Geometric Modeling and Processing

Transcription

1 Geometric Modeling and Processing Tutorial of 3DIM&PVT 2011 (Hangzhou, China) May 16, 2011

2 6. Mesh Simplification

3 Problems High resolution meshes becoming increasingly available 3D active scanners Computer vision methods Meshes extracted from volumetric data Terrain data

4 Simplification Examples

5 Simplification Applications Level-of-detail modeling Generate a family of models for the same object with different polygon counts Select the appropriate model based on estimates of the object's projected size Simulation proxies Run the simulation on a simplified model Interpolate results across a more complicated model to be used for rendering

6 Tradeoff Size Error

7 Level of Detail (LOD) Refined mesh for close objects Simplified mesh for far

8 Performance Requirements Offline Generate model at given level(s) of detail Focus on quality Real-time Generate model at given level(s) of detail Focus on speed Requires preprocessing Time/space/quality tradeoff

9 Quality

10 Algorithms Vertex Removal/Decimation Edge Collapse Appearance-Preserving Simplification

11 Methodology Sequence of local operations Involve near neighbors - only small patch affected in each operation Each operation introduces error Find and apply operation which introduces the least error

12 Simplification Operations (1) Decimation Vertex removal: v v-1 f f-2 Remaining vertices - subset of original vertex set

13 Simplification Operations (2) Decimation Edge collapse v v-1 f f-2 Vertices may move

14 Simplification Operations (3) Decimation Triangle collapse v v-2 f f-4 Vertices may move

15 Simplification Operations (4) Contraction Pair contraction Vertices may move

16 Simplification Operations (5) Contraction Cluster contraction (set of vertices) Vertices may move

17 Error Control Local error: Compare new patch with previous iteration Fast Accumulates error Memory-less Global error: Compare new patch with original mesh Slow Better quality control Can be used as termination condition Must remember the original mesh throughout the algorithm

18 Local vs. Global Error 2000 faces 488 faces 488 faces

19 Local Simplification Algorithm Repeat Select the element with minimal error Perform simplification operation (remove/contract) Update error (local/global) Until mesh size / quality is achieved

20 Simplification Error Metrics

21 Implementation Details Vertices/Edges/Faces data structure Easy access from each element to neighboring elements Use priority queue (e.g. heap) Fast access to element with minimal error Fast update

22 Vertex Clustering Merge all vertices within the same cell

23 Steps Partition space into cells grids [Rossignac-Borrel], spheres [Low-Tan], octrees,... Merge all the vertices falling within a single cell together and replace with a single representative vertex Form triangles with resulting vertices that attempt to preserve the original topology

24 Pros and Cons Advantages Does not require manifold models Can handle multiple objects Fast Disadvantages Low quality Hard to control

25 Questions?

26 7. Surface Editing

27 Building Shapes 1 From zero Create a shape by extrusion or revolution Extrusion (sweeping) Revolution

28 Building Shapes 2 Selecting a base shape Create a shape by extrusion or revolution Editing mesh Selecting and editing vertices Displacing areas of the mesh

29

30 Mesh Deformation Interactive mesh editing Animation Modeling Modeling new models via editing given models

31 Methods Deform the vertex coordinates directly Drag vertices Deform the control points Bézier, NURBS Deform the embedded space FFD, Axial deformation Dem o

32 Vertex Dragging

33 Global Deformation [Barr 84] taper twist bend

34 7.1 Free-form Deformation (FFD)

35 Free-form Deformation (FFD) [Sederberg et al. 86] Embed the object into a domain that is more easily parametrized than the object. Advantages: You can deform arbitrary objects Independent of object representation

36

37 Axial Deformation (AxDf) [1994] Object Define an axis Deform axis Deformed object

38 AxDf

39 Wires [1999]

40 Mean Value Coordinates Extended barycentric coordinates Cage-based editing [2005]

41 7.2 Multiresolution Editing

42 Multiresolution Editing Frequency decomposition Change low frequencies Add high frequency details

43

44 Decimation Editing Reconstruction

45 7.3 Discrete Differential Coordinates Detail Preserving Representation

46 Differential Coordinates 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

47 What s are Details? Detail = surface smooth (surface) Smoothing = averaging

48 Differential Coordinates Differential coordinates are defined by the discrete Laplacian operator: v w v i i j j jn () i average of the neighbors

49 What s the Difference? Absolute Coordinate Relative Coordinate v ( x, y, z ) i i i i v w v i j j i jn () i

50 Weighting Schemes Uniform weight (geometry oblivious) Cotangent weight (geometry aware) Normalization wj 1 w (cot cot ) j w j w j wj j

51 Geometric Meaning DCs represent the local detail / local shape description The direction approximates the normal The size approximates the mean curvature 1 d δi vi v v v i 1 len( ) v 1 lim v vds H ( v ) n len( ) v N() i len( ) 0 i i i v i ds

52 Laplacian Matrix The transition between the and xyz is linear: L 1 2 n x x x ( ) 1 ( ) 2 ( ) x x x n 0 i ij d i j D otherwise otherwise j N i A ij 0 ) ( 1 1 L I D A

53 Reconstruction From relative coordinates to absolute coordinates. Solving a sparse linear system Lv

54 Reconstruction Soft constraints T L Lv T L

55 Variational Viewpoint Laplacian Approximation Gradient Approximation

56 Laplacian matrix The transition between the and xyz is linear: L v x = x L v y = y δ w v v i ij i j jn () i L v z = z

57 Basic properties Rank(L) = n-c (n-1 for connected meshes) We can reconstruct the xyz geometry from delta up to translation Lx δ

58 Laplacian Mesh Editing

59 Laplacian Editing Local detail representation enables detail preservation through various modeling tasks Representation with sparse matrices Efficient linear surface reconstruction

60 Editing framework The spatial constraints will serve as modeling constraints Reconstruct the surface every time the modeling constraints are changed Detail constraints: Modeling constraints: Lx δ x c, j { j, j, j } j j 1 2 k

61 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

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

63 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

64 User Interfaces ROI is bounded by a belt (static anchors) Manipulation through handle(s)

65 Results

66 Results

67 Lots of Invariance Explicit transformation of the differential coordinates prior to surface reconstruction Lipman, Sorkine, Cohen-Or, Levin, Rössl and Seidel [SMI 04], Differential Coordinates for Interactive Mesh Editing, Estimation of rotations from naive reconstruction Yu, Zhou, Xu, Shi, Bao, Guo and Shum [SIGGRAPH 04], Mesh Editing With Poisson-Based Gradient Field Manipulation, Propagation of handle transformation to the rest of the ROI using geodesic distances Zayer, Rössl, Karni and Seidel [EG 05], Harmonic Guidance for Surface Deformation, Propagation of handle transformation to the rest of the ROI using harmonic functions

68 Silhouette Sketch-based Editing [Siggraph 05]

69 Sketching UI Silhouette sketching Feature Sketching sketching a shape can be interpreted as inverse Non- Photorealistic Rendering A sketch-based modeling interface which uses silhouettes and sketches as input, and produces contours, ridges and ravines

70 Silhouette Sketching Approximate sketching Balance weighting between detail and positional constraints

71 Silhouette Sketching Approximate sketching Balance weighting between detail and positional constraints

72 Mesh Editing Methods (Recap) FFD [1986, Sederberg and Parry] Free-form deformation of solid geometric models [1994, Lazarus et al.] Axial deformations: An intuitive deformation technique [2005, Juet al.] Mean value coordinates for closed triangular meshes Multiresolution editing [1995, Eck et al.] Multiresolutionanalysis of arbitrary meshes [1998, Kobbelt et al.] Interactive multi-resolution modeling on arbitrary meshes [2003, Botsch and Kobbelt] Multiresolution surface representation based on displacement volumes Differential editing [2004, Yu et al.] Mesh editing with Poisson-based gradient field manipulation [2004, Sorkineet al.] Laplacian surface editing [2005, Lipman et al.] Linear rotation-invariant coordinates for meshes

73 1. FFD (cage-based) Manipulating Embedding object Pros Simple and intuitive Cons Hard to preserve details Suitable for smooth objects Invariant variables Parametric coordinates

74 2. Multiresolution Editing Manipulating Simplified model Pros Preserving details, scalable Cons Instable reconstruction for large deformation Resampling problem Invariant variables Detail information

75 3. Differential Editing Manipulating Handles Pros Preserving details Enhance user interaction Cons Much computation cost Invariant variables Differential information

76 Questions?