Laplacian Meshes. COS 526 Fall 2016 Slides from Olga Sorkine and Yaron Lipman

Transcription

1 Laplacian Meshes COS 526 Fall 2016 Slides from Olga Sorkine and Yaron Lipman

2 Outline Differential surface representation Ideas and applications Compact shape representation Mesh editing and manipulation Membrane and flattening Generalizing Fourier basis for surfaces

3 Motivation Meshes are great, but: Geometry is represented in a global coordinate system Single Cartesian coordinate of a vertex doesn t say much z x y

4 Laplacian Mesh Editing Meshes are difficult to edit

5 Motivation Meshes are difficult to edit

6 Motivation Meshes are difficult to edit

7 Differential coordinates Represent a point relative to it s neighbors. Represent local detail at each surface point better describe the shape Linear transition from global to differential Useful for operations on surfaces where surface details are important

8 Differential coordinates Local control for mesh morphing, Alexa 01 Detail = surface smooth(surface) Smoothing = averaging 1 δ v v i i j di j N () i 1 δ v v i i j jn () i di

9 Connection to the smooth case The direction of i approximates the normal The size approximates the mean curvature 1 d δ v v v v i i v N() i len( ) 0 i 1 len( ) v 1 lim v v ds H ( v ) n len( ) i i i v i ds

10 Laplacian matrix

11 Weighting schemes Ignore geometry umbrella : w ij = 1 Integrate over circle around vertex mean value : w ij = tan ij /2 + tan ij+1 /2 Integrate over Voronoi region of vertex cotangent : w ij = cot a ij + cot b ij i jn () i w ij i j jn () i v w v ij n a ij ij b ij

12 Laplacian mesh Vertex positions are represented by Laplacian coordinates ( x y z ) L v x = x L v y = y δ w v v i ij i j jn () i L v z = z

13 Basic properties rank(l) = n c (n 1 for connected meshes) We can reconstruct the xyz geometry from up to translation Lx 1 δ L v x = x c x fix L v y = y 1 c y L v z = z 1 c z

14 Reconstruction L v x = x 1 1 c x e x edit fix 1 L v y = y c y 1 e y L v z = z 1 1 c z e z

15 Reconstruction L v x = x 1 c x 1 e x x arg min Lx δx xk c x s1 k 2 2 k

16 Reconstruction L v x = x 1 c x 1 e x A x = b Normal Equations: A T A x = x = A T b (A T A) -1 compute once A T b

17 Cool underlying idea Mesh vertex positions are defined by minimizer of an objective function L v x = x 1 c x 1 e x x arg min Lx δx xk c x s1 k 2 2 k

18 What we have so far Laplacian coordinates Local representation Translation-invariant Linear transition from to xyz can constrain more that 1 vertex least-squares solution

19 Editing using differential coordinates The editing process from the user s point of view: 1) First, a ROI, anchors and a handle vertex should be set. 2) Then the edit is Performed By moving this vertex.

20 Editing using differential coordinates The user moves the handle and interactively the surface changes. The stationary anchors are responsible for smooth transition of the edited part to the rest of the mesh. This is done using increasing weight with geodesic distance in the soft spatial equations.

21 Mesh Editing Example

22 Mesh Editing Example

23 Mesh Editing Example

24 Mesh Editing Example

25 What else can we do with it?.

26 Parameterization Use zero Laplacians. L V 0 = c 1 c 2 c k In 2D:

27 Texture Mapping

28 Texture Mapping [Piponi2000]

29 Feature Preserving Smoothing Weighted positional and smoothing constraints w 1w2w3 V = P L 0

30 Feature Preserving Smoothing Weighted positional and smoothing constraints Original Smoothed

31 Detail transfer Peel the coating of one surface and transfer to another

32 Detail transfer

33 Detail transfer

34 Mixing Laplacians Taking weighted average of i and i

35 Mesh transplanting Geometrical stitching via Laplacian mixing

36 Mesh transplanting Details gradually change in the transition area

37 Mesh transplanting Details gradually change in the transition area

38 The End