Yaron Lipman Thomas Funkhouser. RifR Raif Rustamov. Princeton University. Drew University

Transcription

1 Barycentric Coordinates on Surfaces Yaron Lipman Thomas Funkhouser RifR Raif Rustamov Princeton University Drew University

2 Motivation Barycentric coordinates good for: interpolation shading deformation Bezier surfaces parameterization interior distance image cloning shape retrieval finite i elements

3 Goal Define barycentric coordinates on surfaces: generalize planar intrinsic fast to compute

4 Definition on the plane v 1 v v 2 5 p p = affine comb of vertices v3 v4

5 Challenges on surfaces p belongs to convex hull of vertices

6 Challenges on surfaces Mobius, Alfeld et al, Cabral et al Ju et al Spherical triangles Spherical convex Langer et al Spherical all

7 Challenges on surfaces involves Cartesian coords not intrinsic

8 Our approach

9 Our approach Riemannian

10 Outline Introduction Riemannian Center of Mass Construction Computation Properties Results Applications

11 Usual center of mass Euclidean distance

12 Riemannian center of mass Geodesic distance

13 Karcher s theorem If the points are not too far from each other, the Riemannian center of mass is unique has a unique minimum, which is the unique zero of

14 Construction

15 Construction

16 Construction

17 Construction

18 Computation Summary Pick a kind of planar baryc coords E.g. Mean Value Coords Compute the gradient polygon at each point Explicit formula exists Surface baryc coords = = planar baryc coords wrt gradient polygon

19 Computation of gradients inverse exponential map Schmidt et al. [2006]

20 Computation of gradients Schmidt et al. [2006]

21 Intuitiveness I point towards v1. My length is equal to geod. dist. from p to v1

22 Intuitiveness I am at the correct distance and direction wrt p I point towards v1. My length is equal to geod. dist. from p to v1

23 Discrete setting

24 Discrete setting

25 Discrete setting

26 Discrete setting Pentagon Need five instances of single source, all destinations

27 Properties Defining gproperties p Lagrangian Partition of unity Riemannian center of mass Unique reconstruction from coordinates Due to Karcher s theorem Planar reproduction If surface is plane, get planar coordinates back Similarity invariance Smoothness

28 Properties Edge linearity

29 Properties Isometry invariance isometry invariance + unique reconstruction = = isometry map can be reconstructed

30 Results Darkest red point vertex wrt which coords are computed Dark blue small value Dark red large value Equally spaced isolines Planar baryc coords = =Mean Value Coordinates

31 Effect of surface shape

32 Variety

33 Variety

34 Effect of planar coordinate Mean Value Coordinates Maximum Entropy Coordinates

35 Timing g( (seconds)

36 Application: Interpolation

37 Application: Decal mapping Local parameterizations used for texturing We use the same idea as in image warping

38 Application: Decal mapping

39 App: Correspondence Refinement Based on isometry reconstruction property Correspondence is exact, if true isometry

40 App: Correspondence Refinement

41 App: Correspondence Refinement

42 Summary Definition and construction of barycentric coordinates on surfaces: properly generalize existing planar coordinates insensitive to isometric deformations easy to implement, and fast to compute

43 Future work Better Karcher s theorem Other distances instead of geodesic in Uniqueness of c.m. for larger polygons? More empirical studies of various choices of Distance Planar barycentric coordinates Further applications of surface coordinates

44 Thank you Software Szymon Rusinkiewicz for Trimesh2 Danil Kirsanov for Exact Geodesic 3D models Daniela Giorgi Raison d'être Remy of Ratatouille whose posing for [Joshi et al 2007] got me interested in barycentric coordinates

45 Thank you