Manifolds or why a cow is a sphere

1

Manifoldsor why a cow is a sphere

Cindy Grimm

Media and Machines LabDepartment of Computer Science and Engineering

Washington University in St. Louis

2 Berkeley, 5/5/2005

Overview

• What is a manifold?• Constructive definition

• Building a circle manifold• Simple example showing concepts

• Using a manifold• Surface modeling

• Applications• Environment mapping• Facial animation• Image-based rendering


What is a manifold?History:•Mathematicians 1880’s

• How to analyze complex shapes using simple maps

•Cartographers• World atlas

Informal definition•Map world to pages

• Each page rectangular• Every part of world on at

least one page• Pages overlap

• May not agree exactly• Agree enough to

navigate

Overlap

The world


Traditional definition

A manifold M is an object that is locally Euclidean•For every p in M

• Neighborhood U around p• U maps to Rn • No folding, tearing

Technical note:•M may be a surface embedded in Rm

• n is dimension of surface• m >n

•M may be abstract (topological)

1D2D

Not manifold examples


A mesh is manifold if…

We can construct local maps around points:•Point is in face•Point is on edge

• There are exactly two faces adjacent to each edge

•Point is on vertex• The faces around a vertex v can be flattened

into the plane without folding or tearing• Vertices wi adjacent to v can be ordered

w0,…,wn-1 such that the triangles wi,v,w(i+1)mod n all exist

•Can do on abstract or embedded mesh

2D

P3D

P3D

2D

P

2D

3D


Traditional definition

Given: Manifold MConstruct: Atlas A•Chart

• Region Uc in M (open disk)• Region c in Rn (open disk)• Function c taking Uc to c

•Atlas is collection of charts• Every point in M in at least

one chart• Overlap regions• Transition functions:

ij = j o i-1 smooth

M

01

21

10

12

12

02

0


A circle manifold

How to represent a circle?•Problem:

• All we have are Euclidean operations• In R2, must project onto circle

•Solution:• Represent circle as repeating interval

[0,2)• Point +2k is same for all k integer

(1,0)

)sin,(cos)(1 S

122 yxThese are embeddings

This is not

0 2


/** A point on the circle, represented as an angle value from [0 to 2 pi). If the angle is * bigger (or smaller) it is shifted until it lies in the 0 to 2 pi range */class CirclePoint {protected: double m_dTheta;

public: double GetTheta() const { return m_dTheta; }

CirclePoint( const double in_dTheta ) : m_dTheta( in_dTheta ) {

while ( m_dTheta >= M_PI * 2.0 ) // Bring theta into the 0,2pi range m_dTheta -= 2.0 * M_PI; while ( m_dTheta < 0 ) m_dTheta += 2.0 * M_PI;

}

~CirclePoint() {}};

/// Definition for S^1, an embedding of the circleclass Circle {protected:public: Point2D operator()( const CirclePoint & in_circPt ) const { return Point2D( cos( in_circPt.GetTheta() ), sin( in_circPt.GetTheta() ) ); }

Circle() { } ~Circle() { }};

(1,0)

)sin,(cos)(1 S


Chart on a circleChart specification:•Left and right ends of Uc

• Counter-clockwise order to determine overlap region

•Co-domain c = (-1/2,1/2)•Simplest form for c

• Translate• Center in [0,2)

• Scale-1/2 1/2

LRtscc ))((

)/(1cc st

Uc-1/2 1/2

( )c ( )

L

R

10 Berkeley, 5/5/2005

Chart on a circleChart specification:•Alpha function

• Sort point first• Euclidean distance

is topological distance

•Alpha inverse• Point in [0,2)

ChartPoint Chart::Alpha( const CirclePoint &in_circpt ) const{ const double dTheta = in_circpt.GetTheta(); double dThetaShift = dTheta;

// Find the value for theta (+- 2 PI) that is closest to my chart center if ( fabs( dTheta - m_dThetaCenter ) <= M_PI ) { dThetaShift = dTheta; } else if ( fabs( (dTheta + 2.0 * M_PI) - m_dThetaCenter ) <= M_PI ) { dThetaShift = dTheta + 2.0 * M_PI; } else if ( fabs( (dTheta - 2.0 * M_PI) - m_dThetaCenter ) <= M_PI ) { dThetaShift = dTheta - 2.0 * M_PI; } else { assert( false ); }

const double dT = (dThetaShift - m_dThetaCenter) * m_dScale;

return ChartPoint( this, dT );}

CirclePoint Chart::AlphaInv( const double in_dT ) const{ const double dTheta = in_dT / m_dScale + m_dThetaCenter;

// Converts to 0,2pi range return CirclePoint( dTheta );}

c

( )

11 Berkeley, 5/5/2005

Making a chartconst Chart *Atlas::AddChart( const CirclePoint &in_circptLeft, const CirclePoint &in_circptRight ){ double dThetaMid = 0.0; double dScale = 1.0;

// Does not cross 0, 2pi boundary if ( in_circptLeft.GetTheta() < in_circptRight.GetTheta() ) { dThetaMid = 0.5 * ( in_circptLeft.GetTheta() + in_circptRight.GetTheta() ); dScale = 0.5 / (in_circptRight.GetTheta() - dThetaMid); } else { // Add 2pi to right end point dThetaMid = 0.5 * ( in_circptLeft.GetTheta() + in_circptRight.GetTheta() + 2.0*M_PI ); dScale = 0.5 / (in_circptRight.GetTheta() + 2.0 * M_PI - dThetaMid); }

if ( dThetaMid >= 2.0 * M_PI ) dThetaMid -= 2.0 * M_PI;

if ( fabs( dScale ) < 1e-16 ) return false;

const Chart *opChart = new Chart( m_aopCharts.size(), dThetaMid, dScale ); m_aopCharts.push_back( opChart );

return opChart;}

LR

L R

12 Berkeley, 5/5/2005

An example circle atlas

Four charts•Each covering ½ of circle•Only 4 points not covered by two charts•Check transition functions

• All translates• Overlaps ½ of chart

Chart 0

),0[,/)2/()(0

)2/()(10 tt

-1/2 1/2

Chart 1

-1/2 1/2

)()(11 tt

)2/3,2/[,/)()(1

Chart 2

-1/2 1/2

)2,[,/)2/3()(2

)2/3()(12 tt

Chart 3

-1/2 1/2

)2/,2/[,/)0()(3

)0()(13 tt

atlas.AddChart( 0.0, M_PI );atlas.AddChart( M_PI / 2.0, 3.0 * M_PI / 2.0 );atlas.AddChart( M_PI, 2.0 * M_PI );atlas.AddChart( 3.0 * M_PI / 2.0, 5.0 * M_PI / 2.0 );

Overlap regions

5.0)(1,0 ssChart 0 Chart 1 Chart 2 Chart 3 Chart 0

5.0)(0,1 tt

5.0)(2,1 ss

5.0)(1,2 tt

5.0)(2,3 ss

5.0)(3,2 tt

5.0)(3,0 ss

5.0)(0,3 tt

0 1

0 1

)0,2/1(1, iiu

)2/1,0(1, iiuTransition functions

02, iiu

)2/1,2/1(, iiu

1 ijij

13 Berkeley, 5/5/2005

Why do I care?

Applications:•Joint angle, closed curveAdvantages:•Can pretend everything is Euclidean

• Make chart that overlaps area of interest• Scaled to fit

• Encapsulates 2 shift once and for all• Use existing code as-is

•Derivatives all work (atlas is smooth)• Can move calculation to overlapping chart

• No seams• No boundary conditions

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Closed curve

Create an embedding of the manifold•Embed each chart

• Standard R->R2 problem•Blend between embeddings

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Joint angle example

Reinforcement learning:•Swing pendulum until balanced

• Can apply angular force at each time step•Goal

• Learn how to apply force from any position, velocity, to get to balanced state

• Manifold: Circle X R• Function on manifold: force to apply

• Learn function on each chart• Blend to get force

•Extend to two joints: Circle X circle X R X R

Pendulum

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Writing functions on manifolds

Do it in pieces•Write embed function per chart

• Can use any Rn technique• Splines, RBFs, polynomials…• Doesn’t have to be homogenous!

•Write blend function per chart• k derivatives must go to zero by boundary• Normalize to get partition of unity

• Spline functions get this for free

17 Berkeley, 5/5/2005

Blend functions

Define proto-blend function•Defined on chart•Support equals chart

• Ck – use B-Spline basis function• Cinf – use rational combination of

exponentials•Promote to function on manifold by setting to zero•Normalized blend function

• Divide by sum• Sum not zero by atlas covering

property• Only influenced by overlapping

charts

Accc

ccc

pB

pBpB))((ˆ

))((ˆ)(

cB

-1/2 1/2

1

Normalized blend function

-1/2 1/2

cB̂

Proto blend function

ccc UppB ,0))((ˆ

18 Berkeley, 5/5/2005

Final embedding function

Embedding is weighted sum•Generalization of splines•Define using transition functions

• Derivatives• Embed point in chart c

Continuity is minimum continuity of constituent parts

Ac

ccc pEpBpE ))(()()(

Accccccc pEpBpE ))(())(())(( 11

19 Berkeley, 5/5/2005

2D manifolds

•Sphere• Point as x2 + y2 + z2 = 1• Latitude-longitude mappings

• Each chart ½ sphere (approx)• Nice 6 chart partition

² Bounded by great arcs

• Polar projection• Charts anywhere• Center plus radius

20 Berkeley, 5/5/2005

2D manifolds

Torus•Nine charts

• Each 2/3 of torus•Torus as tiled plane

• Specify 4 corners• Ordering matters The nine charts

relative to .The plane tiled with copies of

(0,0)

(22)

1

(0,0)

(22)

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2D manifolds

Torus•Overlaps

• May be more than one disjoint region

Chart i

Chart j

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2D manifoldsN-holed torus•Hyperbolic disk tiled with 4n-sided polygon

• Linear fractional transforms• Well-defined inverse

dczbazz

dcba

1

acbd

23 Berkeley, 5/5/2005

Hyperbolic disk

Unit disk with hyperbolic geometry• Sum of triangle angles < 180• Angle is defined by tangents

Lines are circle arcs• Circles meet disk perpendicularly

24 Berkeley, 5/5/2005

N-holed torus

Why hyperbolic geometry?•4n-sided polygon

• Each paired edge creates loop• One loop through hole• One loop around

•Tiling• Corners that meet at 2/(4n)

angles

2

01

365

7 4

1a1b

1c

a

1d

b

d

c 0

1

65

4

3

2

7

25 Berkeley, 5/5/2005

N-holed atlas

One chart for each element of polygon•Face•Edge

• Two maps to center polygon•Vertex

• 4n maps to center polygon

02 1

4 37

5 6

)21,

21(

)21,

21( )

21,

21(

)21,

21(

26 Berkeley, 5/5/2005

N-holed arbitrary charts

LFT specifies center and radius•Data structures to keep track of which copies are overlapped

27 Berkeley, 5/5/2005

Surface modeling applications

Parameterization•Use correct topology

• No seams• Well-defined overlaps

• Mip-mapping, texture transfer, fluid-flow, procedural textures

Spherical cow

28 Berkeley, 5/5/2005


Surface reconstruction•Choice of embedding function

• Spline, RBF, etc.•Decide number of charts

• Fit each chart individually• No boundary constraints

Original mesh

Mesh in 1-1 correspondence

with manifold

Base

Ear

Tail

Spline

RBF

29 Berkeley, 5/5/2005


Consistent parameterizations•Selected points on both meshes•Parameterize both so points match

• Fit surface

Fitted surfaces

Source meshes(David Laidlaw,

Brown University)

LeftRight

(reflected)

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Surface modeling

Making new models•Start with manifold of correct topology•User creates sketch (mesh)

• Embed mesh in topology• Creates atlas

• Use mesh geometry to create chart embeddings

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Surface modeling

Editing models•Draw on surface to indicate locations for new charts

• Inverse function (embedding to manifold) well-defined

•Caveat: How do we ensure new charts “over-ride” old ones?

• Surface pasting or hierarchical editing

32 Berkeley, 5/5/2005

Adding charts

Adding new charts•Option 1) Crank up blend function•Option 2) “Zero-out” blend functions in higher level

• Define masking function for each level

)))((1()(ˆ)( ppBpB ccjcij

ic

ic

33 Berkeley, 5/5/2005

Other applications

Environment mapping•Parameterization of the sphereAnimation•Configuration space is a manifoldImage-based rendering•Rotation and “push-broom” camera capture

• Goal is to find single image (manifold)• Lining up images == finding transition functions

34 Berkeley, 5/5/2005

Conclusion

Manifolds:•Provide a natural way to partition complex surfaces

• Simple functions on Rn

• Maintain analytic properties• “Moving” analysis (no seams!)

•Separate topology from geometry• Adjacency relationships• Consistent parameterizations

•Encapsulate non-Euclidean operations• Closest point, geodesics, movement in domain

Manifolds or why a cow is a sphere

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