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Interpolation and elevation zhu ping 07.10.10 zhu ping 07.10.10

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Problem T 0,T 1,T 2,…,T n ∈ R 2, T i ≠T i+1 Find a curve P(t): P:[0,1]—> R 2, satisfying P(t i )=T i i=0,1,…,n

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Paper 1. > Lin Hongwei Science in China 04 2. > Lin Hongwei Computers and Mathematics with Application 05 3. > J.Delgado CAGD 07 4. > Jernej Kozak CAGD 07 5. > Les A.Piegl The Visual Computer 05 6. > Wang Guozhao CAGD07 7. > Takashi Maekawa CAD 07

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Constructing Iterative Non-Unifrom B-spline Curve and Surface to Fit Data Points Problem:Progressive iterative approximation Proposed by Prof.Qi Dongxu and de Boor In 1991, cubic uniform B-spline and convergence Shortage:NURBS,convexity-preserving,explict expression

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Iterative algorithm (1) parameter (2)normal

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Parameter

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Normal

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Deduce:

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Iterative error:

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Iterative convergence

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Convexity-preserving algorithm Polyline L is convexity and if sgn(α i )=sgn(β i ) polyline L’is also convexity Polyline L is convexity and if sgn(α i )=sgn(β i ) polyline L’is also convexity

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Iterative formulae

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Surface iterative

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Example: convexity-preserving property of non-uniform B-spline curve convexity-preserving property of non-uniform B-spline curve 30 times

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Totally Positive Bases and Progressive Iteration Approximation Totally Positive Bases Definition: Given a basis {Bi(t) ≥0|i=0,1,…,n}, an increasing sequence the collocation matrix is Basis {Bi} is totally positive basis if the matrix is a totally positive matrix Basis {Bi} is totally positive basis if the matrix is a totally positive matrix

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Theorem. If the basis is totally positive and its collocation matrix B at {t 0,t 1,…,t n } is non-singular,the curve has progressive iteration appromation. Proof.

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Example: Bezier,B-spline and NUBRS curve and surface 20 times 60 times Bezier

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Progressive iterative approximation and bases with the fastest convergence rates Outline: The normalized B-basis has optimal shape preserving properties and we prove that it satisfies the progressive iterative approximation property with the fastest convergence rates. A similar result for tensor product surfaces is also derived.

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Theorem: Given a space U with an TP basis,the normalized B-basis of U provided a progressive iterative approximation with the fastest convergence rates among all TP bases of U.

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Proof. will be minimum when the smallest eigenvalue of B is maximum. U:unique normalized B-basis (b 0,…,b n ),TP basis (v 0,…,v n ). (v 0,…,v n )=(b 0,…,b n )K. K is a stochastic TP matrix. prove that the smallest eigenvalue of B is greater than the smallest eigenvalue of V

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Consider matrix since is similar to is similar to.We should prove

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Tensor product surfaces: Kronecker product:

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Numerical test: Bernstein Basis VS Said-Ball Basis Definition:

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Comparion:

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Error compare:

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Geometric Method for Hermite Iterpolation by a class of PH Quintics Outline: Geometric relationship among Bezier control points of PH Quintics is developed.And choose the best curve in the resulant PH quintics.

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Theorem 1: A Bezier curve is PH curve when two points A,B exist making

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Hermite Method:

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On the degree elevation of B-spline curves and corner cutting Outline: Our idea is making bi-degree B-spline to elevate B-spline. Old Method:splite into pieces of Bezier curves

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Theorem 1:, are defined and Theorem 2:

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Property:positivity,partition of unity,linear independence globally Theorem 3:

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An alternative method of curve interpolation Author: Les A.Piegl, South Florida University, research in CAD/CAM,geometric modeling,computer graphics Wayne Tiller,in GeomWare, The NURBS Book

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Outline:given a point data set that contains several fairly unevently distributed random points,this paper presents a new paradigm of curve interpolation to fit a curve to the data with end tangent vector constraints.

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Major components: 1. Base curve 2. Localization 3. Constrained shape manipulation 4. Parametrization adjustment

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Definition:a B-spline curve of degree p

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Constrained shaping:

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Parametrization: 1.Obtain a polygonal aproximation of the base curve 2.Project all points onto the polygon and find the closest vertex or polygon leg 3.Obtain an approximation parameter from the parameters of the closest leg or vertex

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Geometric decomposition: 1.Decompose the NURBS curve into piecewise Bezier segments 2.Bezier curve decomposition

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Location: Greville abscissae: degree p degree p-1

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Procedure: 1.Go through the brackets (η i,η i+1 ) i=1,…,n-1 to see how many parameters fall within the brackets 2.If there are two or more parameters,insert a knot and recomputer the brackets 3.Repeat the process until each bracket contains no more than one parameter

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Let w ∈ (η i,η i+1 ) be the new abscissa

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The algorithm: 1.Computer the Grevile brackets 2.Check end conditions and insert knots if necessary 3.While there are brakets with more than one parameter 3.1 Find first offending span 3.2 Average parameters lying inside the span.The average is to become the new Greville abscissa 3.3 Get the knot to be inserted 3.4 If the knot falls on an existing knot,select another parameter representative 3.5 Insert the knot and recompute Greville brackets

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Base curve:two methods (1)1.Obtain a local cubic C 1 interpolation to the points. 2.Sample the interpolant 3.Approximate the smple points by least-squares.

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(2)using Hermite curves.

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The algorithm: 1.Computer an initial parameter 2.For each point 2.1 Computer 2.2 Get 3.Computer and 4.Repeat 4.1 Recomputer parameters by projecting the points onto the Hermite curve 4.2 Recomputer λ Untill no significant change in λis obtained

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The updated parameters

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The whole algorithm: 1.If there is no internal points,generate a Hermite curve defined by 2.Get a base curve: 2.1 If the data is simple: 2.1.1 Computer a Hetmite curve to 2.1.2 Scale tangents to approximate internal points 2.1.3 Iteratively improve parameters and the base curve

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2.2 Otherwise: 2.2.1 Interpolate data with a local method 2.2.2 Sample interpolating curve 2.2.3 Approximate sampling points to eliminate multiple knots 3. Perform constrained shape manipulation to obtain interpolation. 4. While not all points have been interpolated: 4.1 Get/update parameter by projecting points onto the current curve 4.2 If necessary,eliminate point clustering and add additional knots 4.3 Set up vectors Vi by pulling curve points towards Qi by the computered increment

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4.4 Perform shape operation 4.5 Mark data points that have been reached,and exclude them from further operations 5.Output shaped base curve and postprocess

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The End !

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