TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A AA A A A A A A A A  Justin Solomon, Raif Rustamov, Leonidas Guibas Stanford University Adrian Butscher Autodesk Research TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A

Point-to-point “Geodesic” Feature-to-pointFeature-to-feature Torus by M. Irons, signed distance by R. Kolluri, curve distance by C. Wu

“Somewhere over here.” TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A AA A A A A A A A A  TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A

“Exactly here.”

“One of these two places.” Superposition

Which is closer, 1 or 2? Query12

L p norm KL divergence

Query12

Overlap is the wrong measure!

Positive

 Many names Wasserstein distance, transportation distance, Mallows distance  Theoretically sound Regularity properties, continuous and discrete formulations  Popular option Computer vision, machine learning, operations, graphics

Our approach: Think of probabilities like a fluid Probabilities advect along the surface New discretization, optimization, and (consequently) applications!

Scales linearly Total work Theoretical version: “Beckmann problem”

Curl-freeDiv-free New idea!

1. 2.

1. 2.  Piecewise-linear FEM, optimized via ADMM  Spectral approximation (optional) Satisfies triangle inequality!

Iterations are fast and easy to implement!

Proposition: Satisfies triangle inequality. 0 eigenfunctions100 eigenfunctionsBiharmonicGeodesic

Proposition: Satisfies triangle inequality. 0 eigenfunctions100 eigenfunctionsBiharmonicGeodesic Spectral  Geodesic

Use barycentric coordinates (mean value)

Works for negative weights Reduces to geodesic distance

Curl-freeDiv-free

Avoid centerDistance to feature

 Quadratic ground distance  Other representations Point clouds? Polygon soup? Graphs?  Faster optimization

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A AA A A A A A A A A  Thanks! Matlab code online!

Single-source all-targets

All-pairs for sample of M points

PerturbationIsometry and remeshing