. H.
, and
Boston
,
N.
,
2010
, “
Robust and Accurate Curvature Estimation Using Adaptive Line Integrals
,”
EURASIP J. Adv. Signal Process.
,
2010
(1), p. 240309.
27.
Sohn
,
K.
,
Alexander
,
W. E.
,
Kim
,
J. H.
,
Kim
,
Y.
,
Yoon
,
S. H.
,
Park
,
E. H.
, and
Ntuen
,
C. A.
,
1991
, “
Optimal Boundary Smoothing for Curvature Estimation
,”
Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems and Computers
, pp.
1220
1224
.
28.
Lee
,
I.-K.
,
2000
, “
Curve Reconstruction From Unorganized Points
,”
Comput. Aided Des.
,
17
(
2
), pp.
161
177
.
29.
García
,
D.
,
2010
, “
Robust Smoothing of Gridded Data in One and Higher Dimensions With Missing Values
,”
Comput. Stat. Data Anal.
,
54
(
4
), pp.
1167
1178
.
30.
Weinert
,
H. L.
,
2007
, “
Efficient Computation for Whittaker-Henderson Smoothing
,”
Comput. Stat. Data Anal.
,
57
(
2
), pp.
959
974
.
31.
Farin
,
G.
,
2002
,
Curves and Surfaces for CADG: A Practical Guide
,
Morgan Kaufmann Publishers
,
San Francisco, CA
.
32.
Piegl
,
L.
, and
Tiller
,
W.
,
1995
,
The NURBS Book
,
Springer-Verlag
,
New York
.
33.
Savitzky
,
A.
, and
Golay
,
M. J. E.
,
1964
, “
Smoothing and Differentiation of Data by Simplified Least Squares Procedures
,”
Anal. Chem.
,
36
(
8
), pp.
1627
1639
.
34.
Prautzsch
,
H.
,
Boehm
,
W.
, and
Paluszny
,
M.
,
2002
,
Bézier and B-Spline Techniques
,
Springer-Verlag
,
New York
.
35.
Su
,
B. Q.
, and
Liu
,
D. Y.
,
1989
,
Computational Geometry—Curve and Surface Modeling
,
Academic Press
,
Boston, MA
.
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