Surface Applications

1
Surface Applications

• Fitting Manifold Surfaces To 3D Point Clouds,
  Cindy Grimm, David Laidlaw and Joseph Crisco.
  Journal of Biomechanical Engineering, 2002
• Parameterization using Manifolds, Cindy Grimm.
  International Journal of Shape Modeling, 10(1)
  51-80, 2004
• Spherical Manifolds for Adaptive Resolution
  Surface Modeling, Cindy Grimm. In "Graphite",
  pages 161-168, 2005
• WUCSE-2006-29 Smooth Surface Reconstruction
  using Charts for Medical Data, Cindy Grimm and
  Tao Ju. Technical Report 2006-29, Washington
  University in St. Louis, 2006
• Feature Detection Using Curvature Maps and the
  Min-Cut/Max-Flow Algorithm, Timothy Gatzke and
  Cindy Grimm. In "Geometric Modeling and
  processing", 2006

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Surface operations supported by manifolds

• Hierarchical modeling
• Add charts anywhere
• Surface reconstruction
• Continuity guaranteed
• Known topology
• Capturing shape variation
• Consistent parameterization
• Different embedding, same manifold
• Multiple parameterizations
• Color, geometry, electrical

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Manifold representation goals

• Add charts anywhere, at any scale/orientation
• Not restricted to an initial set of charts
• Adjust parameterization
• Slide charts around, change scale/orientation
• Re-use parameterization when fitting new geometry
• Embedding free to change
• Change data samples
• Create new parameterization over existing one
• In correspondence with original

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Surface types, R2 in Rn

• Plane
• Sphere
• Torus
• N-holed torus
• Other surfaces with boundary
• Cut holes

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Canonical manifolds

• Small number of possible types (3 closed)
• Define a canonical manifold for each
• Support equivalent of affine transformations
• Points, lines, triangles, barycentric coordinates
• Define a procedure for making charts
• C8
• Position plus shape control

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Sphere example

• Manifold Unit sphere, x2 y2 z2 1
• Representing geometry on the sphere
• Mesh on the sphere
• Making charts
• Embedding the sphere
• Blend and embed functions on each chart
• Hierarchical modeling
• Over-riding existing charts
• Surface fitting
• Assigning charts
• Multiple parameterizations

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Operations on the unit sphere

• How to represent points, lines and triangles on a
  sphere?
• Point is (x,y,z)
• Given two points p, q, what is (1-t) p t q?
• Solution Gnomonic projection
• Project back onto sphere
• Valid in ½ hemisphere
• Line segments (arcs)
• Barycentric coordinates in spherical triangles
• Interpolate in triangle, project

All points such that
(1-t)p tq
q
p
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Mesh on sphere

• Place vertices on sphere
• Project edges, faces, onto sphere by taking ray
  through point
• Use mid-point barycentric coords for 3 sides
• Valid for faces

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Chart on a sphere

• Chart specification
• Center and radius on sphere Uc
• Range c unit disk
• Simplest form for ac
• Project from sphere to plane
• (stereographic)
• Adjust with projective map
• Affine

Uc
1
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Defining an atlas

• Define overlaps computationally
• Point in chart evaluate ac
• Coverage on sphere (Uc domain of chart)
• Define in reverse as ac-1MD-1(MW-1(D))
• D becomes ellipse after warp, ellipsoidal on
  sphere
• Can bound with cone normal

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Embedding the manifold

• Write embed function per chart (polynomial)
• Write blend function per chart (B-spline basis
  function)
• k derivatives must go to zero by boundary of
  chart
• Guaranteeing continuity
• Normalize to get partition of unity

Normalized blend function
Proto blend function
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Surface editing

• User sketches shape (sketch mesh)
• Create charts
• Embed mesh on sphere
• One chart for each vertex, edge, and face
• Determine geometry for each chart (locally)

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Charts

• Optimization
• Cover corresponding element on sphere
• Dont extend over non-neighboring elements
• Projection center center of element
• Map neighboring elements via projection
• Solve for affine map
• Face big as possible, inside polygon
• Use square domain, projective transform for
  4-sided

Face
Face charts
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Edge and vertex

• Edge cover edge, extend to mid-point of adjacent
  faces
• Vertex Cover adjacent edge mid points, face
  centers

Vertex charts
Edge
Edge charts
Vertex
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Defining geometry

• Fit to original mesh (?)
• 1-1 correspondence between surface and sphere
• Run subdivision on sketch mesh embedded on sphere
  (no geometry smoothing)
• Fit each chart embedding to subdivision surface
• Least-squares Ax b

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Result

• CK analytic surface approximating subdivision
  surface
• Real time editing
• Other closed topologies
• Define manifold and domain to R2 map

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Torus, associated edges

• Cut torus open to make a square
• Two loops (yellow one around, grey one through)
• Each loop is 2 edges on square
• Glue edges together
• Loops meet at a point
• Chart map affine transform

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N-Holed tori

• Similar to torus cut open to make a 4N-sided
  polygon
• Two loops per hole (one around, one through)
• Glue two polygon edges to make loop
• Loops meet at a point
• Polygon vertices glue to same point

Front
Back
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Hyperbolic geometry

• Why is my polygon that funny shape?
• Need corners of polygon to each have 2p / 4N
  degrees (so they fill circle when glued together)
• Tile hyperbolic disk with 4N-sided polygons
• Chart map Linear Fractional Transform

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Hierarchical editing

• Override surface in an area
• Add arms, legs
• User draws on surface
• Smooth blend
• No geometry constraints

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Adding more charts

• User draws new subdivision mesh on surface
• Only in edit area
• Simultaneously specifies region on sphere
• Add charts as before
• Problem need to mask out old surface

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Masking function

• Alter blend functions of current surface
• Zero inside of patch region
• Alter blend functions of new chart functions
• Zero outside of blend area
• Define mask function h on sphere,
• Set to one in blend region, zero outside

1
1
0
0
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Defining mask function

• Map region of interest to plane
• Same as chart mapping
• Define polygon P from user sketch in chart
• Define falloff function f(d) - 0,1
• d is min distance to polygon
• Implicit surface
• Note Can do disjoint regions
• Mask blend functions

0
1
d
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Patches all the way down

• Can define mask functions at multiple levels
• Charts at level i are masked by all ji mask
  functions
• Charts at level i zeroed outside of mask region

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Results
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Surface fitting

• Embed data set on sphere
• Any spherical parameterization
• Group data points (overlapping)
• One chart per group
• Chart centered on group, covers
• Fit chart embeddings to data points

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Consistent parameterization

• Identify features in data
• Align features on sphere
• Build one set of charts
• Embed
• Fit each data set

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Consistent parameterization

• Build manifold once
• Fit to multiple bone point data sets
• Parameterized similarly

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Multiple parameterizations

• Build atlas for each case
• Atlas does not need to cover sphere
• Can be C0

Simulation
Geometry
Texture mapping
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Benefits of canonical approach

• Global manifold
• Link different embeddings, atlases
• Easy to adjust charts
• Charts independent from source data
• Layered charts (different atlases)
• Masking, boundaries, creases
• No special cases
• Not polynomial
• Computationally more expensive
• Topology-dependent algorithms