2.6. BOUNDING VOLUME HIERARCHIES
Overview of different forms of bounding volume hierarchy
Exploration of different types of bounding volume hierarchy
Bounding Volume Hierarchies
Given a total of n objects, a simple brute force approach towards
producing a list of intersecting objects is to consider all object pairings (i.e.
n2/2 comparisons).
Assuming each object can be described in terms of one of more bounding
volumes, by arranging the object’s bounding volumes into a tree-based
hierarchy (known as a bounding volume hierarchy), a logarithmic order
time complexity can be obtained.
BVH Design Issues: Desirable Characteristics
Desirable characteristics of a game-oriented BVH are likely to include:
• The nodes contained in any given sub-tree should be spatially ‘close’ to
each other.
• Each node in the hierarchy should be of minimal volume.
• Pruning a node near the root of the tree should
remove more objects from further
consideration than removal of a deeper node.
• The volume of overlap between sibling nodes
should be minimal.
• The hierarchy should be balanced with respect
to both its node structure and its content.
• The worst-case time for queries should not be
much worse than the average-case query time.
BVH Building Strategies
As the number of possible BVH trees grows exponentially as the number of
objects increase it is not possible to use an exhaustive (optimal) search.
Instead, heuristic rules are used to direct the BVH construction (selecting
the best option from a small number of hopefully near optimal options at
each step).
There are three main categories of tree construction methods:
•
top-down
•
bottom-up
•
insertion
BVH Building Strategies
• Top-down: partitioning input set into two
(or more) subsets, introduce bounds, and
recursing over bounded subsets (typically
does not produce best trees, but is easy to
implement and hence popular).
• Bottom-up: start with leaves of the tree,
group two (or more) to form a bounding
node, progressively grouping bounded
volumes until a single node is reached
(typically produces best quality BVH, but
more difficult to implement).
• Insertion: incrementally insert an object
into the tree so as to minimize some
insertion cost measurement
Forms of BVH Construction
Consult the recommended course text for full details of the
following approaches:
•
•
•
Top-down Construction
Bottom-up Construction
Incremental (Insertion) Construction
Top-down construction is explored next by way of an
example.
Top-down construction of a bounding volume hierarchy
Top-down Construction of a BVH
The process of top-down construction can be algorithmically described as:
void BuildTopDownBVH(BVHNode treeNode, ArrayList objects)
{
Node newNode = new Node(); Create an empty node and add to the tree
treeNode.add(newNode);
Build a suitable bounding volume around
the specified object list
newNode.BoundingVolume = ComputeBoundingVolume(objects);
if( objects.Count <= MIN_OBJECTS_PER_LEAF ) { If needed, simply
newNode.Type = LEAF;
store the object list
within this node
newNode. Objects = objects;
} else {
newNode.Type = NODE;
Sort and split the objects into two partitions
int splitIdx = RearrangeAndPartitionObjects(objects);
BuildTopDownBVH( Recursively build the BVH along the defined left and right sub-trees
newNode.LeftTree, objects.Subset(0,splitIdx));
BuildTopDownBVH(
newNode.RightTree, objects.Subset(splitIdx,objects.Count);
}
}
Top-down partitioning strategy
Other than selecting which bounding volume to use, the top-down
strategy need only define how an input set of objects can be partitioned
into two subsets.
Typically the input set is divided using a splitting hyperplane.
Any primitives intersected by the
splitting plane are commonly
assigned to the subset within which
its centroid belongs, alongside
extending subset bounding volume
by half the width of the primitive
(thereby fully enclosing the
primitive and minimising overlap
between sibling bounding volumes).
Top-down partitioning strategy
A number of different partitioning strategies
might be adopted (depending upon the
performance/memory needs, etc. of the
application), including:
• Minimize the sum of the volumes (or surface
areas) of the child volumes.
• Minimize the volume (surface area) of the
Assuming spherical bounding volumes
Minimal child volume, but high overlap
volume
intersection of the child volumes.
• Maximize the separation of child volumes.
• Divide primitives equally between the child
volumes (known as a median-cut algorithm,
and tends to produced balanced trees)
• Combinations of above strategies.
Larger child volume, but smaller
overlap volume
Top-down partitioning: Halting recursion
The recursive partitioning stops (thereby forming a leaf node) when
some termination condition is reached. This can include:
• The node contains less than some k primitives.
• The volume of the bounding volume falls below a cut-off limit.
• The depth of the node has reached a predefined cut-off depth.
Partitioning might also fail because:
• All primitives fall on one side of the split plane.
• One or both child volumes end up with as many (or nearly as
many) primitives as the parent volume.
• Both child volumes are (almost) as large as the parent volume.
In these cases, it is reasonable to try other partitioning criteria
before terminating the recursion.
Top-down partitioning: Choice of axis
The choice of which separating axis to use is
often restricted to the following:
• Local x, y, and z coordinate axes (easy to use,
also form orthogonal set, i.e. good
coverage).
• Axes from some aligned bounding volume
(e.g. from a reference k-DOP).
• Axes of the parent bounding volume
• Axis along which variance is greatest (using
the dimension with largest spread serves to
minimise the size of the child volumes).
Top-down partitioning: Choice of split point
The choice of which split point to use is
(also) often restricted to the following:
• Object median: splitting
at the middle object
(thereby evenly
distributing the primitives
and providing a balanced
tree).
• Object mean:
splitting at the
mean object offset
(e.g. along the axis
with greatest
variance).
• Bounding volume
median: splitting at the
middle of the bounding
volume (thereby
splitting into two equal
parts).
(a) Splits at object median, (b) splits at object mean, (c) splits at spatial median
Approaches to traversing bounding volumes hierarchies
Hierarchy Traversal
Given two BVHs a descent rule is used to determine
overlap status when the top-level volumes overlap.
Approaches include breadth-first (BFS) and depthfirst (DFS) searches. BFS explores all nodes at a
given depth before descending. DFS descends down
into the tree, backtracking up once a leaf is hit.
A decent method is blind (or uninformed) if it only
uses the tree structure to select the next node.
Informed methods examine node data (e.g. using
heuristics) to select the next node (e.g. a best-first
search maintains a list of best-scoring moves).
For collision detection systems, DFS (possibly a
simple heuristic – i.e. basically a best-first approach)
is typically favoured over BFS.
Descent Rules
Given two BVHs the following descent strategies might be adopted:
1.Descend A before B (or vice versa). Fully descend into the leaves
of A before starting to descend into B. This will be very expensive
if B’s root bound fully contains A and/or if A contains lots of nodes.
2.Descend the larger volume. Dynamically determine which BVH
node is currently larger and descend into (avoiding the problems
of Approach 1).
3.Descend A and B simultaneously. Similar to
Approach 2 but without any cost evaluation
overhead (but it may not prune space as efficiently).
4.Descend based on overlap. Similar to Approach 2,
priortise descent on degree of overlap between BVH
regions.
Generic Informed Depth-first traversal
An example informed depth-first traversal algorithm is:
CollisionResult is assumed to store relevant collision detection/contact information
void BVHInformedDFS(CollisionResult r, BVHNode a, BVHNode b)
{ If the bounding volumes do not overlap then simply return
if (!BVOverlap(a, b)) return;
}
if (IsLeaf(a) && IsLeaf(b)) { If two leaf nodes are found, then perform
CollidePrimitives(r, a, b); collision detection on the primitives
} else {
Descent rule to determine which BVH to follow
if (DescendA(a, b)) {
BVHInformedDFS (r, a->left, b);
BVHInformedDFS (r, a->right, b);
} else {
BVHInformedDFS (r, a, b->left);
BVHInformedDFS (r, ,a, b->right);
}
bool DescendA(BVHNode a, BVHNode b)
}
{
return IsLeaf(b) || Descend into the larger volume
(!IsLeaf(a) && (SizeOfBV(a) >= SizeOfBV(b)));
}
Descent Rules
See the source text book for:
• A non-recursive, faster, implementation of the generic
informed depth-first traversal
• Simultaneous depth-first traversal algorithm
• Optimised leaf-directed depth-first traversal
Directed mathematical reading
Directed reading
• Read Chapter 6 of Real Time
Collision Detection (pp235-284)
• Related papers can be found from:
http://realtimecollisiondetection.net/books/rtcd/references/
Summary
Today we
explored:
 Bounding
volume
hierarchies
 Overview of
top-down
construction
approaches