Consistent Mesh Parameterizations

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Consistent Mesh Parameterizations

• Emil Praun (Princeton)
• Wim Sweldens (Bell Labs)
• Peter Schröder (Bell Labs)

Woefully butchered by Chris Lutz
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The University of Virginias GLunch Cavalcade
Variety Hour Presents
Perhaps a better title screen would be...
LUTZ-O THE MAGNIFICENT
A REAL LIFE Comical Dancing ! INTELLECTUAL MIDGET
!
With his Travelling Show of Pretty Pictures
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The Problem (Seriously)

• We would like to operate on a set of dis/similar
  models...
• ...But how do you average two heads?
• This is tough with 3D geometry
• Geometry is non-Euclidean (?)
• Differing sampling patterns and connectivity
• We need a base parameterization for a group of
  models to provide a point of comparison

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The Problem (Not So Seriously)
Why, just think of all the wonderful things we
could do! We could... ADD UP HEADS Learn
Math and Anatomy at the same time! Save hours!
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More Tepid Drivel
CLOTHE THE ANIMALS No longer will Mr. Ed
gallivant unclad, gaily flashing his uncovered
loins for all to behold!
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Last One, Folks. Dont Worry.
and CREATE HIDEOUS MUTANT BOVINE/HUMAN
HYBRIDS Chuckle at their malformed
limbs! Laugh at their twisted spines! Mock their
pathetic cries for mercy!
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Consistent Parameterizations

• Establish relationships between models
• Enforce the same connectivity across dissimilar
  models
• If each vertex has a unique corresponding vertex
  in every model, a whole bunch of wacky operations
  can be performed (see the previous 3 slides for
  review)

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Contributions of this Paper

• Given a base model and a set of meshes with
  identified features
• we can create a consistent parameterization for
  all models
• Then we can create remeshes of the original
  models and make mutants
• In case you already forgot

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Related Work

• Lots of algorithms on creating parameterized
  meshes already out there
• Lee et. al. created meta-meshes
• These are huge and wouldnt scale well
• Marschner et. al. worked on embedding a prototype
  layout into similar models (faces) and used a
  least squares fitting approach
• Models must be geometrically similar for this to
  work

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And Heres Where I Lose It

• Terminology
• A mesh M (P, K)
• P set of N points in 3-space
• K abstract simplicial complex (duh) set of
  subsets of vertices, edges, and faces
• S set of meshes M0MN
• We need to create a new coarsest mesh for all
  meshes M0MN by applying base connectivity to
  individual base meshes

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Step 1 (Take Valium)

• We need
• A set of meshes that have feature points defined
  (either by hand or with a feature detection
  algorithm)
• The connectivity L0 between the feature points
  (each triangle in L0 corresponds to a patch on
  each of the meshes, and edges define patch
  boundaries)
• Now, just trace the boundaries on the models
  while preserving L0 topology and outlining fair
  patches! No problem.

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Topological Equivalence

• Need to satisfy 2 conditions
• Two patch boundaries can only intersect at a
  feature vertex
• Each feature vertex must keep the same rotational
  ordering of its edges
• Shortest path wont work

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More Topological Equivalence

• Instead of shortest path, use a modified brush
  fire algorithm to trace the path
• Restrictions
• Cant cross previously traced path
• New paths must be attached in the right place
• Be wary of encircling (1st create spanning tree)

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Fair Boundary Curves

• Ok, so we have an equivalent mapping however, it
  could look really weird
• Specifically, we want
• Patches of equal surface area
• Smooth patch boundaries
• Fair patch boundaries (i.e. no swirling effects)
• The first two can be achieved through
  relaxation (close your eyes -- you are on a
  tropical beach)

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Why Swirling Is Evil

• Reason 1 Nasty, twisted meshes
• Reason 2 Nasty, twisted math
• No swirls global min. of fairness functional
• But that leads to a non-convex, mixed
  discrete-continuous global optimization problem
  (The Horror! The Horror!)
• So we cheat with a bunch of heuristics

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How to Thwart Swirlies

• Parameterization
• Traced curves should try to follow iso-parameter
  lines (basically, patch boundaries repel each
  other)
• Tracing Curves
• Trace a curve along the local maximum of the
  objective function
• Priority Queues Spanning Trees
• Trace a tentative curve using CBF algorithm put
  into a priority queue by length
• Upon removal from the queue, check to see if the
  tentative path is valid (doesnt cross any paths
  and connects correctly)
• If it is not valid, retrace the curve and stick
  it back in the queue
• If it is valid, run the swirl detector from both
  adjacent patches
• If one or both fails, put it back into the queue
  with a penalty
• If both pass, add the curve to the net

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The Swirl Detector

• Parametrically trace a line from c directly to
  the potential path a, b
• If it arrives on the wrong side, somethings
  wrong (which makes sense, I suppose)
• Postpone tracing a, b until a, b, and c
  are linked into the rest of the net (stick it
  back into the queue with a penalty)

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The Subtle Glories ofEdge-Straitening

• We now have a complete net of curves equivalent
  to L0
• Do some crazy mapping of the submeshes composed
  of all triangles which are incident to lines
  connecting to feature point i onto an n-gon
  inscribed in a unit circle
• Straighten the lines and map it back into
  model-space, creating new triangles as necessary

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That Wasnt So Bad, Was It?

• All examples take about the same time to process
• 900Mhz P-III
• tracing curve net 5 sec
• crude ? org 2 sec
• smoothing 1.5 min
• remeshing (43K triangles) 6 minutes

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Applications Beyond Mutants

• Principle Mesh Components one can compute
  eigenheads from which one can compose new faces
  from different features
• Texture Transfer Mr. Ed is finally decent
• Transfer of Wavelet Details One can
  mix-and-match low frequencies and high
  frequencies from different models
• Shape Blending Look for SimMutant at a store
  near you

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Future Work

• Extend the method to handle higher-genus models
• Applications in animation perhaps a new way to
  make a soda-can dance?
• Model database searching by computing principle
  components
• More efficient compression of multiple shapes

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You Were Supposed to Be a Thing of Beauty...