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Knot placement in B-spline curve approximation Reporter:Cao juan Date:2006.54.5
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Outline: Introduction Some relative paper discussion
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Introduction: BBackground: TThe problem is…
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It is a multivarate and multimodal nonlinear optimization problem
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The NURBS Book Author:Les Piegl & Wayne Tiller
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They are iterative processes : 1.Start with the minimum or a small number of knots 2.Start with the maximum or many knots
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Use chordlength parameterization and average knot:
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Disadvantage: Time-consuming Relate to initial knots
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Knot Placement for B-spline Curve Approximation Author: Anshuman Razdan (Arizona State University, Technical Director, PRISM)
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Assumptions: A parametric curve evaluated at arbitrary discrete values Goals: closely approximate with B-spline
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Estimate the number of points required to interpolate (ENP)
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Adaptive Knot Sequence Generation (AKSG)
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Based on curvature only Using origial tangents
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The Pre-Processing of Data Points for Curve Fitting in Reverse Engineering Author: Ming-Chih Huang & Ching-Chih Tai Department of Mechanical Engineering, Tatung University, Taipei, Taiwan Advanced Manufacturing Technology 2000
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Chord length parameter:
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Problem: data are noise & unequal distribution Aim: reconstruction (B-spline curve with a “good shape”)
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Characters: approximate the curve once
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Data fitting with a spline using a real-coded genetic algorithm Author:Fujiichi Yoshimoto, Toshinobu Harada, Yoshihide Yoshimoto Wakayama University CAD(2003)
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About GA: 60’s by J.H,Holland some attractive points: Global optimum Robust... fitness
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Fitness function: Bayesian information criterion Initial population:
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Example of two-point crossover:
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Mutation method: for each individual for counter = 1 to individual length Generate a random number Generate a random number add a gene randomlyDelete a gene randomly >Pm Y N Counter + 1 >0.5 NY
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Character: insert or delete knots adaptively Quasi-multiple knots Don’t need error tolerance Independent with initial estimation of the knot locations Only one –dimensional case
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Adaptive knot placement in B- spline curve approximation author: Weishi Li, Shuhong Xu, Gang Zhao, Li Ping Goh CAD(2005)
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a heuristic rule for knot placement Su BQ,Liu DY: > approximation interpolation best select points
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Algorithm: smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate
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smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate inflection points
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smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate curvature integration
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smooth the discrete curvature divide into several subsets iteratively bisect each segment till satisfy the heuristic rule check the adjacent intervals that joint at a feature point Interpolate curvature integration
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Example:
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character: smooth discrete curvature automatically sensitive to the variation of curvature torsion? arc length?
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summary: torsion arc length multi-knots (discontinue,cusp)
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reference: Piegl LA, Tiller W. The NURBS book. New York: Springer; 1997. Razdan A. Knot Placement for B-spline curve approximation. Tempe,AZ: Arizona State University; 1999 http://3dk.asu.edu/archives/publication/publication.html Huang MC, Tai CC. The pre-processing of data points for curve fittingin reverse engineering. Int J Adv Manuf Technol 2000;16:635–42 Yoshimoto F, Harada T, Yoshimoto Y. Data fitting with a spline using a real-coded genetic algorithm. Comput Aided Des 2003;35:751–60. Weishi Li,Shuhong Xu,Gang Zhao,Li Ping Goh.Adaptive knot placement in B-spline curve approximation.Computr-Aided Design.2005;37:791-797
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