Graphics I Faramarz Samavati UNIVERSITY OF CALGARY Modeling Curves and Surfaces part I Graphics II Faramarz Samavati UNIVERSITY OF CALGARY Making objects 3d scanner /3d digitizer (copying) Hard to edit a very high quality mesh There is no original object!
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Graphics IFaramarz Samavati
UNIVERSITY OF
CALGARY
ModelingCurves and Surfaces
part I
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Making objects
3d scanner /3d digitizer (copying)
Hard to edit a very high quality mesh
There is no original object!
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Mathematical Models
Line and arcs (from ancient world geometrician)
A short review on:
Graph of functions
implicit representation
parametric representation
parametric polynomial curves
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Mathematical Models
S
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY
Point in or
Bezier ModelP.Bezier,
1960, Unisurf in Renault automobile designing.
Mathematical definition
Old CAD and graphics software
Postscript
METAFONT
Modeling
Q(u)
curve
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Bezier Curve Definition
u is the parameter
Berenstein polynomial of order d
:Control points ( inputs)
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Bezier Curve Details
Bezier curve of order 1
Bezier curve of order 3
Bezier curve of order 2
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Cubic Bezier Curve
properties
1. is a polynomial curve with the domain [0,1]
2.
3.
4.
5.
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY
Non-negative polynomial of order d
Properties of
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Convex Hull
What is the Convex Hull?
Given a set of points (in )
The smallest convex polyhedral that includes all the points
given
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Implementation (brute force)
Input P[i], d //P[i]: control point, d : degree , u : parameter
for u=0 to 1 step 0.01for i=0 to d
b = Berenstein( i , d , u )q = q + b * P[i]
endplot(q);
end
This program isn’t efficientRedundant computations
High degree polynomialHigh number of control pointsHigh degree polynomial Unstable computation
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY de Casteljau AlgorithmAvoid direct evaluating of polynomials
Geometric interpretation
Consider a planer cubic at
column by column updating rule
u1-u
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Pseudo CodeInput P[j] , d , u//P[j] : control point, d : degree , u : parameter //output will be Q(u)
for i = 1 to dfor j = 0 to d-i
P[j] =(1-u)*P[j] + u*P[j+1]end
endOutput ???
Why is this algorithm correct?
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARYSome Observation on Cubic Case
After one step of deCasteljue algorithm for
Obtaining
4 green points
4 blue points
Where
Small left Bezier curve:
Small right Bezier curve: Original Bezier curve
Divided and conquer method
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Matrix Relation
Unit summation of any row.
Non negative elements.
Banded structure.
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Another View
We start with 4 points
We will obtain 8 new points 7 new points
New points:
New points are closer to the curve
We can repeat for any 4 points in the new points sets
And initial polyline is replaced by a finer polyline
Subdivision method
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARYPseudo Code for SubdivisionMethodPolyline 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])/2O[j+5]=(P[i+2] + P[i+3])/2t = (P[i+1] + P[i+2])/2
O[j+2]=(t + O[j+1])/2O[j+4]=(t + O[j+5])/2O[j+3]=(O[j+2] + O[j+4])/2j=j+6;
endforReturn O
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Weakness of Bezier curves
High degree polynomial
How can we increase the controlling on the curve without adding control points?
Composite Bezier curve: join Bezier curve segments.Apply some constraints at the connection points.What are these constrains?
Any other weakness?
o Positional Continuity = zero degree continuityo Same direction tangents = first degree continuity
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY
Easy and local algorithm
Two magic! numbers and
Corner cutting
Piecewise quadratic polynomials:Chaikin Algorithm
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Convergencesmooth limit curve (no corner), quadratic B-spline!
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Subdivision Matrix
Coarse points fine points
F= PC
Iterative process
P has a regular banded structure
subdivision
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Faber Subdivision
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Subdivision Matrix
What is the limit curve?Obvious interpretation in the resolution enhancement of images Periodic case
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Image Example
each row
each column
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Image application
Increasing the resolution of image
Traditional method(Faber)
½ left + ½ right
Chaikin ruleRepeating algorithm for every row
Graphics IIFaramarz Samavati
UNIVERSITY OF
CALGARY Comparing the results
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