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Computing and Processing Correspondences with Functional Maps SIGGRAPH 2017 course Maks Ovsjanikov , Etienne Corman, Michael Bronstein, Emanuele Rodol, Mirela BenChen , Leonidas Guibas, Frederic Chazal, Alex Bronstein Functional Vector Fields

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Computing and Processing Correspondences with Functional Maps

Maks Ovsjanikov, Etienne Corman, Michael Bronstein, Emanuele Rodolà, Mirela Ben‐Chen, Leonidas Guibas, Frederic Chazal, Alex Bronstein

SIGGRAPH 2017 course

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Functional Vector Fields

Mirela Ben‐Chen Technion, Israel Institute of Technology

ERC Project 714776 (OPREP)

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So far

Computing and analyzing FMAPs Using FMAPs for function transfer

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[Ovsjanikov, BC, Solomon, Butscher, Guibas, SIGGRAPH 2012]

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SLIDE 4

This part

Computing FMAPs using vector fields Using FMAPs for vector field transfer

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[Azencot, BC, Chazal, Ovsjanikov, SGP 2013]

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What are vector fields?

  • Smooth assignment of “arrow”

per point

  • Only tangent vector fields
  • Visualize with texture
  • Can see direction but not length

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SLIDE 6

Vector fields and maps – symbiosis

Fluid Simulation Quad remeshing Map Improvement

Azencot, Corman, BC, Ovsjanikov, SIGGRAPH 2017 Corman, Ovsjanikov, Chambolle, SGP 2015 Azencot, Vantzos, Wardetzky, Rumpf, BC, SCA 2015

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Transporting data with point‐to‐point maps

Mapping scalars Mapping vectors

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∈ ∈

  • ∈

∈

  • ∈

∈

  • ?
  • Tangent plane of at
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The Map Differential

  • Start a curve from with

tangent

  • Map the curve to
  • is the tangent of the mapped

curve at

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  • ∈

∈

  • ∈

∈

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SLIDE 9

Transporting data with point‐to‐point maps

Mapping scalars Mapping vectors

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∈ ∈

  • ∈

∈

  • ∈

∈

  • Tangent plane of at
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Transporting data with functional maps

Mapping functions Mapping vector fields

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: → : →

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The Map Differential

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  • : 0,1 →

: 0,1 → .5 1 .5 1

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The functional Map Differential

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  • : 0,1 →

: 0,1 → .5 1 .5 1

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The functional Map Differential

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The functional Map Differential

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: →

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The functional Map Differential

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: → : →

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The functional Map Differential

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: → : →

  • 0 0

′0 〈 , 〉 ′0 〈 , 〉

, 〈 , 〉

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The functional Map Differential

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Corresponding vector fields

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  • For all
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Corresponding vector fields

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  • For all
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Functional Vector Fields

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Linear Complete*

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Corresponding FVFs

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  • , 〈, 〉
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Corresponding FVFs commute with FMap

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Application – Joint vector field design

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  • constraints

smoothness map consistency

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Application – Joint vector field design

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constraints + smoothness + consistency

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Application ‐ Joint Quad‐remeshing

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Wed 9am, Room 152

[Azencot, Corman, BC, Ovsjanikov, SIGGRAPH 2017]

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Map “animation”

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∈ ∈

  • One map

Map sequence

∈

  • ∈

1 .5

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A 1‐parameter family of maps

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Back to vector fields

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Back to vector fields, self maps

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.5 ∈

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Back to vector fields, self maps

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∈

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From maps to vector fields

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∈ ,

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From vector fields to trajectories

  • We know: map → vector fields
  • How to do: vector fields → map?
  • Given solve for such that

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The flow PDE

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From vector fields to functional trajectories

  • Given (stationary) ,

solve for such that

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  • ∘
  • The flow:

The functional flow:

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The functional flow

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The functional flow

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〈 , 〉

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The discrete functional flow

  • The surface is a triangle mesh
  • The function is represented using a linear basis , e.g. hat basis
  • is a vector, is a matrix
  • A matrix ODE

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The discrete functional flow

  • A matrix ODE
  • Closed form solution

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Implementation

  • Mesh : vertices, faces.
  • Represent

in the hat basis ‐ ∈

  • Represent in the hat basis – a sparse matrix ∈
  • Compute t expmv, ,
  • expmv: https://www.mathworks.com/matlabcentral/fileexchange/29576‐matrix‐exponential‐times‐a‐vector?focused=5172371&tab=function

Interpolate to vertices vector per face gradient operator

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Applications

Maps from vector fields

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[Azencot, Vantzos, BC, SGP 2016]

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Applications

Incompressible flow on surfaces

  • Numerically simulate the equation
  • Vector field changes, more complicated
  • Total vorticity conserved by construction

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  • [Azencot, Weißmann, Ovsjanikov, Wardetzky, BC, SGP 2014]
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Applications

Incompressible flow on surfaces

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Applications

Viscous thin films on surfaces Total volume of fluid conserved by construction

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[Azencot, Vantzos, Wardetzky, Rumpf, BC, SCA 2015]

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Applications

Inferring vector fields

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is distance preserving commutes with Δ

  • , ,
  • ||Δ Δ||
  • commutes with Δ

||Δ Δ||

  • [Azencot, BC, Chazal, Ovsjanikov, SGP 2013]
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SLIDE 44

Applications

Inferring vector fields

is area preserving is orthogonal is a rotation in functional space

  • is anti‐symmetric

is divergence free

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(some) conclusions

  • Vector fields and maps relate through tangents to trajectories
  • Can be used to transport vector fields with maps or

to compute maps from vector fields

  • The functional approach allows to use linear algebra instead of

geometric tracing for trajectory related problems

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References

  • Ovsjanikov, BC, Solomon, Butscher, Guibas, SIGGRAPH 2012, “Functional Maps: A

Flexible Representation of Maps Between Shapes”

  • Azencot, BC, Chazal, Ovsjanikov, SGP 2013, “An Operator Approach to Tangent

Vector Field Processing”

  • Azencot*, Weißmann*, Ovsjanikov, Wardetzky, BC, SGP 2014, “Functional Fluids
  • n Surfaces”
  • Corman, Ovsjanikov, Chambolle, SGP 2015, “Continuous Matching via Vector Field

Flow”

  • Azencot, Vantzos, Wardetzky, Rumpf, BC, SCA 2015, “Functional Thin Films on

Surfaces”

  • Azencot, Vantzos, BC, SGP 2016, “Advection‐Based Function Matching on

Surfaces”

  • Azencot, Corman, BC, Ovsjanikov, SIGGRAPH 2017, “Consistent Functional Cross

Field Design for Mesh Quadrangulation”

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