Discrete Conformal Structures

Transcription

1 Discrete Conformal Structures Boris Springborn (TUB) Ulrich Pinkall (TUB) Peter Schröder (Caltech) 1

2 The Problem Find nice texture maps simplicial surface metric data satisfy triangle inequality 2

3 The Problem Find nice texture maps simplicial surface metric data new (flat) metric 3

4 The Problem Find nice texture maps simplicial surface metric data new (flat) metric 4

5 The Problem Find nice texture maps simplicial surface metric data new (flat) metric 5

6 Ansatz Seek conformally equivalent metric data: simpl. complex & lengths 0utput: new metric (i.e., lengths) new metric conformal factor original metric ignore boundary for the moment 6

7 Ansatz Seek conformally equivalent metric data: simpl. complex & lengths 0utput: new metric (i.e., lengths) variables at vertices 7

8 Ansatz Seek conformally equivalent metric data: simpl. complex & lengths 0utput: new metric (i.e., lengths) variables at vertices goal: desired angle sums target vertex angle sums 8

9 Non-Linear Problem Find to satisfy angle sum targets from lengths to angles watch out for triangle inequality! 9

10 Non-Linear Problem Find to satisfy angle sum targets from lengths to angles a miracle occurs This system of equations can be integrated! 10

11 The Energy Find minimum of a convex energy Milnor s Lobachevsky function

12 The Energy Find minimum of a convex energy Milnor s Lobachevsky function for each triangle logarithmic lengths logarithmic input lengths 12

13 The Energy Find minimum of a convex energy Milnor s Lobachevsky function for each triangle 13

14 The Energy Properties convex: Hessian is pos. semi-def. only one term for boundary edges 14

15 The Energy Properties convex: Hessian is pos. semi-def. solution exists is unique gradient flow is curvature flow target curvature 15

16 The Energy Properties convex: Hessian is pos. semi-def. solution exists is unique gradient flow is curvature flow what about triangle inequality?! 16

17 Domain of Definition Not just any u value is cool triangle inequality 2 legal range

18 Domain of Definition Not just any u value is cool triangle inequality extend definition Real part only -1 Functional remains C 1 in u 18

19 Domain of Definition Not just any u value is cool triangle inequality extend definition 2 1 minimum may occur at illegal values conditions for existence guarantee? 19

20 Boundary Conditions Fixing variables fix u i let Θ i vary u i = 0 : isometric bndry. nice for cut-boundaries! unknown cone angles arbitrary topology fix Θ i let u i vary rectangle, disk 20

21 Practicalities Convex optimization Newton-Steihaug trust region

22 Practicalities Convex optimization Newton-Steihaug trust region Petsc/TAO library SSOR precon for cotan system layout: dual spanning tree achieves 10-9 to acc. alternatively: Dirichlet problem 22

23 Conformal Equivalence From continuous to pair of meshes equivalently: 23

24 Conformal Equivalence From continuous to pair of meshes equivalently: Length cross ratios are preserved 24

25 Why u = 0 on Boundary? In the continuous setting conformally equivalent metric flat choose least varying : 25

26 Why u = 0 on Boundary? In the continuous setting conformally equivalent metric flat choose least varying : 26

27 Dual Functional Angles/lengths are dual variables functional in angles original logarithmic lengths 27

28 Dual Functional Angles/lengths are dual variables functional in angles length cross ratios invariant 28

29 The Big Picture Discrete conformal structure simplicial mesh preserve: phase: circle patterns can t read off angles directly magnitude: new functional CAN read off lengths directly! 29

30 TODO List Future work conditions for existence intrinsic Delaunay not required! automatic cone singularity placement u provides potential hook first Newton step sparse approximation problem 30