Segmentation & Constraints

Transcription

1 Siggraph Course Mesh Parameterization Theory and Practice Segmentation & Constraints

2 Segmentation Necessary for closed and high genus meshes Reduce parametric distortion Chartification Texture Atlas

3 Segmentation Goals Large Charts Low Distortion

4 Segmentation Single Chart Gu et al Sheffer & Hart 2002 High genus surface topological disk Improve the cuts

5 Segmentation Multiple Charts MCGIM LSCM Iso-charts D-charts

6 Iso-charts Fast and effective atlas generator large charts with bounded parametric distortion UVAtlas in DirectX 9 D3DXUVAtlasCreate, D3DXUVAtlasPartition, D3DXUVAtlasPack Light map compression, Halo

7 Iso-charts Demo

8 Iso-charts Overview Surface spectral analysis Stretch optimization Surface spectral clustering Optimize chart boundary Recursively split charts until stretch criterion is met Atlas packing

9 Iso-charts Spectral Analysis x N y N x 2 y 2 d 1 N d 2 N d 12 x 1 min N N i1 j1 ( y i d Geodesic Distance Distortion (GDD) y j y 1 ij 2 )

10 Iso-charts Spectral Analysis d 1 N x 1 d 2 N d 12 x N x 2 D N d d d N1 d d d N 2 d d d 2 1N 2 2N 2 NN Step 1: Construct the matrix of squared distance Surazhsky et al. 2005

11 Iso-charts Spectral Analysis x N 1 d 1 N d 2 N d 12 x 2 B N where J N 2 J I N D 1 N N J T 11 N x 1 Step 2: Perform centering and normalization

12 Iso-charts Spectral Analysis Step 3: Perform eigen-analysis to get the embedding coordinates T n n T T N v v v y y y x 1 x 2 x N N d 1 N d 2 d 12

13 Iso-charts Spectral Analysis Zigelman et al Produces triangle flips Only handles single-chart models

14 Iso-charts Stretch Optimization Sander et al nonlinear optimization 2D texture domain singular values: γ, Γ surface in 3D

15 Iso-charts Stretch Optimization [Sander01], L 2 = 1.04, 222s [Sander02], L 2 = 1.03, 39s Spectral, L 2 = 1.04, 2s Spectral+Optimization, L 2 = 1.03, 6s

16 Iso-charts Spectral Clustering Analysis Clustering

17 Iso-charts Boundary Optimization S T Create nonjaggy cut, through crease edges Minimize embedding distortion

18 Iso-charts Atlas Packing A Tetris algorithm [Lévy02] V U

19 Iso-charts Acceleration Landmark algorithm [Silva et al. 2003] reduce the size of the eigenanalysis Only compute the top 10 eigenvalues over 95% of the squared energy

20 Constraints Enforce specific point-to-point correspondences 3D mesh 2D image constrained texture mapping

21 Constrained Texture Mapping Soft constraints [Lévy 2001] approximate constraints generate foldovers for large constraint sets Hard constraints Lagrange multipliers [Desbrun et al. 2002] Matchmaker [Kraevoy et al. 2003] TextureMontage [Zhou et al. 2005]

22 Matchmaker Key Idea Reduce constrained parameterization problem to unconstrained one by Split mesh into patches s.t. constrained vertices V F are on patch boundaries Split 2D domain into matching convex pieces s.t. matching positions P F are on their boundaries Map each patch to corresponding convex using barycentric embedding 4 3

23 Matchmaker Algorithm 1. Input: 3D mesh + (V F, P F ) 2. Compute unconstrained 2D embedding M 3. Virtual boundary: 1. Triangulate region between M and its bounding rectangle 2. M*= M+new triangles mesh+v F P F 3. Add boundary vertices to V F M M*

24 Matchmaker Algorithm 4. Find matching triangulations Adding non-intersecting edges/paths M* T M* (V F ) P F T(P F )

25 Matchmaker Algorithm Path matching Compute all shortest paths between vertices in V F While paths exist Get shortest path s=(v i,v j ) v k p k Test if s is legal If legal (p i,p j ) does not intersect existing edges in T(P F ) (v i,v j ) does not block future paths Add s to T M* (V F ) & (p i,p j ) to T(P F ) Recompute non-tested paths that intersect s T M* (V F ) T(P F )

26 Matchmaker Algorithm Introducing Steiner vertices allow the creation of a valid path between any pairs of features Perform Dijkstra searches on both the mesh vertices and the edge midpoints T(P F ) Schreiner et al T M* (V F )

27 Matchmaker Algorithm 5. Map each patch S i T M* (V F ) to matching triangle T i T(P F ) using barycentric coordinates T M* (V F ) T(P F ) Barycentric mapping M Texture using M

28 Matchmaker Algorithm 6. Constrained smoothing M Minimize metric distortion compared to original 3D mesh Maintain validity M Smoothed M

29 Matchmaker Results + =

30 TextureMontage Texturing arbitrary surface from multiple images Stanford Bunny 6 photographs of another bunny

31 TextureMontage Key Issues Texturing arbitrary surface from multiple images Partition the model and images automatically and simultaneously Ensure continuity across patch boundaries with few correspondences Fill in non-texture areas

32 TextureMontage Algorithm 1. Preprocessing Adjust exposure differences and color balances Segment background regions (e.g. LazySnapping [Li04]) Compute and store the distance to the nearest non-background pixel for each pixel

33 TextureMontage Algorithm 2. Partition mesh and images Add valid path-edge pairs (similar to Matchmaker) Prevent path-edge pairs from crossing background regions

34 TextureMontage Algorithm 3. Generate progressive mesh Half-edge collapse simplification [Hoppe 96] v j v i v j

35 TextureMontage Algorithm 4. Create coarse texture map Map triangles of the base mesh to corresponding texture triangles in images

36 TextureMontage Algorithm 5. Coarse-to-fine map construction Insert deleted vertices back one by one v j v j v i For inner vertices, minimize geometric stretch energy [Sander et al. 2001] For boundary vertices, minimize texture mismatch

37 TextureMontage Algorithm Texture coords optimization for boundary vertices v m p v j v j v i v k v l v m p v i p v k v l v j p v j G A (p) G B (p) v i v k Image 1 Image 2

38 TextureMontage Algorithm Texture coords optimization for boundary vertices Evaluate texture mismatch energy Minimize both geometric stretch and texture mismatch Fix one border and optimize the other n k k k k k tex s s s s E 1 ' 2 ' ) ( G ) G ( ) (1 ) ( I ) ( I E geo E tex E ) 1 (

39 TextureMontage Algorithm Texture coords optimization for boundary vertices Original Model Unconstrained Optimization Color Matching Enforced Color and Gradient

40 TextureMontage Algorithm 6. Surface texture inpainting Unassigned region due to lack of correspondences

41 TextureMontage Algorithm 6. Surface texture inpainting Smoothly fill in hole regions based on Poisson equation s.t. * f d f f f : unknown color values of vertices in hole regions [Polthier and Preuss 2000] [Pérez et al. 2003] f * d : : : color values of vertices around hole regions Laplace operator a controlling scalar field

42 TextureMontage Algorithm 6. Surface texture inpainting (a) (b) (c)

43 TextureMontage Algorithm 7. Atlas packing (UVAtlas)

44 TextureMontage Demo

45 Siggraph Course Mesh Parameterization Theory and Practice Segmentation & Constraints