Evolutionary Morphing and Shape Distance

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Evolutionary Morphing and Shape Distance

• Nina Amenta
• Computer Science, UC Davis

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Collaborators

• Physical Anthropology
• Eric Delson, Steve Frost, Lissa Tallman, Will
  Harcourt-Smith
• Morphometrics
• F. James Rohlf
• Computer Science and Math
• Katherine St. John, David Wiley, Deboshmita
  Ghosh, Misha Kazhdan, Owen Carmichael, Joel Hass,
  David Coeurjolly

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Outline

• Application of 3D Procrustes tangent space
  analysis in primate evolution
• Some issues with the shape space
• An idea

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Evolutionary Trees
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Computing Trees
Trees on extant species come from genomic data.
Papio Macaca Cercocebus Cercopithecus Allenopithec
us
Tree inference method
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Estimating morphology

• Using 3D data for extant species, and tree,
  estimate cranial shapes for the hypothetical
  ancestors.

3D input data
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Estimating morphology

• Generalized least-squares, covariance matrix
  derived from weighted tree edges.

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Evolutionary Morphing
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Fossils

• Genomic trees dont include fossils.
• Primates 200 extinct genera, 60 extant.
• Fossils have to be added based on shape and
  meta-data.

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Fossil Restoration
symmetrization
reflection
fossil
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Sahelanthropos
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Fossil Restoration
TPS
restored fossil
template surface
reconstructed specimen
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Improve Estimated Morphology
synthetic basal node
repaired Victoriapithecus
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Improve Estimated Morphology
improved basal node
repaired Victoriapithecus
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Parapapio, a more recent fossil
Template is root of subtree where we believe it
falls
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Placement of Parapaio
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User-defined landmarks
We optimize for correspondence only within
surface patches (Bookstein sliding, does not
work
well).
Our users want to specify or edit landmarks, but
more automation is clearly needed.
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Procrustes Distance

• DEuc(A,B) Euclidean distance in R3n
• Choose transformation T (scale, trans, rot)
  producing minimum DEuc
• DProc(A,B) min DEuc(T(A), B)
• T
• We work in Euclidean tangent space.

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Example
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Features are not aligned
..even starting with optimal correspondence.
Procrustes distance emphasizes big change, misses
similarity of parts.
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Features are not aligned
Changing the details might even reduce DProc.
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Features are not aligned
Optimizing correspondence under DProc will not
lead to intuitively better correspondence.
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Complex Shapes
All parts cannot be simultaneously aligned by
linear deformations. Deformation really is
non-linear.
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Edge-length Distance
Proposal represent correspondence as
corresponding triangle meshes instead of
corresponding point samples.
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Edge-length Distance

• Li is Euclidean length of edge ei
• Shape feature vector v is (L1 Lk)
• DEL DEuc(v(A), v(B))
• This represents a mesh as a discrete metric set
  of lengths on a triangulated graph, respecting
  the triangle inequality

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Information Loss

• In 2D, this does not make much sense.
• But in 3D, almost all triangulated polyhedra are
  rigid. So a discrete metric has a finite number
  of rigid realizations.

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Not a New Idea

• Euclidean Distance Matrix Analysis, Lele and
  Richtmeier, 2001 use the complete distance
  matrix as shape rep.
• Truss metrics include only enough edges to
  get rigidity.

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Quote

• the arbitrary choice of a subset of linear
  distances could accentuate the influence of
  certain linear distances in the comparison of
  forms, while masking the influence of others. -
  Richtsmeier, Deleon, and Lele, 2002.
• Not an issue in R3!

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Nice Properties

• Rotation and translation invariant
• Invariant to rotations and translations of parts
  (isometries).
• Any convex combination of specimens gives another
  vector of Li obeying triangle inequalities. So
  we can do statistics in a convex region of
  Euclidean space.

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Scale

• Can normalize to produce scale invariance, as
  with Procrustes distance.
• Choosing scale so that S Li 1 keeps all
  specimens in a linear subspace.

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Degrees of Freedom

• Dimension of Kendall shape space is 3n-7
• Number of edges for a triangulated object
  homeomorphic to a sphere is 3n-6
  (Eulertriangulation constraints), -1 for scale
  3n-7

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Scale

• But this does not solve the problem of matching
  parts getting different scales.
• What if we apply local scale factors at each
  vertex?

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Local Scale?

• We could add a scale factor at each vertex,
  producing a discrete conformal representation
  (Springborn, Schoeder, Bobenko, Pinkall)but this
  has way too many degrees of freedom.
• Q1 How to incorporate the right amount of local
  scale?

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Drawback

• Isometric surfaces have distance zero.
• Complicates reconstruction of interpolated
  shapes. Q2.

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More Questions

• Q3 Given a discrete metric formed as a convex
  combination of specimens, how to choose the right
  3D realization for visualization?
• Q4 How to optimize correspondence so as to
  minimize DEL? How to weight by area?

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Thank you!
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Correspondence-base Metrics

• Correspondence diffeomorphism or sampling
  therof. Usually point samples.

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Invariance

• to some set of transformations of the input (eg,
  rot, trans, scale).
• DRMS (sum-squared difference) is notbut it gives
  a Euclidean shape space.
• DTPS (bending energy) isbut does not give a
  Euclidean shape space, and has other problems.

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Commercialization
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Shape Metric

• Requires a shape metric.
• Much easier in Euclidean shape-space.

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Shape Is Invariant

• ...to transformations of the input (rot, trans,
  scale, correspondence).

n landmarks
B
A
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Tangent Space

• DProc does not give a Euclidean shape space.
• But computing the Karcher mean M and aligning all
  specimens to M does gives a nearby Euclidean
  shape space.

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Invariance is not enough

• We could remove nuisance parameters from lots of
  metrics, achieving invariance, and we would still
  not necessarily be measuring shape.
• Consistency is important, but does not
  necessarily imply we are measuring shape
  distance.

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Philosophy

• But, we feel like DProc should be a shape metric
  because
• DEuc is a good metric for shapes distorted by
  noise
• The transformation T that we find seems correct
  (except for correspondence)
• It works OK in applications

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Where did we go wrong?

• In the initial choice of DE

What should we do?
Take the derivative!