Computational Design. Stelian Coros

Transcription

1 Computational Design Stelian Coros

2 Schedule for presentations February March April Send me: ASAP: 3 choices for dates + approximate topic (scheduling) 1-2 weeks before your presentation: list of papers you plan to talk about Day before each presentation: 3 questions for one of the papers that will be discussed

3 Next class Visit the Digital Fabrication Lab

4 Mini-assignment

5 Computational Design Need a parameterized model Mathematical formulation that predicts behavior of a system Examples? CAD systems Expose design parameters Use mathematical model to preview outcome What do you want from a CAD system?

6 Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration of design spaces Inverse design: automatically infer design parameters from functional specifications Optimization-based

7 Forward Design Editing 3D Models as an example

8 Editing 3D Models Options: Directly edit mesh vertices Finding the right level of abstraction is key!

9 Editing 3D Models Options: Directly edit mesh vertices Finding the right level of abstraction is key! Cage-based editing

10 Cage-based mesh editing Embed 3D model in a coarse mesh (cage) Edit cage vertices, deform model with it automatically

11 Cage-based mesh editing Computing Appropriate Weights Harmonic coordinates, mean value coordinates, bounded biharmonic weights, etc. Want: Smoothness Monotonicity Non-negativity Partition of unity Locality and sparsity Simple approach: Barycentric coordinates (volumetric cages)

12 Barycentric coordinates Ratio of areas (volumes in 3D) Conceptually the same in higher dimensions

13 Cage-based mesh editing can t precisely control surface properties

14 Editing 3D Models Options: Directly edit mesh vertices Finding the right level of abstraction is key! Cage-based editing Skeletal Rigs

15 Skeletal Rigs

16 Skeletal Rigs

17 Skeletal Rigs

18 Skeletal Rigs Recent research aims to automate each step construct rig compute weights choose transforms

19 Skeletal Rigs Recent research aims to automate each step construct rig compute weights choose transforms

20 From shape to skeleton Skeleton implies shape and shape implies skeleton Medial Axis (topological skeleton)

21 Surface flow degenerates to skeleton approximating medial axis

22 Medial axis [Au et al. 2008]

23 Skeletal Rigs Recent research aims to automate each step construct rig compute weights choose transforms

24 Weight computation The closer to a bone a vertex is, the larger the weight should be

25 Weight computation

26 Weight computation Want a measure of distance that is shape-aware!

27 Weight computation

28 Skeletal Rigs Recent research aims to automate each step construct rig compute weights choose transforms

29 Editing 3D Models Options: Directly edit mesh vertices Finding the right level of abstraction is key! Cage-based editing Skeletal Rigs High-level surface editing

30 As-Rigid-As-Possible Surface Modeling (Sorkine & Alexa, 2007) What do we want? Smooth deformations Precise control over specific features As-rigid-as-possible deformations for details Ease of use

31 As-Rigid-As-Possible Surface Modeling (Sorkine & Alexa, 2007) Main idea: Preserve shape of surface patches as much as possible (subject to user constraints) Surface patches should overlap to prevent bending

32 As-Rigid-As-Possible Surface Modeling (Sorkine & Alexa, 2007) Surface patches 1 ring neighbor for each vertex

33 As-Rigid-As-Possible Deformations Approach 1: differential coordinates the relative/differential coordinate vector average of the neighbors

34 Why differential coordinates? They represent the local detail / local shape description The direction approximates the normal The size approximates the mean curvature

35 Reconstruction Reconstruct position of each vertex using differential coordinates and neighborhood positions: Find v i such that as close as possible to

36 Reconstruction Or, in matrix form:

37 Reconstruction

38 Problems? Not rotation invariant!!! Can we do better?

39 v i v j 1 Rotation-invariant deformation energy Ask each 1-ring neighborhood to transform rigidly up to some rotation R v j 2 min ( v i v j ) Ri ( vi v j ) j N () i 2

40 Deformations and Rotations If v, v are known then R i is uniquely defined v j 2 v j2 v i v j1 v i v j1 R i Compute deformation gradient F s.t. V = F V Polar Decomposition: F = R D F = U W T ; R = UW T Positive semi-definite, symmetric matrix representing deformations (stretch, shear)

41 Energy formulation Can formulate overall energy as: n min ( v v ) R ( v v ) v i 1 j N ( i) i j i i j s. t. v c, j C j j 2 v and R treated as separate sets of variables

42 Iterative solver Alternating iterations Given initial guess v 0, find optimal rotations R i Easy to parallelize Keep R i fixed, minimize the energy by finding new v n min ( v v ) R ( v v ) v i 1 j N ( i) i j i i j 2

43 Iterative solver Alternating iterations Given initial guess v 0, find optimal rotations R i Easy to parallelize Keep R i fixed, minimize the energy by finding new v Lv b Uniform mesh Laplacian

44 Iterative solver Each iteration decreases the energy (or at least guarantees not to increase it) The matrix L stays fixed Precompute factorization back-substitute in each iteration (+ the SVD computations) Lv b

45 As-Rigid-As-Possible Surface Modeling

46 Surface vs volumetric models

47 Editing 3D Models Options: Directly edit mesh vertices Finding the right level of abstraction is key! Cage-based editing Skeletal Rigs High-level surface editing

48 Creating new models

49 Creating new models

50 Creating new models Finding the right level of abstraction is key Restrict design space to some extent Trade-off between flexibility and ease of use Alternatives? Explore design space

51 Design Space Exploration

52 Design Space Exploration Sample parameter space Poisson sampling Present design space in a manageable way Cluster similar designs Visualize designs exhibiting greatest variation Hierarchical refinement

53 Design Space Exploration - Examples Eric Brochu, Tyson Brochu and Nando de Freitas. A Bayesian Interactive Optimization Approach to Procedural Animation Design. ACM SIGGRAPH/Eurographics Symposium on Computer Animation, 2010

54 Design Space Exploration - Examples Many-Worlds Browsing for Control of Multibody Dynamics Twigg and James, SIGGRAPH 2007

55 Many Worlds Browsing Sampling Plausible Worlds (parameter choices) v t+1 ` v t compute and apply impulse [O Sullivan et al., 2003]

56 Many Worlds Browsing Interactive Browsing various criteria

57 Many Worlds Browsing

58 Questions?