Least Squares Curves, Rational Representations, Splines and Continuity

Slides:

Advertisements

Similar presentations

1 Dr. Scott Schaefer Least Squares Curves, Rational Representations, Splines and Continuity.

Advertisements

Numeriska beräkningar i Naturvetenskap och Teknik Today’s topic: Approximations Least square method Interpolations Fit of polynomials Splines.

P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems.

Cubic Curves CSE167: Computer Graphics Instructor: Steve Rotenberg UCSD, Fall 2006.

CS 445/645 Fall 2001 Hermite and Bézier Splines. Specifying Curves Control Points –A set of points that influence the curve’s shape Knots –Control points.

Selected from presentations by Jim Ramsay, McGill University, Hongliang Fei, and Brian Quanz Basis Basics.

1 Chapter 4 Interpolation and Approximation Lagrange Interpolation The basic interpolation problem can be posed in one of two ways: The basic interpolation.

08/30/00 Dinesh Manocha, COMP258 Hermite Curves A mathematical representation as a link between the algebraic & geometric form Defined by specifying the.

1 Curves and Surfaces. 2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized.

Slide 127 October 1999CS Computer Graphics (Top Changwatchai) Review of Spline Concepts Sections 10-6 to in Hearn & Baker Splines can be 2D.

CS CS 175 – Week 9 B-Splines Blossoming, Bézier Splines.

Rational Bezier Curves

09/04/02 Dinesh Manocha, COMP258 Bezier Curves Interpolating curve Polynomial or rational parametrization using Bernstein basis functions Use of control.

1 Dr. Scott Schaefer Curves and Interpolation. 2/61 Smooth Curves How do we create smooth curves?

Modeling of curves Needs a ways of representing curves: Reproducible - the representation should give the same curve every time; Computationally Quick;

1 Dr. Scott Schaefer The Bernstein Basis and Bezier Curves.

Degree Reduction for NURBS Symbolic Computation on Curves Xianming Chen Richard F. Riesenfeld Elaine Cohen School of Computing, University of Utah.

09/09/02 Dinesh Manocha, COMP258 Properties of Bezier Curves Invariance under affine parameter transformation P i B i,n (u) = P i B i,n ((u –a)/(b-a))

Curves Locus of a point moving with one degree of freedom

1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation.

ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 24 Regression Analysis-Chapter 17.

Lecture 10: Support Vector Machines

COEN Computer Graphics I

Section 5.5 – The Real Zeros of a Rational Function

Splines By: Marina Uchenik.

11/19/02 (c) 2002, University of Wisconsin, CS 559 Last Time Many, many modeling techniques –Polygon meshes –Parametric instancing –Hierarchical modeling.

CS 376 Introduction to Computer Graphics 04 / 23 / 2007 Instructor: Michael Eckmann.

Regression analysis Control of built engineering objects, comparing to the plan Surveying observations – position of points Linear regression Regression.

1 Dr. Scott Schaefer Smooth Curves. 2/109 Smooth Curves Interpolation  Interpolation through Linear Algebra  Lagrange interpolation Bezier curves B-spline.

Notes Over 6.7 Finding the Number of Solutions or Zeros

1 Dr. Scott Schaefer Blossoming and B-splines. 2/105 Blossoms/Polar Forms A blossom b(t 1,t 2,…,t n ) of a polynomial p(t) is a multivariate function.

Vector Computer Graphic. Vector entities Line Circle, Ellipse, arc,… Curves: Spline, Bezier’s curve, … … Areas Solids Models.

1 Dr. Scott Schaefer Coons Patches and Gregory Patches.

Curve-Fitting Regression

Ch. 3: Geometric Camera Calibration

Lecture 16 - Approximation Methods CVEN 302 July 15, 2002.

CS 376 Introduction to Computer Graphics 04 / 25 / 2007 Instructor: Michael Eckmann.

Representation of Curves & Surfaces Prof. Lizhuang Ma Shanghai Jiao Tong University.

Cubic Spline Interpolation. Cubic Splines attempt to solve the problem of the smoothness of a graph as well as reduce error. Polynomial interpolation.

Parametric Curves CS 318 Interactive Computer Graphics John C. Hart.

CS 325 Computer Graphics 04 / 30 / 2010 Instructor: Michael Eckmann.

Fourier Approximation Related Matters Concerning Fourier Series.

Computing & Information Sciences Kansas State University Lecture 30 of 42CIS 636/736: (Introduction to) Computer Graphics Lecture 30 of 42 Wednesday, 09.

Rendering Bezier Curves (1) Evaluate the curve at a fixed set of parameter values and join the points with straight lines Advantage: Very simple Disadvantages:

1 Chapter 4 Interpolation and Approximation Lagrange Interpolation The basic interpolation problem can be posed in one of two ways: The basic interpolation.

Math 495B Polynomial Interpolation Special case of a step function. Frederic Gibou.

CS552: Computer Graphics Lecture 19: Bezier Curves.

Object Modeling: Curves and Surfaces CEng 477 Introduction to Computer Graphics.

Introduction to Parametric Curve and Surface Modeling.

APPROXIMATION SPLINE Does not necessarily pass through all control points General representation is P(u) = Σ Fi(n,u) Pi where- Fi is the blending function.

Curve & Surface.

Bezier Triangles and Multi-Sided Patches

Approximation of circular arcs by quartic splines

Notes Over 3.4 The Rational Zero Test

Creating Polynomials Given the Zeros.

Chapter 12 Curve Fitting : Fitting a Straight Line Gab-Byung Chae

© University of Wisconsin, CS559 Spring 2004

Curves and Surfaces.

The Bernstein Basis and Bezier Curves

Spline Interpolation Class XVII.

Linear regression Fitting a straight line to observations.

11.1 – Polynomial Approximations of Functions

Parametric Line equations

Coons Patches and Gregory Patches

Basis Expansions and Generalized Additive Models (1)

2.5 The Real Zeros of a Polynomial Function

ANSWER THE FOLLOWING BRIEFLY BUT COMPREHENSIVELY.

Introduction to Parametric Curve and Surface Modeling

6.7 Using the Fundamental Theorem of Algebra

Presentation transcript:

Least Squares Curves, Rational Representations, Splines and Continuity Dr. Scott Schaefer

Degree Reduction Given a set of coefficients for a Bezier curve of degree n+1, find the best set of coefficients of a Bezier curve of degree n that approximate that curve

Degree Reduction

Degree Reduction

Degree Reduction

Degree Reduction

Degree Reduction

Degree Reduction Problem: end-points are not interpolated

Least Squares Optimization

Least Squares Optimization

Least Squares Optimization

Least Squares Optimization

Least Squares Optimization

Least Squares Optimization

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

The PseudoInverse What happens when isn’t invertible?

Constrained Least Squares Optimization

Constrained Least Squares Optimization Solution Constraint Space Error Function F(x)

Constrained Least Squares Optimization

Constrained Least Squares Optimization

Constrained Least Squares Optimization

Constrained Least Squares Optimization

Constrained Least Squares Optimization

Least Squares Curves

Least Squares Curves

Least Squares Curves

Least Squares Curves

Degree Reduction Problem: end-points are not interpolated

Degree Reduction

Degree Reduction

Rational Curves Curves defined in a higher dimensional space that are “projected” down

Rational Curves Curves defined in a higher dimensional space that are “projected” down

Rational Curves Curves defined in a higher dimensional space that are “projected” down

Rational Curves Curves defined in a higher dimensional space that are “projected” down

Why Rational Curves? Conics

Why Rational Curves? Conics

Why Rational Curves? Conics

Why Rational Curves? Conics

Derivatives of Rational Curves

Derivatives of Rational Curves

Derivatives of Rational Curves

Derivatives of Rational Curves

Splines and Continuity Ck continuity:

Splines and Continuity Ck continuity:

Splines and Continuity Ck continuity:

Splines and Continuity Ck continuity:

Splines and Continuity Ck continuity:

Splines and Continuity Assume two Bezier curves with control points p0,…,pn and q0,…,qm

Splines and Continuity Assume two Bezier curves with control points p0,…,pn and q0,…,qm C0: pn=q0

Splines and Continuity Assume two Bezier curves with control points p0,…,pn and q0,…,qm C0: pn=q0 C1: n(pn-pn-1)=m(q1-q0)

Splines and Continuity Assume two Bezier curves with control points p0,…,pn and q0,…,qm C0: pn=q0 C1: n(pn-pn-1)=m(q1-q0) C2: n(n-1)(pn-2pn-1+pn-2)=m(m-1)(q0-2q1+q2) …

Splines and Continuity Geometric Continuity A curve is Gk if there exists a reparametrization such that the curve is Ck

Splines and Continuity Geometric Continuity A curve is Gk if there exists a reparametrization such that the curve is Ck

Splines and Continuity Geometric Continuity A curve is Gk if there exists a reparametrization such that the curve is Ck

Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve

Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain Ck continuity!!!

Problems with Bezier Curves More control points means higher degree Moving one control point affects the entire curve Solution: Use lots of Bezier curves and maintain Ck continuity!!! Difficult to keep track of all the constraints. 

B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control

B-spline Curves Not a single polynomial, but lots of polynomials that meet together smoothly Local control