Computing and Processing Correspondences with Functional Maps - PowerPoint PPT Presentation

Computing and Processing Correspondences with Functional Maps SIGGRAPH 2017 course Maks Ovsjanikov , Etienne Corman, Michael Bronstein, Emanuele Rodolà, Mirela Ben‐Chen , Leonidas Guibas, Frederic Chazal, Alex Bronstein

Functional Vector Fields Mirela Ben‐Chen Technion, Israel Institute of Technology ERC Project 714776 (OPREP)

So far Computing and analyzing FMAPs Using FMAPs for function transfer 3 [Ovsjanikov, BC, Solomon, Butscher, Guibas, SIGGRAPH 2012]

This part Computing FMAPs using vector fields Using FMAPs for vector field transfer 4 [Azencot, BC, Chazal, Ovsjanikov, SGP 2013]

What are vector fields? • Smooth assignment of “arrow” per point • Only tangent vector fields • Visualize with texture • Can see direction but not length 5

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

� � � Tangent plane of � at � Transporting data with point‐to‐point maps Mapping scalars Mapping vectors ���� ������� ���� ���� � ∈ � � ∈ � � ∈ � � ∈ � � � ∈ � � �? � ∈ � � � 7

The Map Differential • Start a curve from � with tangent � • Map the curve to � ���� ���� • � is the tangent of the mapped curve at � � ∈ � � ∈ � � ∈ � � � � ∈ � � � 8

� � � Tangent plane of � at � Transporting data with point‐to‐point maps Mapping scalars Mapping vectors ���� ������� ���� ���� � ∈ � � ∈ � � ∈ � � ∈ � � � ∈ � � � �� � ��� � ∈ � � � 9

Transporting data with functional maps Mapping functions Mapping vector fields � � �: � → � �: � → � 10

� The Map Differential ��1� ��1� � � ��.5� ��.5� ��0� ��0� ���� ���� �: 0,1 → � �: 0,1 → � 11

� The functional Map Differential ��1� ��1� ��.5� ��.5� ��0� ��0� ��� � � ��� � � �: 0,1 → � �: 0,1 → � 12

� The functional Map Differential � � ���� ���� 13

� � The functional Map Differential � ��� � � �: � → � 14

The functional Map Differential � � � � ��� � � ��� � � �: � → � �: � → � 15

� � 0 � � � �0� �′�0� � 〈�� , �〉 The functional Map Differential �′�0� � 〈�� , �〉 � � �� , � � 〈�� , �〉 ��� � � ��� � � �: � → � �: � → � � 16

The functional Map Differential � � � � � �

Corresponding vector fields For all � � 18

Corresponding vector fields For all � � 19

Linear Complete * Functional Vector Fields 20

� �, �� � 〈�, �����〉 Corresponding FVFs � � 21

Corresponding FVFs commute with FMap � � 22

Application – Joint vector field design smoothness � � � � � � map consistency constraints 23

Application – Joint vector field design + consistency constraints + smoothness 24

Application ‐ Joint Quad‐remeshing Wed 9am, Room 152 25 [Azencot, Corman, BC, Ovsjanikov, SIGGRAPH 2017]

Map “animation” One map Map sequence � � ��1� ��.5� ��0� � � ∈ � � ∈ � � � ∈ � � ∈ � 27

A 1‐parameter family of maps � � 28

Back to vector fields � � 29

Back to vector fields, self maps � .5 ∈ �� 30

Back to vector fields, self maps � � ∈ �� 31

� �, ���� � �� �� ��� From maps to vector fields � � ∈ �� 32

From vector fields to trajectories • We know: map → vector fields • How to do: vector fields → map? • Given ���� solve for ���� such that The flow PDE 33

From vector fields to functional trajectories • Given (stationary) � , � � solve for ���� such that The flow: �� �� � � ∘ ���� � 0 � �� The functional flow: �� � � 34

The functional flow � 35

� � ��� � �〈�� � , �〉 The functional flow � 36

� � ��� � �� � ���� 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 37

The discrete functional flow • A matrix ODE • Closed form solution 38

Implementation • Mesh � : � vertices, � faces. � in the hat basis ‐ � � ∈ � � • Represent � • Represent � � in the hat basis – a sparse matrix � � ∈ � ��� � ��� ���� ���� Interpolate to vertices vector per face gradient operator • Compute � t � expmv��, � � , � � � expmv: https://www.mathworks.com/matlabcentral/fileexchange/29576‐matrix‐exponential‐times‐a‐vector?focused=5172371&tab=function

Applications Maps from vector fields � � 40 [Azencot, Vantzos, BC, SGP 2016]

Applications Incompressible flow on surfaces • Numerically simulate the equation � • Vector field changes, more complicated • Total vorticity � � conserved by construction 41 [Azencot, Weißmann, Ovsjanikov, Wardetzky, BC, SGP 2014]

Applications Incompressible flow on surfaces 42

Applications Viscous thin films on surfaces Total volume of fluid conserved by construction 43 [Azencot, Vantzos, Wardetzky, Rumpf, BC, SCA 2015]

� Applications Inferring vector fields � is distance preserving � commutes with Δ � � commutes with Δ � � � � � � , � � � , ||�Δ � � Δ � �|| ��� ||� � Δ � Δ� � || ��� ��� � � � � 44 [Azencot, BC, Chazal, Ovsjanikov, SGP 2013]

� Applications Inferring vector fields is area preserving is orthogonal � is anti‐symmetric ��� � � � �� � � �� � � �������� � � � � � is a rotation in is divergence free functional space � � �

(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 46

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 on 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” 47