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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 Tree inference method Papio Macaca Cercocebus Cercopithecus Allenopithecus Trees on extant species come from genomic data.

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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 don’t 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 fossil symmetrization reflection

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Sahelanthropos

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Fossil Restoration restored fossil template surface reconstructed specimen TPS

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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 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).

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Procrustes Distance D Euc (A,B) = Euclidean distance in R 3n Choose transformation T (scale, trans, rot) producing minimum D Euc D Proc (A,B) = min D Euc (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 D Proc.

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Features are not aligned Optimizing correspondence under D Proc 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 L i is Euclidean length of edge e i Shape feature vector v is (L 1 … L k ) D EL = D Euc (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, Not an issue in R 3 !

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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 L i 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 L i = 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 (Euler+triangulation 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 D EL ? 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). D RMS (sum-squared difference) is not…but it gives a Euclidean shape space. D TPS (bending energy) is…but 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 A B

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Tangent Space D Proc 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 D Proc should be a shape metric because: –D Euc 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 D E What should we do? Take the derivative!

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