FastLSM: Fast Lattice Shape Matching for Robust Real-Time Deformation Alec RiversDoug James Cornell University

Soft body simulation 2

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

SpeedStability Range of deformation Simplicity Physical accuracy Explicit methods √X√√√ Implicit methods; corotational; invertible FEM [Terzopoulos & Fleischer 1988; Mueller et al. 2002; Irving et al. 2004] X√√X√ Reduced models [Pentland & Williams 1989; Debunne et al. 2001; Grinspun et al. 2002; Barbic & James 2005] √√XX√ Shape matching [Mueller et al. 2005] √√X√X FastLSM √√√√X 3

Shape matching [Mueller et al. 2005] Particles at mesh vertices Save initial positions as rest configuration Move particles independently Match rest configuration to particles Push particles towards goal positions 4

Shape matching – cont. Quadratic shape matching Multiple clusters 5

Shape matching – limitations Limited range of deformation Can be slow with many clusters Boundary issues No interior volumes 6

Shape matching – limitations Limited range of deformation Can be slow with many clusters Boundary issues No interior volumes 6

Shape matching – limitations Limited range of deformation Can be slow with many clusters Boundary issues No interior volumes 6

Our contributions Lattice shape matching –New framework for shape matching-based deformation –Addresses many of these concerns Fast summation optimization 7

Voxelize mesh Many overlapping small regions Lattice shape matching 8

One region centered at each lattice index Breadth-first search to depth w Region generation w = 1 w = 2 w = 3w = 4 9

Dynamics: Goal position for each particle is average of goal positions relative to each region Flexibility: –Many rigid regions > few quadratic LSM Dynamics 10

Shape matching comparison 11

Material modeling Control rigidity by tuning w Can also tune timesteps / frame 12

Optimization Cost scales with number of particles in each region –O(w 3 ) Define O(1) fast summation operator: 13

Optimization Cost scales with number of particles in each region –O(w 3 ) Define O(1) fast summation operator: ––– O(w 3 ) Naïve O(w) Intermediate ······ O(1) FastLSM w 13

Fast summation 14

… 6.1 … … Fast summation 14

… 6.1 … … Fast summation 14

… 6.1 … … … = … = 39.7 Fast summation 14

… 6.1 … … = = …= 24.6 Fast summation 14

Fast summation 14

Fast summation 14

Fast summation 15

Fast summation – 3D Regions Plates Bars 16

Fast summation After collapsing: n regions,w plates each ~n unique plates,w bars each ~n unique bars,w particles each Total cost: –nw + nw + nw = 3nw = O(nw) –O(w) per region (instead of O(w 3 ) ) Related work: [Crow 1984; Weiss 2006] 17

Constant-time fast summation 18

Constant-time fast summation 18

Constant-time fast summation With constant time: –Each bar, plate, region: add + subtract (2 flops) Total cost: –2n + 2n + 2n = 6n –O(n), independent of w 19

Rigid shape matching using fast summations Definitions: –x i 0 Rest positions –x i Deformed positions –m i Particle masses –m i m i / |R i | –Mr= Translation: Rotation: Goals: 20

Extensions Fast polar decompositions –Track eigenvectors of A r to warm start –Average cost decrease: 2.5 μs → 0.45 μs Fast damping –[Mueller et al. 2006] –Accelerated for LSM using fast summation 21

Timings breakdown Fast summation Polar decompositions Shape matching Damping 22

Extensions – cont. Fracture –High range of deformation allows fracture –Sever links based on strain tolerances Hardware-accelerated rendering –Store mesh in GPU –Upload just particle positions each frame –Deform geometry on GPU ; 23

Results: Complex objects Solid Buddha: 57,626 particles w = ms / timestep 1,680 ms / frame Shell Buddha: 19,957 particles w = ms / timestep 484 ms / frame 24

Results: Complex objects 25

Results: Articulated characters 2,570 particles w = 2 15 ms / timestep 26

Results: Articulated characters 27

Results: Speed 150 particles w = ms / timestep 28

Results: Speed 5 FPS; 0.28 ms/penguin (w=2) 150 penguins w/ 150 particles/penguin (22,500 particles total) “Peng-chinko” 29

Conclusion Advantages: –Fast –Large range of deformation –Stable –Easy to implement 30

Conclusion Disadvantages: –Not physically accurate –Poor control over material properties –Does not conserve volume 31

Future work Theoretical foundations –Material modeling Different particle frameworks –Irregular samplings Apply FastLSM smoothing to other geometric problems 32

Demos and source code available at

Acknowledgements Jernej Barbič, Chris Twigg, Chris Cameron, Giovanni Tummarello Anonymous reviewers Funding and support: –National Science Foundation (CAREER, EMT) –Alfred P. Sloan Foundation –NIH –Pixar –NVIDIA (hardware, graduate fellowship) –Intel –Autodesk –The Boeing Company –The Link Foundation 34

Thanks! Demos and source code available at