Evolution-based least-squares fitting using Pythagorean hodograph spline curves Speaker: Ying.Liu November

Institute of Applied Geometry, Jphannes Kepler University,Linz, Austira u.at Martin Aigner Bert Juttler

Author: Martin Aigner: –Dr. Mag., research assistant – martin. jku.at Zbynek Sir: –Dr.; research assistant at FWF-Projekt P17387-N12 FWF –Alumni

Author: Bert Juttler Selected scientific activities: –Since 2003:associated editor of CAGD –Organizer of various Mini symposia –Member of program committees of numerous conferences of numerous conferences Research interests: –CAGD, Applied Geometry, Kinematics, Robotics, Differential Geometry

Introduction Using PH spline curves to evoluted fitting a given set of data points or a curve For example:

Steps: Introduce a general framework for abstract curve fitting Apply this framework to PH curves Discuss the relationship between this method and Gauss-Newton iteration

An abstract framework for curve fitting via evolution Parameterized family of curves: （ s, u ） -> –u is the curve parameter –s is the vector of shape parameters Let s depend smoothly on an evolution parameter t, s( t)=( ) Approximately compute the limit

An abstract framework for curve fitting via evolution Each point travels with the velocity: Normal velocity of the inner points:

An abstract framework for curve fitting via evolution Assume a set of data points is given. Let and Expected to toward their associated data points if then

An abstract framework for curve fitting via evolution

Time derivatives of the shape parameters satisfied the following equation in least-squares sense Necessary condition for a minimum

An abstract framework for curve fitting via evolution Definition: –A given curve: –a set of parameters U is said to be regular: –A set parameters: that and –Unit normal vectors That the matrix has a maximal rank

An abstract framework for curve fitting via evolution Lemma: in a regular case and if all closet points are neither singular nor boundary points, then any solution of the usual least-squares fitting is a stationary point of the differential equation derived from the evolution process is a stationary point of the differential equation derived from the evolution process

Evolution of PH splines Ordinary PH curves c (u)=[x ( u),y (u)] satisfied the following conditions: Regular PH curves: let w=1. The difference : gcd (x ’ ( u ),y ’ (u)) is a square of a polynomial called preimage curve called preimage curve

Evolution of PH splines Proposition: if a regular PH curve c (u) and then: –Smooth field of unit tangent vectors for all u –Parametric speed and arc-length are polynomial functions –Its offsets are rational curves

Evolution of PH splines Let an open integral B-spline curve, and Let

Evolution of PH splines In the evolution we fix the knot vector, so the shape parameters are the velocity The unit normals

Evolution of PH splines The length of PH spline: The regularization term: Which forces the length to converge to some constant value

Examples of PH splines evolution Simple example: –fitting two circular arcs with radius 1. –Two cubic PH segments depending on 8 shape parameters –Initial position: straight line

Examples of PH splines evolution

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Example of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial: two straight segments For the global shape =8, Gradually raised length to 14 Fix end points Insert knots

Examples of PH splines Initial value by Hermite interpolation –Split data points at estimated inflections

Speed of convergence Lemma: the Euler update of the shape parameters for the evolution with step h is equivalent to a Gauss- Newton step with the same h of the problem Provided that

Speed of convergence

Quadratic convergence of the method

Concluding remarks Least-squares fitting by PH spline cuves is not necessarily more complicated than others Future work is devoted to using the approximation procedure in order to obtain more compact representation of NC tool paths

Q&A

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