Efficient Bounds for PointBased Animations

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Efficient Bounds for Point-Based Animations

• Submitted to Point-Based Graphics 2007
• Denis Steinemann, Miguel Otaduy

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Overview

• Motivation
• Related Work
• Point-Based Animation, BVH
• Computing Bounds
• General
• Efficient evaluation
• Results
• Where to go next?

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Motivation

• Collision detection a key part of animation of
  deformable objects
• In general, still rather expensive
• Accelerate with spatial partioning, BVH
• Data structures must be updated to account for
  deformation, cost depends on surface complexity

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Motivation

• In many deformation models, surface deformation
  defined by small set of n degrees-of-freedom
• Reduced linear deformations
• Skinning
• Low-res FEM
• Point-based animation
• Exploit it to update BVH in O(n)

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Related Work

• Reduced linear deformations (BD-Tree JPo4)
• p pUq
• Compute conservative bound per deformation field,
  add them up ? tight only for low of def. fields
• Linear-blend Skinning (KZ05)
• Few joints that define deformation
• Vertices convex combinations of joint transforms
• Exploit limited convex weights to get tight
  bounds

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Related Work

• Low-res FEM (OGRG07)
• Surface embedded in tet mesh defining deformation
• Surface vertices convex combinations of
  simulation nodes
• Point-Based Animation (AKP05)
• Adaptation of BD-Tree
• Tightness depends on dofs

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Overview

• Motivation
• Related Work
• Point-Based Animation, BVH
• Computing Bounds
• General
• Efficient evaluation
• Results
• Where to go next?

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Point-Based Animation

• Elastic deformations derived from continuum
  mechanics MKN04
• Deformation field defined at a discrete set of
  simulation nodes
• Position of a vertex/surfel

deformation field u
x
xu
simulation nodes xj
wkj in 0,1, add up to 1
vertices vk
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Point-Based Animation

• As wk is a vector of convex weights
• Deformed vertex convex combination of n affine
  transforms of undeformed vertex

Aj
tj
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Bounding Volume Hierarchy

• Construction Initialization
• Use AABBs
• Top-down splitting in reference position
• For every AABB, store
• n influencing simulation nodes
• Do not store m vertices
• Center, extensions in ref. pos
• Max/min weight per node

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Bounding Volume Hierarchy

• Run-time update
• Usually bottom-up
• O(verts)
• On-demand (top-down)
• Update AABB before testing for intersection
• O(sim.nodes)

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Overview

• Motivation
• Related Work
• Point-Based Animation, BVH
• Computing Bounds
• General
• Efficient evaluation
• Results
• Where to go next?

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Computing Bounds

• Goal
• We want to tightly re-fit any AABB using only
  information from simulation nodes and not the
  surface itself

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Definitions

• Set of convex weights in Rn

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Definitions

• Definition 1 Transformed AABB
• Lemma 1 Transf. vertex bounded by transf. AABB

Parallelepiped def. by transformed corners
vk0
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Definitions

• Definition 2 Convex combination of AABBs
• Lemma 2 Convex combination of AABBs Bj
  another AABB with extrema defined by same convex
  combination of extrema of Bj

B1
B2
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Overview

• Motivation
• Related Work
• Point-Based Animation, BVH
• Computing Bounds
• General
• Efficient evaluation
• Results
• Where to go next?

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AABB-Refitting

• Bounding a single vertex

Lemma 1
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AABB-Refitting

• Bounding a set of vertices
• Bound convex hull
• Finally,

From prev. slide
Swapping sums
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AABB-Refitting

• Bounding a set of vertices
• Bound convex hull
• Finally,

From prev. slide
Swapping sums
Simply another AABB! (Lemma 2)
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AABB-Refitting

• We could bound vertices in O(n) by computing
  and simply bound them
• Assuming
• Results in loose bounds
• Exploit
• cannot have any value in 0,1
• Bounded by

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Limited Convex Combinations

• Set of limited convex weights
• Is a subset of Wn and Rn
• Our vector of convex weights

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Limited Convex Combinations

• How does Wn look?
• Example for n3

1
0
1
0
1
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Limited Convex Weight Space

• Wn Limited Convex Weight Space
• Hyperplane of convex weights
• bounded by n sets of 2 parallel hyperplanes
• Can be represented as convex comb. of corners

Weight vectors of vertices
corner
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Constructing Corners

• Start with (n-1)-dim simplex representing Wn
• Cut off min-hyperplanes
• Results in a smaller (n-1)-d simplex
• Cut n corners of simplex with max-hyperplanes
• Cut n-1 lines meeting at corner ? n(n-1) corners
  total
• Each corner defined by 1 , (n-2) , 1x

n4
n3
n5
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Evaluating Corners

• Remember Corners simply represent some weight
  vectors in
• We could evaluate all corners and bound resulting
  AABBs
• Is related to what KZ05 do
• Would have to evaluate O(n2) corners ? too
  expensive for our deformation model!

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Evaluating Corners

• Since we use AABB, were only interested in
  extremes
• Lemma 2 simply another AABB, where
• Our problem reduces to finding the two extreme
  scalar (!) values along each axis x,y,z, using
  extreme corners

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Evaluating Extreme Corners

• Sort scalar values along

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Evaluating Extreme Corners

• Sort scalar values along

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Evaluating Extreme Corners

• Sort scalar values along

Extreme corner
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Evaluating Extreme Corners

• Finally, evaluate extreme along
• Same for max/min, just switch signs

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Evaluating Extreme Corners

• It turns out that
• In Wn, weight vector of extreme corner along
  direction is defined as
• j1 and j2 indices of largest and second largest
  values ? can be found in O(n)

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Algorithm Summary

• Given
• a set of m vertices (v1vm)
• bounded by AABB B0
• influenced by n simulation nodes
• Transform B0 with Tj for every node ? Bj
• Bound Bj with a new AABB
• For each direction x,y,z, find with the
  largest extrema, evaluate bounds using extreme
  corners

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Overview

• Motivation
• Related Work
• Point-Based Animation, BVH
• Computing Bounds
• General
• Efficient evaluation
• Results
• Where to go next?

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Results

• Tightness of AABB

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Results

• Tightness

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Results

• Scalability
• Evaluating root of BVH (in microsecs)

independent of surface complexity
O(n)
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Results

• Performance
• Colliding santas

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Results

• Performance (in ms)

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Conclusion

• Efficiently Update AABBs depending on dof of
  deformation model
• Tighter bounds than for previous approaches
• Collision detection with speed-up of more than
  one order of magnitude

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Where to go next?

• Is the approach more general?
• Does it have to be convex comb. of affine
  transforms?
• Original BV can be any convex set
• We can transform corners of BV, will bound
  transformed point
• Find extreme along arbitrary direction in Rn
• Project point onto direction
• Results in convex combination of scalar values

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Where to go next?

• Corners, extreme corner evaluation
• Will this always work? Sort of proved it
• Relation to simplex algorithm?
• Problem of finding extrema of a limited convex
  combination can be transformed to general LP
• maximize

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Where to go next?

• General linear combinations
• Affine combinations?
• Any linear combination?
• Bound set of m points in Rn with convex polytope,
  define them as convex combinations of polytope
  corners
• How many corners to transform?
• Will extreme corner evaluation still work?
• Examples RBF, PCA, Deformable Modelling,,?

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Where to go next?

• Self-collisions
• Can we exploit convexity in our approach to
  efficiently prune BVH in collision queries?
• Other applications
• Where we are interested in computing bounds using
  reduced-dim space
• Screen-space bounds? Projections?
• ???

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• sf