1. Manifolds or why a cow is a sphere
Cindy Grimm
Media and Machines Lab
Department of Computer Science and Engineering
Washington University in St. Louis

2. 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
Berkeley, 5/5/2005

3. What is a manifold? The world Overlap 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
The world
Overlap
Berkeley, 5/5/2005

4. 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) 1D 2D
Not manifold examples
Berkeley, 5/5/2005

5. 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 P 3D 2D P 2D 3D
Berkeley, 5/5/2005

6. Traditional definition
Given: Manifold M
Construct: Atlas A
Chart
Region Uc in M (open disk)
Region c in Rn (open disk)
Function ac taking Uc to c
Atlas is collection of charts
Every point in M in at least one chart
Overlap regions
Transition functions:
yij = aj o ai-1 smooth M
Berkeley, 5/5/2005

7. 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 [0,2p)
Point q+2pk is same for all k integer
(1,0)
These are embeddings p
This is not
Berkeley, 5/5/2005

8. double GetTheta() const { return m_dTheta; }
/** 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 circle
class Circle {
protected:
public:
Point2D operator()( const CirclePoint & in_circPt ) const {
return Point2D( cos( in_circPt.GetTheta() ),
sin( in_circPt.GetTheta() ) ); }
Circle() { }
~Circle() { } };
(1,0)
Berkeley, 5/5/2005

9. Chart on a circle Chart specification: Left and right ends of Uc
Counter-clockwise order to determine overlap region
Co-domain c = (-1/2,1/2)
Simplest form for ac
Translate
Center in [0,2p)
Scale qR qc Uc
-1/ /2
( )
( ) qL
-1/2
1/2 qR qL
Berkeley, 5/5/2005

10. Chart on a circle ( ) Chart specification: Alpha function
Sort point first
Euclidean distance is topological distance
Alpha inverse
Point in [0,2p)
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 * M_PI) - m_dThetaCenter ) <= M_PI ) {
dThetaShift = dTheta * M_PI;
} else if ( fabs( (dTheta * M_PI) - m_dThetaCenter ) <= M_PI ) {
dThetaShift = dTheta * M_PI;
} else {
assert( false ); }
const double dT = (dThetaShift - m_dThetaCenter) * m_dScale;
return ChartPoint( this, dT ); qc q
( )
q-2p
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 ); }
Berkeley, 5/5/2005

11. Making a chart 2p qL qR 2p qR qL
const 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() * 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; qL qR 2p qL qR 2p
Berkeley, 5/5/2005

12. An example circle atlas
atlas.AddChart( 0.0, M_PI );
atlas.AddChart( M_PI / 2.0, * M_PI / 2.0 );
atlas.AddChart( M_PI, * M_PI );
atlas.AddChart( 3.0 * M_PI / 2.0, 5.0 * M_PI / 2.0 );
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
-1/2
1/2
Chart 1
Chart 2
Chart 3
Overlap regions
Chart 0
Chart 1
Chart 2
Chart 3 1
Transition functions
Berkeley, 5/5/2005

13. Why do I care? Applications: Joint angle, closed curve Advantages:
Can pretend everything is Euclidean
Make chart that overlaps area of interest
Scaled to fit
Encapsulates 2p 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
Berkeley, 5/5/2005

14. Closed curve Create an embedding of the manifold Embed each chart
Standard R->R2 problem
Blend between embeddings
Berkeley, 5/5/2005

15. 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
Berkeley, 5/5/2005

16. 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
Berkeley, 5/5/2005

17. 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
-1/2
1/2
Proto blend function
-1/2
1/2 1
Normalized blend function
Berkeley, 5/5/2005

18. 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
Berkeley, 5/5/2005

19. 2D manifolds Sphere Point as x2 + y2 + z2 = 1
Latitude-longitude mappings
Each chart ½ sphere (approx)
Nice 6 chart partition
Bounded by great arcs
Polar projection
Charts anywhere
Center plus radius
Berkeley, 5/5/2005

20. 2D manifolds Torus Nine charts Each 2/3 of torus Torus as tiled plane
Specify 4 corners
Ordering matters
The plane tiled with copies of t.
(0,0)
(2p,2p ) t
The nine charts relative to t.
(0,0)
(2p,2p )
Berkeley, 5/5/2005

21. 2D manifolds Torus Overlaps May be more than one disjoint region
Chart i
Chart j
Berkeley, 5/5/2005

22. 2D manifolds N-holed torus Hyperbolic disk tiled with 4n-sided polygon
Linear fractional transforms
Well-defined inverse t t
Berkeley, 5/5/2005

23. 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
Berkeley, 5/5/2005

24. N-holed torus Why hyperbolic geometry? 4n-sided polygon2 1 3 6 5 7 4
Why hyperbolic geometry?
4n-sided polygon
Each paired edge creates loop
One loop through hole
One loop around
Tiling
Corners that meet at 2p/(4n) angles 1 6 5 4 3 2 7
Berkeley, 5/5/2005

25. N-holed atlas One chart for each element of polygon Face Edge
Two maps to center polygon
Vertex
4n maps to center polygon 2 1 4 3 7 5 6
Berkeley, 5/5/2005

26. N-holed arbitrary charts
LFT specifies center and radius
Data structures to keep track of which copies are overlapped
Berkeley, 5/5/2005

27. Surface modeling applications
Parameterization
Use correct topology
No seams
Well-defined overlaps
Mip-mapping, texture transfer, fluid-flow, procedural textures
Spherical cow
Berkeley, 5/5/2005

28. Surface modeling applications
Surface reconstruction
Choice of embedding function
Spline, RBF, etc.
Decide number of charts
Fit each chart individually
No boundary constraints
Original mesh
Ear
Spline
Base
Tail
RBF
Mesh in 1-1 correspondence with manifold
Berkeley, 5/5/2005

29. Surface modeling applications
Consistent parameterizations
Selected points on both meshes
Parameterize both so points match
Fit surface
Right (reflected)
Source meshes
(David Laidlaw, Brown University)
Left
Fitted surfaces
Berkeley, 5/5/2005

30. 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
Berkeley, 5/5/2005

31. 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
Berkeley, 5/5/2005

32. Adding charts Adding new charts Option 1) Crank up blend function
Option 2) “Zero-out” blend functions in higher level
Define masking function h for each level
Berkeley, 5/5/2005

33. Other applications Environment mapping Parameterization of the sphere
Animation
Configuration space is a manifold
Image-based rendering
Rotation and “push-broom” camera capture
Goal is to find single image (manifold)
Lining up images == finding transition functions
Berkeley, 5/5/2005

34. 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
Berkeley, 5/5/2005