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CALGARY

ModelingCurves and Surfaces

part I

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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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CALGARY Mathematical Models

Line and arcs (from ancient world geometrician)

A short review on:

Graph of functions

implicit representation

parametric representation

parametric polynomial curves

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CALGARY Mathematical Models

S

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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

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CALGARY Bezier Curve Definition

u is the parameter

Berenstein polynomial of order d

:Control points ( inputs)

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CALGARY Bezier Curve Details

Bezier curve of order 1

Bezier curve of order 3

Bezier curve of order 2

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CALGARY Cubic Bezier Curve

properties

1. is a polynomial curve with the domain [0,1]

2.

3.

4.

5.

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CALGARY

Non-negative polynomial of order d

Properties of

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CALGARY Convex Hull

What is the Convex Hull?

Given a set of points (in )

The smallest convex polyhedral that includes all the points

given

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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

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CALGARY de Casteljau AlgorithmAvoid direct evaluating of polynomials

Geometric interpretation

Consider a planer cubic at

column by column updating rule

u1-u

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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?

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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

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CALGARY Matrix Relation

Unit summation of any row.

Non negative elements.

Banded structure.

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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

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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

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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

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CALGARY

Easy and local algorithm

Two magic! numbers and

Corner cutting

Piecewise quadratic polynomials:Chaikin Algorithm

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CALGARY Convergencesmooth limit curve (no corner), quadratic B-spline!

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CALGARY Subdivision Matrix

Coarse points fine points

F= PC

Iterative process

P has a regular banded structure

subdivision

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CALGARY Faber Subdivision

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CALGARY Subdivision Matrix

What is the limit curve?Obvious interpretation in the resolution enhancement of images Periodic case

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CALGARY Image Example

each row

each column

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CALGARY Image application

Increasing the resolution of image

Traditional method(Faber)

½ left + ½ right

Chaikin ruleRepeating algorithm for every row

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CALGARY Comparing the results