Manifolds or why a cow is a sphere

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Manifoldsor why a cow is a sphere

• Cindy Grimm
• Media and Machines Lab
• Department of Computer Science and Engineering
• Washington University in St. Louis

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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

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What is a manifold?

• History
• Mathematicians 1880s
• 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
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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)

Not manifold examples
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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(i1)mod
  n all exist
• Can do on abstract or embedded mesh

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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
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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 q2pk is same for all k integer

These are embeddings
0 2p
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/ 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 CirclePoint()
/// Definition for S1, 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()
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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
( )
qL
-1/2
1/2
qR
qL
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Chart on a circle

• Chart specification
• Alpha function
• Sort point first
• Euclidean distance is topological distance
• Alpha inverse
• Point in 0,2p)

ChartPoint ChartAlpha( 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 )
dThetaShift dTheta else if ( fabs(
(dTheta 2.0 M_PI) - m_dThetaCenter )
) dThetaShift dTheta 2.0 M_PI
else if ( fabs( (dTheta - 2.0 M_PI) -
m_dThetaCenter )
dTheta - 2.0 M_PI else
assert( false ) const double dT
(dThetaShift - m_dThetaCenter) m_dScale
return ChartPoint( this, dT )
CirclePoint ChartAlphaInv( const double in_dT )
const const double dTheta in_dT /
m_dScale m_dThetaCenter // Converts to
0,2pi range return CirclePoint( dTheta )
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Making a chart
const Chart AtlasAddChart( 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() 2.0M_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 ) const Chart opChart new Chart(
m_aopCharts.size(), dThetaMid, dScale )
m_aopCharts.push_back( opChart ) return
opChart
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An example circle atlas
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 )

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

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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

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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

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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
• Doesnt 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

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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
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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

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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

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2D manifolds

• Torus
• Nine charts
• Each 2/3 of torus
• Torus as tiled plane
• Specify 4 corners
• Ordering matters

The nine charts relative to t.
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2D manifolds

• Torus
• Overlaps
• May be more than one disjoint region

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2D manifolds

• N-holed torus
• Hyperbolic disk tiled with 4n-sided polygon
• Linear fractional transforms
• Well-defined inverse

t
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Hyperbolic disk

• Unit disk with hyperbolic geometry
• Sum of triangle angles
• Angle is defined by tangents
• Lines are circle arcs
• Circles meet disk perpendicularly

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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 2p/(4n) angles

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N-holed atlas

• One chart for each element of polygon
• Face
• Edge
• Two maps to center polygon
• Vertex
• 4n maps to center polygon

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N-holed arbitrary charts

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

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Surface modeling applications

• Parameterization
• Use correct topology
• No seams
• Well-defined overlaps
• Mip-mapping, texture transfer, fluid-flow,
  procedural textures

Spherical cow
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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
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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
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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

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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

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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

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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