Presentation is loading. Please wait.

Presentation is loading. Please wait.

Engineering Research Center for Computer Integrated Surgical Systems and Technology 1 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor.

Published byMae Hawkins Modified over 9 years ago

Similar presentations

Presentation on theme: "Engineering Research Center for Computer Integrated Surgical Systems and Technology 1 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor."— Presentation transcript:

1 Engineering Research Center for Computer Integrated Surgical Systems and Technology 1 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Interpolation and Deformations A short cookbook

2 Engineering Research Center for Computer Integrated Surgical Systems and Technology 2 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Linear Interpolation 10

3 Engineering Research Center for Computer Integrated Surgical Systems and Technology 3 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Linear Interpolation

4 Engineering Research Center for Computer Integrated Surgical Systems and Technology 4 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Linear Interpolation 1

5 Engineering Research Center for Computer Integrated Surgical Systems and Technology 5 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bilinear Interpolation

6 Engineering Research Center for Computer Integrated Surgical Systems and Technology 6 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bilinear Interpolation

7 Engineering Research Center for Computer Integrated Surgical Systems and Technology 7 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bilinear Interpolation

8 Engineering Research Center for Computer Integrated Surgical Systems and Technology 8 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor N-linear Interpolation

9 Engineering Research Center for Computer Integrated Surgical Systems and Technology 9 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Barycentric Interpolation

10 Engineering Research Center for Computer Integrated Surgical Systems and Technology 10 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Barycentric Interpolation

11 Engineering Research Center for Computer Integrated Surgical Systems and Technology 11 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Barycentric Interpolation

12 Engineering Research Center for Computer Integrated Surgical Systems and Technology 12 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Barycentric Interpolation

13 Engineering Research Center for Computer Integrated Surgical Systems and Technology 13 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Interpolation of functions

14 Engineering Research Center for Computer Integrated Surgical Systems and Technology 14 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Fitting of interpolation curves The discussion below follows (in part) G. Farin, Curves and surfaces for computer-aided geometric design, a practical guide, Academic Press, Boston, 1990, chapter 10 and pp 281-284.

15 Engineering Research Center for Computer Integrated Surgical Systems and Technology 15 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 1-D Interpolation

16 Engineering Research Center for Computer Integrated Surgical Systems and Technology 16 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 1-D Interpolation

17 Engineering Research Center for Computer Integrated Surgical Systems and Technology 17 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bezier and Bernstein Polynomials Excellent numerical stability for 0<v<1 There exist good ways to convert to more conventional power basis

18 Engineering Research Center for Computer Integrated Surgical Systems and Technology 18 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Barycentric Bezier Polynomials Excellent numerical stability for c<0<1 There exist good ways to convert to more conventional power basis

19 Engineering Research Center for Computer Integrated Surgical Systems and Technology 19 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bezier Curves

20 Engineering Research Center for Computer Integrated Surgical Systems and Technology 20 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Bezier Curves: de Casteljau Algorithm

21 Engineering Research Center for Computer Integrated Surgical Systems and Technology 21 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Iterative Form of deCasteljau Algorithm

22 Engineering Research Center for Computer Integrated Surgical Systems and Technology 22 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Advantages of Bezier Curves Numerically very robust Many nice mathematical properties Smooth “Global” (may be viewed as a disadvantage)

23 Engineering Research Center for Computer Integrated Surgical Systems and Technology 23 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor B-splines

24 Engineering Research Center for Computer Integrated Surgical Systems and Technology 24 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor deBoor Algorithm

25 Engineering Research Center for Computer Integrated Surgical Systems and Technology 25 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor B-spline basis functions

26 Engineering Research Center for Computer Integrated Surgical Systems and Technology 26 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor Some advantages of B-splines Efficient Numerically stable Smooth Local

27 Engineering Research Center for Computer Integrated Surgical Systems and Technology 27 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 2D Interpolation

28 Engineering Research Center for Computer Integrated Surgical Systems and Technology 28 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 2D Interpolation: Finding the best fit

29 Engineering Research Center for Computer Integrated Surgical Systems and Technology 29 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 2D Interpolation: Finding the best fit

30 Engineering Research Center for Computer Integrated Surgical Systems and Technology 30 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor 2D Interpolation: Finding the best fit There are a number of caveats to the “grid” method on the previous slide. (E.g., you need enough data for each of the least squares problems). But where applicable the method can save computation time since it replaces a number of m and n variable least squares problems for one big m x n problem Note that there is a similar trick that you can play by grouping all the common u i elements together. Note that the y’s and the c’s do not have to be scalar numbers. They can be Vectors, Matrices, or other objects that have appropriate algebraic properties

Download ppt "Engineering Research Center for Computer Integrated Surgical Systems and Technology 1 600.445 Fall 2000; Updated: 12 September 2015 Copyright © R. H. Taylor."

Similar presentations

Interpolating curves.

Interpolating curves.
Computer Graphics (Spring 2008) COMS 4160, Lecture 6: Curves 1

Computer Graphics (Spring 2008) COMS 4160, Lecture 6: Curves 1
Parametric Curves Ref: 1, 2.

Parametric Curves Ref: 1, 2.
Interpolation A method of constructing a function that crosses through a discrete set of known data points. .

Interpolation A method of constructing a function that crosses through a discrete set of known data points. .
Cubic Curves CSE167: Computer Graphics Instructor: Steve Rotenberg UCSD, Fall 2006.

Cubic Curves CSE167: Computer Graphics Instructor: Steve Rotenberg UCSD, Fall 2006.
09/25/02 Dinesh Manocha, COMP258 Triangular Bezier Patches Natural generalization to Bezier curves Triangles are a simplex: Any polygon can be decomposed.

09/25/02 Dinesh Manocha, COMP258 Triangular Bezier Patches Natural generalization to Bezier curves Triangles are a simplex: Any polygon can be decomposed.
Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces.

Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces.
Cubic Curves CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2005.

Cubic Curves CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2005.
KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., Spline Methods in CAGD Lee Byung-Gook Dongseo Univ.

KMMCS, April. 2003, Lee Byung-Gook, Dongseo Univ., Spline Methods in CAGD Lee Byung-Gook Dongseo Univ.
If this (...) leaves you a bit wondering what multivariate splines might be, I am pleased. For I don’t know myself. Carl de Boor.

If this (...) leaves you a bit wondering what multivariate splines might be, I am pleased. For I don’t know myself. Carl de Boor.
Bezier Curves and Splines David Eno MAT 499 Fall ‘06.

Bezier Curves and Splines David Eno MAT 499 Fall ‘06.
MIT EECS 6.837, Durand and Cutler Curves & Surfaces.

MIT EECS 6.837, Durand and Cutler Curves & Surfaces.
Lecture 29 of 42 Bezier Curves and Splines Wednesday, 02 April 2008

Lecture 29 of 42 Bezier Curves and Splines Wednesday, 02 April 2008
MIT EECS 6.837, Durand and Cutler Curves & Surfaces.

MIT EECS 6.837, Durand and Cutler Curves & Surfaces.
CS175 2003 1 CS 175 – Week 9 B-Splines Blossoming, Bézier Splines.

CS CS 175 – Week 9 B-Splines Blossoming, Bézier Splines.
A Bezier Based Approach to Unstructured Moving Meshes ALADDIN and Sangria Gary Miller David Cardoze Todd Phillips Noel Walkington Mark Olah Miklos Bergou.

A Bezier Based Approach to Unstructured Moving Meshes ALADDIN and Sangria Gary Miller David Cardoze Todd Phillips Noel Walkington Mark Olah Miklos Bergou.
CS175 2003 1 CS 175 – Week 9 B-Splines Definition, Algorithms.

CS CS 175 – Week 9 B-Splines Definition, Algorithms.
09/04/02 Dinesh Manocha, COMP258 Bezier Curves Interpolating curve Polynomial or rational parametrization using Bernstein basis functions Use of control.

09/04/02 Dinesh Manocha, COMP258 Bezier Curves Interpolating curve Polynomial or rational parametrization using Bernstein basis functions Use of control.
Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 12: Spline Curves (review) Ravi Ramamoorthi Most material.

Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 12: Spline Curves (review) Ravi Ramamoorthi Most material.
Modeling of curves Needs a ways of representing curves: Reproducible - the representation should give the same curve every time; Computationally Quick;

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

Similar presentations

Ads by Google