+Bounding volume hierarchies¶
+Basic concept¶
+Bounding Volume Hierarchies (BVHs) are comparatively simple data structures that can accelerate closest-point searches by orders of magnitude. +BVHs are tree structures where the regular nodes are bounding volumes that enclose all geometric primitives (e.g. polygon faces) further down in the hierarchy. +There are two types of nodes in a BVH:
+-
+
Regular nodes, which do not contain any of the primitives/objects, but stores references to child nodes.
+Leaf nodes, which contain a subset of the primitives.
+
Both types of nodes contain a bounding volume which closes all geometric primitives in the subtree below the node.
+Fig. 4 shows a concept of BVH partitioning of a set of triangles. +Here, \(P\) is a regular node whose bounding volume encloses all geometric primitives in it’s subtree.
+
+Fig. 4 Example of BVH partitioning for enclosing triangles. The regular node \(P\) contains two leaf nodes \(L\) and \(R\) which contain the primitives (triangles).¶
+There is no fundamental limitation to what type of primitives/objects can be enclosed in BVHs, which makes BVHs useful beyond triangulated data sets. +For example, analytic signed distance functions can also be embedded in BVHs, provided that we can construct a bounding volume that encloses the object.
+Note
+BVHs are not limited to binary trees. +EBGeometry supports \(k\) -ary trees where each regular node has \(k\) children nodes.
+Construction¶
+BVHs have extremely flexible rules regarding their construction. +Since they are hierarchical structures, the only requirement when dividing a leaf node is that each child node gets at least one primitive.
+Although the rules for BVH construction are flexible, performant BVHs are completely reliant on having balanced trees with the following heuristic properties:
+-
+
Bounding volumes should enclose primitives as tightly as possible.
+There should be a minimal overlap between the bounding volumes.
+The BVH trees should be approximately balanced.
+
Construction of a BVH is usually done recursively, from top to bottom (so-called top-down construction). +Alternative construction methods also exist, but are not used in EBGeometry. +In this case one can represent the BVH construction of a \(k\) -ary tree is done through a single function:
+where \(\vec{O}\) is an input a list of objects/primitives, which is partitioned into \(k\) new list of primitives. +Note that the lists \(\vec{O}_i\) do not contain duplicates. +Top-down construction can thus be illustrated as a recursive procedure:
+topDownConstruction(Objects):
+ partitionedObjects = Partition(Objects)
+
+ forall p in partitionedObjects:
+ child = insertChildNode(newObjects)
+
+ if(enoughPrimitives(child)):
+ child.topDownConstruction(child.objects)
+In practice, the above procedure is supplemented by more sophisticated criteria for terminating the recursion, as well as routines for creating the bounding volumes around the newly inserted nodes.
+Signed distance function¶
+When computing the signed distance function to objects embedded in a BVH, one takes advantage of the hierarcical embedding of the primitives. +Consider the case in Fig. 5, where the goal of the BVH traversal is to minimize the number of branches and nodes that are visited. +We consider the following steps:
+-
+
When descending from node \(P\) we determine that we first investigate the left subtree (node \(A\)) since its bounding volume is closer than the bounding volumes for the other subtree. +The other subtree will be investigated later.
+Since \(A\) is a leaf node, we find the signed distance from \(\mathbf{x}\) to the primitives in \(A\).
+When moving back to \(P\), we find that the distance to the primitives in \(A\) is shorter than the distance from \(\mathbf{x}\) to the bounding volume that encloses nodes \(B\) and \(C\). +Consequently, the entire subtree containing \(B\) and \(C\) can be pruned.
+
+Fig. 5 Example of BVH tree pruning.¶
+
