Bounding Volume Hierarchy

1
Bounding Volume Hierarchy

• Efficient Distance Computation Between
  Non-Convex Objects
• Sean Quinlan
• Stanford, 1994
• Presented
• by Mathieu Brédif

2
Simplifying Assumptions

• Surface analysis only
• Decomposition of objects into sets of convex
  surfaces
• Easy in graphics all surfaces are composed
  of triangles
• Not necessarily easy in general

3
Basis Convex Objects

• Efficient distance computation between
  convex polyhedra
• Decomposition of two objects into m and n
  convex polyhedra.
• O(mn) distance computation

4
Algorithm Overview

• PreprocessingApproximation by Bounding Volume
  Hierarchies of Spheres
• Query Branch and Bound in both trees to find
  minimum distance

5
The Sphere Hierarchy

• Binary tree
• Root node objects bounding sphere
• Leaf nodes are tiny spheres their union
  approximates the objects surface
• Every nodes sphere contains entirely its
  descendents

6
Creating the Leaves

• Cover the object surface with tiny spheres (leaf
  nodes).
• Radius user-determined.

7
Creating a balanced Tree

• Divide halfway in the major axis of a rectangular
  bounding box.
• Recurse until 1 leaf per set
• (Root down to leaves)

8
Decorating the Tree with Spheres

• Leaves up to root create bounding spheres for
  each node.
• Best of 2 methods
• Find the minimal sphere that contains the two
  spheres of the child nodes
• Determine a sphere directly from the leaf nodes
  descended from this node

9
Resulting Tree Representation
10
Resulting Tree Representation
11
Resulting Tree Representation
12
Resulting Tree Representation
13
Query Computing Distances

• Depth-first search on the binary tree
• Keep an updated minimum distance
• Depth-first ? more pruning in search
• Prune search on branches that wont reduce
  minimum distance
• Once leaf node is reached, examine underlying
  convex polygon for exact distance (former method)

14
Simple Example

• Set initial distance value to infinity

Start at the root node. 20 lt infinity, so
continue searching
15
Simple Example

• Set initial distance value to infinity

Start at the root node. 20 lt infinity, so
continue searching.
40 lt infinity, so continue searching recursively.

• Choose the nearest of the two child spheres to
  search first

16
Simple Example

• Eventually search reaches a leaf node

40 lt infinity examine the polygon to which the
leaf node is attached.
17
Simple Example

• Eventually search reaches a leaf node

Call algorithm to find exact distance to the
segment. Replace infinity with new minimum
distance (42 in this case).
40 lt infinity examine the segment to which the
leaf node is attached.
18
Simple Example

• Continue depth-first search

45 gt 42 dont search this branch any further
19
Simple Example

• Continue depth-first search

60 gt 42 we can prune this half of our tree from
the search
45 gt 42 dont search this branch any further
20
Efficiency Building the Tree

• Roughly balanced binary tree
• Expected time O(n log n)
• Worst case time O(n 2)
• (n leaves nodes)
• Precomputed Built only once!

21
Efficiency Searching the Tree

• Full search
• O(n) to traverse the tree
• O(p) time to compute distance to each polygon
  in the underlying model
• The algorithm allows a pruned search
• Worst case no change! (close objects)
• Best case O(log n) a single polygon comparison

22
Collision Detection

• BVH Two sphere trees
• 2 spheres intersects gt descend further
• May need to descend to polygon level (line
  segment in 2D)

23
Collision Detection

• Quick
• Widely separated
• Collided
• Slower
• Close but not overlapping
• Close at many points

24
Extension Multiple Trees

• If 2nd object is not a single point
• Start at root of both trees
• Branch split the larger sphere
• The closest child to the unsplit sphere is
  searched first
• Deformable Object precomputation of solid links,
  meta-tree computation before queries

25
Extension Relative Error

• New minimum distance d Registration
• d (1-a)d
• 0?alt1 user-determined relative error
• d0 iff d0 correct collision detection
• Better pruning gt performance speedup
• Exact distance in d,d/(1-a)
• Trade off efficiency/exactitude

26
Empirical Results

• Relative error of 20
• more pruning in search
• speedup of 2 orders of magnitude
• Objects close together
• less pruning in search
• less efficient

27
Conclusions

• Simple and intuitive way to speed up distance
  calculations
• Gives more information than collision detection
  algorithms
• Built on top of algorithms that compute distances
  between convex polygons
• Applications to path planning (robotobject1,
  obstaclesobject2)