Evolutionary Morphing and Shape Distance

Nina Amenta

Computer Science, UC Davis

Collaborators

Physical AnthropologyEric Delson, Steve Frost, Lissa Tallman, Will

Harcourt-Smith

MorphometricsF. James Rohlf

Computer Science and MathKatherine St. John, David Wiley, Deboshmita Ghosh,

Misha Kazhdan, Owen Carmichael, Joel Hass, David Coeurjolly

Outline

• Application of 3D Procrustes tangent space analysis in primate evolution

• Some issues with the shape space

• An idea

Evolutionary Trees

Computing Trees

Tree inference method

Papio

Macaca

Cercocebus

Cercopithecus

Allenopithecus

Trees on extant species come from genomic data.

Estimating morphology

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

3D input data

Estimating morphology

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

Evolutionary Morphing

Fossils

Genomic trees don’t include fossils.

Primates: ~200 extinct genera, ~60 extant.

Fossils have to be added based on shape and meta-data.

Fossil Restoration

fossil symmetrization reflection

Sahelanthropos

Fossil Restoration

restored fossil

template surface

reconstructed specimen

TPS

Improve Estimated Morphology

synthetic basal node

repairedVictoriapithecus

Improve Estimated Morphology

improved basal node

repairedVictoriapithecus

Parapapio, a more recent fossil

Template is root of subtree where we believe it falls

Placement of Parapaio

User-defined landmarks

Our users want to specify or edit landmarks, but more automation is clearly needed.

We optimize for correspondence only within surface patches (Bookstein sliding, does not work well).

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.

Example

Features are not aligned

..even starting with optimal correspondence. Procrustes distance emphasizes big change, misses similarity of parts.

Features are not aligned

Changing the details might even reduce DProc.

Features are not aligned

Optimizing correspondence under DProc will not lead to intuitively better correspondence.

Complex Shapes

All parts cannot be simultaneously aligned by linear deformations. Deformation really is non-linear.

Edge-length Distance

Proposal: represent correspondence as corresponding triangle meshes instead of corresponding point samples.

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

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.

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.

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!

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.

Scale

Can normalize to produce scale invariance, as with Procrustes distance.

Choosing scale so that Li = 1 keeps all specimens in a linear subspace.

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 (Euler+triangulation constraints), -1 for scale = 3n-7

Scale

But this does not solve the problem of matching parts getting different scales.

What if we apply local scale factors at each vertex?

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?

Drawback

Isometric surfaces have distance zero.

Complicates reconstruction of interpolated shapes. Q2.

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?

Thank you!