CSC 305 More Modelling

Transcription

1 CSC 305 More Modelling Faramarz Samavati & Brian Wyvill The University of Victoria Graphics Group

2 Bernstein Polynomials any degree University of Victoria Island Graphics Lab. CSC page 2

3 Implementation (brute force) Input P[i], d //P[i]: control point, d : degree, u : parameter for u=0 to 1 step 0.01 for i=0 to d b = Berenstein( i, d, u ) q = q + b * P[i] end plot(q); end v This program isn t efficient Redundant computations High number of control points High degree polynomial High degree polynomial Unstable computation University of Victoria Island Graphics Lab. CSC page 3

4 Convex Hull What is the Convex Hull? Given a set of points (in ) The smallest convex polyhedral that includes all the points given University of Victoria Island Graphics Lab. CSC page 4

5 de Casteljau Algorithm Avoid direct evaluating of polynomials Geometric interpretation Consider a planer cubic at u 1-u v column by column updating rule University of Victoria Island Graphics Lab. CSC page 5

6 Pseudo Code Input P[j], d, u //P[j] : control point, d : degree, u : parameter //output will be Q(u) for i = 1 to d for j = 0 to d-i P[j] =(1-u)*P[j] + u*p[j+1] end end Output??? University of Victoria Island Graphics Lab. CSC page 6

7 Some Observations on Cubic Case v After one step of decastlejeau algorithm for v Obtaining v 4 green points v 4 blue points v Where v Small left Bezier curve: v Small right Bezier curve: Original Bezier curve v Divided and conquer method University of Victoria Island Graphics Lab. CSC page 7

8 Matrix Relation v Unit summation of any row. v Non negative elements. v Banded structure. University of Victoria Island Graphics Lab. CSC page 8

9 Another View We start with 4 points We will obtain 8 new points 7 new points v New points: v New points are closer to the curve v We can repeat for any 4 points in the new points sets v And initial polyline is replaced by a finer polyline v Subdivision method University of Victoria Island Graphics Lab. CSC page 9

10 Pseudo Code for Subdivision Method Polyline subbezier (polyline P) //cubic Bezier, d=3, //uniform subdivision, u=1/2 //Input is polyline P // n is number of P points (coarse polyline) // O is output polyline (fine polyline) j=0; for ( i=0 ; i<n ; i=+3) O[j]=P[i]; O[j+1]=(P[i] + P[i+1])/2 O[j+5]=(P[i+2] + P[i+3])/2 t = (P[i+1] + P[i+2])/2 O[j+2]=(t + O[j+1])/2 O[j+4]=(t + O[j+5])/2 O[j+3]=(O[j+2] + O[j+4])/2 j=j+6; endfor Return O University of Victoria Island Graphics Lab. CSC page 10

11 Weakness of Bezier curves l High degree polynomial l How can we obtain better control of the curve without adding control points? v Composite Bezier curve: join Bezier curve segments. v Apply some constraints at the connection points. v What are these constraints? o Positional Continuity = zero degree continuity o Same direction tangents = first degree continuity Any other weakness? University of Victoria Island Graphics Lab. CSC page 11

12 l Easy and local algorithm Piecewise quadratic polynomials: Chaikin Algorithm l Two magic! numbers & l Corner cutting University of Victoria Island Graphics Lab. CSC page 12

13 Convergence v smooth limit curve (no corner), quadratic B-spline! University of Victoria Island Graphics Lab. CSC page 13

14 Subdivision Matrix l Coarse points l F= PC l Iterative process subdivision fine points l P has a regular banded structure University of Victoria Island Graphics Lab. CSC page 14

15 Faber Subdivision v W 4 University of Victoria Island Graphics Lab. CSC page 15

16 Subdivision Matrix v What is the limit curve? v Obvious interpretation in the resolution enhancement of images v Periodic case University of Victoria Island Graphics Lab. CSC page 16

17 Image Example Lena (Lenna) from the November 1972 issue of Playboy Magazine. each column each row University of Victoria Island Graphics Lab. CSC page 17

18 Image application v Increasing the resolution of image v Traditional method(faber) v ½ left + ½ right v Chaikin rule Repeating algorithm for every row University of Victoria Island Graphics Lab. CSC page 18

19 Comparing the results

20 Lena (Lenna) a famous early digitized image. Photo stolen from the November 1972 issue of Playboy Magazine. University of Victoria Island Graphics Lab. CSC page 20