## 7.3 Bounding Volume Hierarchies

Bounding volume hierarchies (BVHs) are an approach for ray intersection acceleration based on primitive subdivision, where the primitives are partitioned into a hierarchy of disjoint sets. (In contrast, spatial subdivision generally partitions space into a hierarchy of disjoint sets.) Figure 7.3 shows a bounding volume hierarchy for a simple scene. Primitives are stored in the leaves, and each node stores a bounding box of the primitives in the nodes beneath it. Thus, as a ray traverses through the tree, any time it does not intersect a node’s bounds, the subtree beneath that node can be skipped.

One property of primitive subdivision is that each primitive appears in the hierarchy only once. In contrast, a primitive may overlap multiple spatial regions with spatial subdivision and thus may be tested for intersection multiple times as the ray passes through them. Another implication of this property is that the amount of memory needed to represent the primitive subdivision hierarchy is bounded. For a binary BVH that stores a single primitive in each leaf, the total number of nodes is , where is the number of primitives. (There are leaf nodes and interior nodes.) If leaves store multiple primitives, fewer nodes are needed.

BVHs are more efficient to build than kd-trees, and are generally more numerically robust and less prone to missed intersections due to round-off errors than kd-trees are. The BVH aggregate, BVHAggregate, is therefore the default acceleration structure in pbrt.

<<BVHAggregate Definition>>=
class BVHAggregate { public: <<BVHAggregate Public Types>>
enum class SplitMethod { SAH, HLBVH, Middle, EqualCounts };
<<BVHAggregate Public Methods>>
BVHAggregate(std::vector<Primitive> p, int maxPrimsInNode = 1, SplitMethod splitMethod = SplitMethod::SAH); static BVHAggregate *Create(std::vector<Primitive> prims, const ParameterDictionary &parameters); Bounds3f Bounds() const; pstd::optional<ShapeIntersection> Intersect(const Ray &ray, Float tMax) const; bool IntersectP(const Ray &ray, Float tMax) const;
private: <<BVHAggregate Private Methods>>
BVHBuildNode *buildRecursive(ThreadLocal<Allocator> &threadAllocators, pstd::span<BVHPrimitive> bvhPrimitives, std::atomic<int> *totalNodes, std::atomic<int> *orderedPrimsOffset, std::vector<Primitive> &orderedPrims); BVHBuildNode *buildHLBVH(Allocator alloc, const std::vector<BVHPrimitive> &primitiveInfo, std::atomic<int> *totalNodes, std::vector<Primitive> &orderedPrims); BVHBuildNode *emitLBVH(BVHBuildNode *&buildNodes, const std::vector<BVHPrimitive> &primitiveInfo, MortonPrimitive *mortonPrims, int nPrimitives, int *totalNodes, std::vector<Primitive> &orderedPrims, std::atomic<int> *orderedPrimsOffset, int bitIndex); BVHBuildNode *buildUpperSAH(Allocator alloc, std::vector<BVHBuildNode *> &treeletRoots, int start, int end, std::atomic<int> *totalNodes) const; int flattenBVH(BVHBuildNode *node, int *offset);
<<BVHAggregate Private Members>>
int maxPrimsInNode; std::vector<Primitive> primitives; SplitMethod splitMethod; LinearBVHNode *nodes = nullptr;
};

Its constructor takes an enumerator value that describes which of four algorithms to use when partitioning primitives to build the tree. The default, SAH, indicates that an algorithm based on the “surface area heuristic,” discussed in Section 7.3.2, should be used. An alternative, HLBVH, which is discussed in Section 7.3.3, can be constructed more efficiently (and more easily parallelized), but it does not build trees that are as effective as SAH. The remaining two approaches use even less computation but create fairly low-quality trees. They are mostly useful for illuminating the superiority of the first two approaches.

<<BVHAggregate Public Types>>=
enum class SplitMethod { SAH, HLBVH, Middle, EqualCounts };

In addition to the enumerator, the constructor takes the primitives themselves and the maximum number of primitives that can be in any leaf node.

<<BVHAggregate Method Definitions>>=
BVHAggregate::BVHAggregate(std::vector<Primitive> prims, int maxPrimsInNode, SplitMethod splitMethod) : maxPrimsInNode(std::min(255, maxPrimsInNode)), primitives(std::move(prims)), splitMethod(splitMethod) { <<Build BVH from primitives>>
<<Initialize bvhPrimitives array for primitives>>
std::vector<BVHPrimitive> bvhPrimitives(primitives.size()); for (size_t i = 0; i < primitives.size(); ++i) bvhPrimitives[i] = BVHPrimitive(i, primitives[i].Bounds());
<<Build BVH for primitives using bvhPrimitives>>
<<Declare Allocators used for BVH construction>>
std::vector<Primitive> orderedPrims(primitives.size()); BVHBuildNode *root; <<Build BVH according to selected splitMethod>>
std::atomic<int> totalNodes{0}; if (splitMethod == SplitMethod::HLBVH) { root = buildHLBVH(alloc, bvhPrimitives, &totalNodes, orderedPrims); } else { std::atomic<int> orderedPrimsOffset{0}; root = buildRecursive(threadAllocators, pstd::span<BVHPrimitive>(bvhPrimitives), &totalNodes, &orderedPrimsOffset, orderedPrims); } primitives.swap(orderedPrims);
<<Convert BVH into compact representation in nodes array>>
bvhPrimitives.resize(0); nodes = new LinearBVHNode[totalNodes]; int offset = 0; flattenBVH(root, &offset);
}

<<BVHAggregate Private Members>>=
int maxPrimsInNode; std::vector<Primitive> primitives; SplitMethod splitMethod;

### 7.3.1 BVH Construction

There are three stages to BVH construction in the implementation here. First, bounding information about each primitive is computed and stored in an array that will be used during tree construction. Next, the tree is built using the algorithm choice encoded in splitMethod. The result is a binary tree where each interior node holds pointers to its children and each leaf node holds references to one or more primitives. Finally, this tree is converted to a more compact (and thus more efficient) pointerless representation for use during rendering. (The implementation is easier with this approach, versus computing the pointerless representation directly during tree construction, which is also possible.)

<<Build BVH from primitives>>=
<<Initialize bvhPrimitives array for primitives>>
std::vector<BVHPrimitive> bvhPrimitives(primitives.size()); for (size_t i = 0; i < primitives.size(); ++i) bvhPrimitives[i] = BVHPrimitive(i, primitives[i].Bounds());
<<Build BVH for primitives using bvhPrimitives>>
<<Declare Allocators used for BVH construction>>
std::vector<Primitive> orderedPrims(primitives.size()); BVHBuildNode *root; <<Build BVH according to selected splitMethod>>
std::atomic<int> totalNodes{0}; if (splitMethod == SplitMethod::HLBVH) { root = buildHLBVH(alloc, bvhPrimitives, &totalNodes, orderedPrims); } else { std::atomic<int> orderedPrimsOffset{0}; root = buildRecursive(threadAllocators, pstd::span<BVHPrimitive>(bvhPrimitives), &totalNodes, &orderedPrimsOffset, orderedPrims); } primitives.swap(orderedPrims);
<<Convert BVH into compact representation in nodes array>>
bvhPrimitives.resize(0); nodes = new LinearBVHNode[totalNodes]; int offset = 0; flattenBVH(root, &offset);

For each primitive to be stored in the BVH, an instance of the BVHPrimitive structure stores its complete bounding box and its index in the primitives array.

<<Initialize bvhPrimitives array for primitives>>=
std::vector<BVHPrimitive> bvhPrimitives(primitives.size()); for (size_t i = 0; i < primitives.size(); ++i) bvhPrimitives[i] = BVHPrimitive(i, primitives[i].Bounds());

<<BVHPrimitive Definition>>=
struct BVHPrimitive { BVHPrimitive(size_t primitiveIndex, const Bounds3f &bounds) : primitiveIndex(primitiveIndex), bounds(bounds) {} size_t primitiveIndex; Bounds3f bounds; <<BVHPrimitive Public Methods>>
Point3f Centroid() const { return .5f * bounds.pMin + .5f * bounds.pMax; }
};

A simple method makes the centroid of the bounding box available.

<<BVHPrimitive Public Methods>>=
Point3f Centroid() const { return .5f * bounds.pMin + .5f * bounds.pMax; }

Hierarchy construction can now begin. In addition to initializing the pointer to the root note of the BVH, root, an important side effect of the tree construction process is that a new array of Primitives is stored in orderedPrims; this array stores the primitives ordered so that the primitives in each leaf node occupy a contiguous range in the array. It is swapped with the original primitives array after tree construction.

<<Build BVH for primitives using bvhPrimitives>>=
<<Declare Allocators used for BVH construction>>
std::vector<Primitive> orderedPrims(primitives.size()); BVHBuildNode *root; <<Build BVH according to selected splitMethod>>
std::atomic<int> totalNodes{0}; if (splitMethod == SplitMethod::HLBVH) { root = buildHLBVH(alloc, bvhPrimitives, &totalNodes, orderedPrims); } else { std::atomic<int> orderedPrimsOffset{0}; root = buildRecursive(threadAllocators, pstd::span<BVHPrimitive>(bvhPrimitives), &totalNodes, &orderedPrimsOffset, orderedPrims); } primitives.swap(orderedPrims);

Memory for the initial BVH is allocated using the following Allocators. Note that all are based on the C++ standard library’s pmr::monotonic_buffer_resource, which efficiently allocates memory from larger buffers. This approach is not only more computationally efficient than using a general-purpose allocator, but also uses less memory in total due to keeping less bookkeeping information with each allocation. We have found that using the default memory allocation algorithms in the place of these uses approximately 10% more memory and takes approximately 10% longer for complex scenes.

Because the pmr::monotonic_buffer_resource class cannot be used concurrently by multiple threads without mutual exclusion, in the parts of BVH construction that execute in parallel each thread uses per-thread allocation of them with help from the ThreadLocal class. Non-parallel code can use alloc directly.

<<Declare Allocators used for BVH construction>>=

If the HLBVH construction algorithm has been selected, buildHLBVH() is called to build the tree. The other three construction algorithms are all handled by buildRecursive(). The initial calls to these functions are passed all the primitives to be stored. Each returns a pointer to the root of a BVH for the primitives they are given, which is represented with the BVHBuildNode structure and the total number of nodes created, which is stored in totalNodes. This value is represented by a std::atomic variable so that it can be modified correctly by multiple threads executing in parallel.

<<Build BVH according to selected splitMethod>>=
std::atomic<int> totalNodes{0}; if (splitMethod == SplitMethod::HLBVH) { root = buildHLBVH(alloc, bvhPrimitives, &totalNodes, orderedPrims); } else { std::atomic<int> orderedPrimsOffset{0}; root = buildRecursive(threadAllocators, pstd::span<BVHPrimitive>(bvhPrimitives), &totalNodes, &orderedPrimsOffset, orderedPrims); } primitives.swap(orderedPrims);

Each BVHBuildNode represents a node of the BVH. All nodes store a Bounds3f that represents the bounds of all the children beneath the node. Each interior node stores pointers to its two children in children. Interior nodes also record the coordinate axis along which primitives were partitioned for distribution to their two children; this information is used to improve the performance of the traversal algorithm. Leaf nodes record which primitive or primitives are stored in them; the elements of the BVHAggregate::primitives array from the offset firstPrimOffset up to but not including are the primitives in the leaf. (This is why the primitives array needs to be reordered—so that this representation can be used, rather than, for example, storing a variable-sized array of primitive indices at each leaf node.)

<<BVHBuildNode Definition>>=
struct BVHBuildNode { <<BVHBuildNode Public Methods>>
void InitLeaf(int first, int n, const Bounds3f &b) { firstPrimOffset = first; nPrimitives = n; bounds = b; children[0] = children[1] = nullptr; } void InitInterior(int axis, BVHBuildNode *c0, BVHBuildNode *c1) { children[0] = c0; children[1] = c1; bounds = Union(c0->bounds, c1->bounds); splitAxis = axis; nPrimitives = 0; }
Bounds3f bounds; BVHBuildNode *children[2]; int splitAxis, firstPrimOffset, nPrimitives; };

We will distinguish between leaf and interior nodes by whether their child pointers have the value nullptr or not, respectively.

<<BVHBuildNode Public Methods>>=
void InitLeaf(int first, int n, const Bounds3f &b) { firstPrimOffset = first; nPrimitives = n; bounds = b; children[0] = children[1] = nullptr; }

The InitInterior() method requires that the two child nodes already have been created, so that their pointers can be passed in. This requirement makes it easy to compute the bounds of the interior node, since the children bounds are immediately available.

<<BVHBuildNode Public Methods>>+=
void InitInterior(int axis, BVHBuildNode *c0, BVHBuildNode *c1) { children[0] = c0; children[1] = c1; bounds = Union(c0->bounds, c1->bounds); splitAxis = axis; nPrimitives = 0; }

In addition to the allocators used for BVH nodes and the array of BVHPrimitive structures, buildRecursive() takes a pointer totalNodes that is used to track the total number of BVH nodes that have been created; this value makes it possible to allocate exactly the right number of the more compact LinearBVHNodes later.

The orderedPrims array is used to store primitive references as primitives are stored in leaf nodes of the tree. It is initially allocated with enough entries to store all the primitives, though all entries are nullptr. When a leaf node is created, buildRecursive() claims enough entries in the array for its primitives; orderedPrimsOffset starts at 0 and keeps track of where the next free entry is. It, too, is an atomic variable so that multiple threads can allocate space from the array concurrently. Recall that when tree construction is finished, BVHAggregate::primitives is replaced with the ordered primitives array created here.

<<BVHAggregate Method Definitions>>+=
BVHBuildNode *BVHAggregate::buildRecursive( ThreadLocal<Allocator> &threadAllocators, pstd::span<BVHPrimitive> bvhPrimitives, std::atomic<int> *totalNodes, std::atomic<int> *orderedPrimsOffset, std::vector<Primitive> &orderedPrims) { Allocator alloc = threadAllocators.Get(); BVHBuildNode *node = alloc.new_object<BVHBuildNode>(); <<Initialize BVHBuildNode for primitive range>>
++*totalNodes; <<Compute bounds of all primitives in BVH node>>
Bounds3f bounds; for (const auto &prim : bvhPrimitives) bounds = Union(bounds, prim.bounds);
if (bounds.SurfaceArea() == 0 || bvhPrimitives.size() == 1) { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
} else { <<Compute bound of primitive centroids and choose split dimension dim>>
Bounds3f centroidBounds; for (const auto &prim : bvhPrimitives) centroidBounds = Union(centroidBounds, prim.Centroid()); int dim = centroidBounds.MaxDimension();
<<Partition primitives into two sets and build children>>
if (centroidBounds.pMax[dim] == centroidBounds.pMin[dim]) { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
} else { int mid = bvhPrimitives.size() / 2; <<Partition primitives based on splitMethod>>
switch (splitMethod) { case SplitMethod::Middle: { <<Partition primitives through node’s midpoint>>
Float pmid = (centroidBounds.pMin[dim] + centroidBounds.pMax[dim]) / 2; auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [dim, pmid](const BVHPrimitive &pi) { return pi.Centroid()[dim] < pmid; }); mid = midIter - bvhPrimitives.begin(); if (midIter != bvhPrimitives.begin() && midIter != bvhPrimitives.end()) break;
} case SplitMethod::EqualCounts: { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
break; } case SplitMethod::SAH: default: { <<Partition primitives using approximate SAH>>
if (bvhPrimitives.size() <= 2) { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
} else { <<Allocate BVHSplitBucket for SAH partition buckets>>
constexpr int nBuckets = 12; BVHSplitBucket buckets[nBuckets];
<<Initialize BVHSplitBucket for SAH partition buckets>>
for (const auto &prim : bvhPrimitives) { int b = nBuckets * centroidBounds.Offset(prim.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; buckets[b].count++; buckets[b].bounds = Union(buckets[b].bounds, prim.bounds); }
<<Compute costs for splitting after each bucket>>
constexpr int nSplits = nBuckets - 1; Float costs[nSplits] = {}; <<Partially initialize costs using a forward scan over splits>>
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }
<<Finish initializing costs using a backward scan over splits>>
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }
<<Find bucket to split at that minimizes SAH metric>>
int minCostSplitBucket = -1; Float minCost = Infinity; for (int i = 0; i < nSplits; ++i) { <<Compute cost for candidate split and update minimum if necessary>>
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }
} <<Compute leaf cost and SAH split cost for chosen split>>
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();
<<Either create leaf or split primitives at selected SAH bucket>>
if (bvhPrimitives.size() > maxPrimsInNode || minCost < leafCost) { auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [=](const BVHPrimitive &bp) { int b = nBuckets * centroidBounds.Offset(bp.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; return b <= minCostSplitBucket; }); mid = midIter - bvhPrimitives.begin(); } else { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
}
}
break; } }
BVHBuildNode *children[2]; <<Recursively build BVHs for children>>
if (bvhPrimitives.size() > 128 * 1024) { <<Recursively build child BVHs in parallel>>
ParallelFor(0, 2, [&](int i) { if (i == 0) children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); else children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims); });
} else { <<Recursively build child BVHs sequentially>>
children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims);
}
node->InitInterior(dim, children[0], children[1]); }
}
return node; }

If bvhPrimitives has only a single primitive, then the recursion has bottomed out and a leaf node is created. Otherwise, this method partitions its elements using one of the partitioning algorithms and reorders the array elements so that they represent the partitioned subsets. If the partitioning is successful, these two primitive sets are in turn passed to recursive calls that will themselves return pointers to nodes for the two children of the current node.

<<Initialize BVHBuildNode for primitive range>>=
++*totalNodes; <<Compute bounds of all primitives in BVH node>>
Bounds3f bounds; for (const auto &prim : bvhPrimitives) bounds = Union(bounds, prim.bounds);
if (bounds.SurfaceArea() == 0 || bvhPrimitives.size() == 1) { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
} else { <<Compute bound of primitive centroids and choose split dimension dim>>
Bounds3f centroidBounds; for (const auto &prim : bvhPrimitives) centroidBounds = Union(centroidBounds, prim.Centroid()); int dim = centroidBounds.MaxDimension();
<<Partition primitives into two sets and build children>>
if (centroidBounds.pMax[dim] == centroidBounds.pMin[dim]) { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
} else { int mid = bvhPrimitives.size() / 2; <<Partition primitives based on splitMethod>>
switch (splitMethod) { case SplitMethod::Middle: { <<Partition primitives through node’s midpoint>>
Float pmid = (centroidBounds.pMin[dim] + centroidBounds.pMax[dim]) / 2; auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [dim, pmid](const BVHPrimitive &pi) { return pi.Centroid()[dim] < pmid; }); mid = midIter - bvhPrimitives.begin(); if (midIter != bvhPrimitives.begin() && midIter != bvhPrimitives.end()) break;
} case SplitMethod::EqualCounts: { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
break; } case SplitMethod::SAH: default: { <<Partition primitives using approximate SAH>>
if (bvhPrimitives.size() <= 2) { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
} else { <<Allocate BVHSplitBucket for SAH partition buckets>>
constexpr int nBuckets = 12; BVHSplitBucket buckets[nBuckets];
<<Initialize BVHSplitBucket for SAH partition buckets>>
for (const auto &prim : bvhPrimitives) { int b = nBuckets * centroidBounds.Offset(prim.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; buckets[b].count++; buckets[b].bounds = Union(buckets[b].bounds, prim.bounds); }
<<Compute costs for splitting after each bucket>>
constexpr int nSplits = nBuckets - 1; Float costs[nSplits] = {}; <<Partially initialize costs using a forward scan over splits>>
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }
<<Finish initializing costs using a backward scan over splits>>
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }
<<Find bucket to split at that minimizes SAH metric>>
int minCostSplitBucket = -1; Float minCost = Infinity; for (int i = 0; i < nSplits; ++i) { <<Compute cost for candidate split and update minimum if necessary>>
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }
} <<Compute leaf cost and SAH split cost for chosen split>>
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();
<<Either create leaf or split primitives at selected SAH bucket>>
if (bvhPrimitives.size() > maxPrimsInNode || minCost < leafCost) { auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [=](const BVHPrimitive &bp) { int b = nBuckets * centroidBounds.Offset(bp.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; return b <= minCostSplitBucket; }); mid = midIter - bvhPrimitives.begin(); } else { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
}
}
break; } }
BVHBuildNode *children[2]; <<Recursively build BVHs for children>>
if (bvhPrimitives.size() > 128 * 1024) { <<Recursively build child BVHs in parallel>>
ParallelFor(0, 2, [&](int i) { if (i == 0) children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); else children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims); });
} else { <<Recursively build child BVHs sequentially>>
children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims);
}
node->InitInterior(dim, children[0], children[1]); }
}

The primitive bounds will be needed regardless of whether an interior or leaf node is created, so they are computed before that determination is made.

<<Compute bounds of all primitives in BVH node>>=
Bounds3f bounds; for (const auto &prim : bvhPrimitives) bounds = Union(bounds, prim.bounds);

At leaf nodes, the primitives overlapping the leaf are appended to the orderedPrims array and a leaf node object is initialized. Because orderedPrimsOffset is a std::atomic variable and fetch_add() is an atomic operation, multiple threads can safely perform this operation concurrently without further synchronization: each one is able to allocate its own span of the orderedPrimitives array that it can then safely write to.

<<Create leaf BVHBuildNode>>=
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;

For interior nodes, the collection of primitives must be partitioned between the two children’s subtrees. Given primitives, there are in general possible ways to partition them into two non-empty groups. In practice when building BVHs, one generally considers partitions along a coordinate axis, meaning that there are about candidate partitions. (Along each axis, each primitive may be put into the first partition or the second partition.)

Here, we choose just one of the three coordinate axes to use in partitioning the primitives. We select the axis with the largest extent of bounding box centroids for the primitives in bvhPrimitives. (An alternative would be to try partitioning the primitives along all three axes and select the one that gave the best result, but in practice this approach works well.) This approach gives good partitions in many scenes; Figure 7.4 illustrates the strategy.

The general goal is to select a partition of primitives that does not have too much overlap of the bounding boxes of the two resulting primitive sets—if there is substantial overlap, then it will more frequently be necessary to traverse both children’s subtrees when traversing the tree, requiring more computation than if it had been possible to more effectively prune away collections of primitives. This idea of finding effective primitive partitions will be made more rigorous shortly, in the discussion of the surface area heuristic.

<<Compute bound of primitive centroids and choose split dimension dim>>=
Bounds3f centroidBounds; for (const auto &prim : bvhPrimitives) centroidBounds = Union(centroidBounds, prim.Centroid()); int dim = centroidBounds.MaxDimension();

If all the centroid points are at the same position (i.e., the centroid bounds have zero volume), then recursion stops and a leaf node is created with the primitives; none of the splitting methods here is effective in that (unusual) case. The primitives are otherwise partitioned using the chosen method and passed to two recursive calls to buildRecursive().

<<Partition primitives into two sets and build children>>=
if (centroidBounds.pMax[dim] == centroidBounds.pMin[dim]) { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
} else { int mid = bvhPrimitives.size() / 2; <<Partition primitives based on splitMethod>>
switch (splitMethod) { case SplitMethod::Middle: { <<Partition primitives through node’s midpoint>>
Float pmid = (centroidBounds.pMin[dim] + centroidBounds.pMax[dim]) / 2; auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [dim, pmid](const BVHPrimitive &pi) { return pi.Centroid()[dim] < pmid; }); mid = midIter - bvhPrimitives.begin(); if (midIter != bvhPrimitives.begin() && midIter != bvhPrimitives.end()) break;
} case SplitMethod::EqualCounts: { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
break; } case SplitMethod::SAH: default: { <<Partition primitives using approximate SAH>>
if (bvhPrimitives.size() <= 2) { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
} else { <<Allocate BVHSplitBucket for SAH partition buckets>>
constexpr int nBuckets = 12; BVHSplitBucket buckets[nBuckets];
<<Initialize BVHSplitBucket for SAH partition buckets>>
for (const auto &prim : bvhPrimitives) { int b = nBuckets * centroidBounds.Offset(prim.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; buckets[b].count++; buckets[b].bounds = Union(buckets[b].bounds, prim.bounds); }
<<Compute costs for splitting after each bucket>>
constexpr int nSplits = nBuckets - 1; Float costs[nSplits] = {}; <<Partially initialize costs using a forward scan over splits>>
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }
<<Finish initializing costs using a backward scan over splits>>
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }
<<Find bucket to split at that minimizes SAH metric>>
int minCostSplitBucket = -1; Float minCost = Infinity; for (int i = 0; i < nSplits; ++i) { <<Compute cost for candidate split and update minimum if necessary>>
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }
} <<Compute leaf cost and SAH split cost for chosen split>>
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();
<<Either create leaf or split primitives at selected SAH bucket>>
if (bvhPrimitives.size() > maxPrimsInNode || minCost < leafCost) { auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [=](const BVHPrimitive &bp) { int b = nBuckets * centroidBounds.Offset(bp.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; return b <= minCostSplitBucket; }); mid = midIter - bvhPrimitives.begin(); } else { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
}
}
break; } }
BVHBuildNode *children[2]; <<Recursively build BVHs for children>>
if (bvhPrimitives.size() > 128 * 1024) { <<Recursively build child BVHs in parallel>>
ParallelFor(0, 2, [&](int i) { if (i == 0) children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); else children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims); });
} else { <<Recursively build child BVHs sequentially>>
children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims);
}
node->InitInterior(dim, children[0], children[1]); }

The two recursive calls access independent data, other than when they allocate space in the orderedPrims array by incrementing orderedPrimsOffset, which we already have seen is thread safe. Therefore, when there are a reasonably large number of active primitives, those calls can be performed in parallel, which improves the performance of BVH construction.

<<Recursively build BVHs for children>>=
if (bvhPrimitives.size() > 128 * 1024) { <<Recursively build child BVHs in parallel>>
ParallelFor(0, 2, [&](int i) { if (i == 0) children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); else children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims); });
} else { <<Recursively build child BVHs sequentially>>
children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims);
}

A parallel for loop over two items is sufficient to expose the available parallelism. With pbrt’s implementation of ParallelFor(), the current thread will end up handling the first recursive call, while another thread, if available, can take the second. ParallelFor() does not return until all the loop iterations have completed, so we can safely proceed, knowing that both children are fully initialized when it does.

<<Recursively build child BVHs in parallel>>=
ParallelFor(0, 2, [&](int i) { if (i == 0) children[0] = buildRecursive(threadAllocators, bvhPrimitives.subspan(0, mid), totalNodes, orderedPrimsOffset, orderedPrims); else children[1] = buildRecursive(threadAllocators, bvhPrimitives.subspan(mid), totalNodes, orderedPrimsOffset, orderedPrims); });

The code for the non-parallel case, <<Recursively build child BVHs sequentially>>, is equivalent, just without the parallel for loop. We have therefore not included it here.

We also will not include the code fragment <<Partition primitives based on splitMethod>> here; it just uses the value of BVHAggregate::splitMethod to determine which primitive partitioning scheme to use. These three schemes will be described in the following few pages.

A simple splitMethod is Middle, which first computes the midpoint of the primitives’ centroids along the splitting axis. This method is implemented in the fragment <<Partition primitives through node’s midpoint>>. The primitives are classified into the two sets, depending on whether their centroids are above or below the midpoint. This partitioning is easily done with the std::partition() C++ standard library function, which takes a range of elements in an array and a comparison function and orders the elements in the array so that all the elements that return true for the given predicate function appear in the range before those that return false for it. std::partition() returns a pointer to the first element that had a false value for the predicate. Figure 7.5 illustrates this approach, including cases where it does and does not work well.

If the primitives all have large overlapping bounding boxes, this splitting method may fail to separate the primitives into two groups. In that case, execution falls through to the SplitMethod::EqualCounts approach to try again.

<<Partition primitives through node’s midpoint>>=
Float pmid = (centroidBounds.pMin[dim] + centroidBounds.pMax[dim]) / 2; auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [dim, pmid](const BVHPrimitive &pi) { return pi.Centroid()[dim] < pmid; }); mid = midIter - bvhPrimitives.begin(); if (midIter != bvhPrimitives.begin() && midIter != bvhPrimitives.end()) break;

When splitMethod is SplitMethod::EqualCounts, the <<Partition primitives into equally sized subsets>> fragment runs. It partitions the primitives into two equal-sized subsets such that the first half of the of them are the with smallest centroid coordinate values along the chosen axis, and the second half are the ones with the largest centroid coordinate values. While this approach can sometimes work well, the case in Figure 7.5(b) is one where this method also fares poorly.

This scheme is also easily implemented with a standard library call, std::nth_element(). It takes a start, middle, and ending iterator as well as a comparison function. It orders the array so that the element at the middle iterator is the one that would be there if the array was fully sorted, and such that all the elements before the middle one compare to less than the middle element and all the elements after it compare to greater than it. This ordering can be done in time, with the number of elements, which is more efficient than the cost of completely sorting the array.

<<Partition primitives into equally sized subsets>>=
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });

### 7.3.2 The Surface Area Heuristic

The two primitive partitioning approaches described so far can work well for some distributions of primitives, but they often choose partitions that perform poorly in practice, leading to more nodes of the tree being visited by rays and hence unnecessarily inefficient ray–primitive intersection computations at rendering time. Most of the best current algorithms for building acceleration structures for ray tracing are based on the “surface area heuristic” (SAH), which provides a well-grounded cost model for answering questions like “which of a number of partitions of primitives will lead to a better BVH for ray–primitive intersection tests?” or “which of a number of possible positions to split space in a spatial subdivision scheme will lead to a better acceleration structure?”

The SAH model estimates the computational expense of performing ray intersection tests, including the time spent traversing nodes of the tree and the time spent on ray–primitive intersection tests for a particular partitioning of primitives. Algorithms for building acceleration structures can then follow the goal of minimizing total cost. Typically, a greedy algorithm is used that minimizes the cost for each single node of the hierarchy being built individually.

The ideas behind the SAH cost model are straightforward: at any point in building an adaptive acceleration structure (primitive subdivision or spatial subdivision), we could just create a leaf node for the current region and geometry. In that case, any ray that passes through this region will be tested against all the overlapping primitives and will incur a cost of

where is the number of primitives and is the time to compute a ray–object intersection with the th primitive.

The other option is to split the region. In that case, rays will incur the cost

(7.1)

where is the time it takes to traverse the interior node and determine which of the children the ray passes through, and are the probabilities that the ray passes through each of the child nodes (assuming binary subdivision), and are the indices of primitives in the two child nodes, and and are the number of primitives that overlap the regions of the two child nodes, respectively. The choice of how primitives are partitioned affects the values of the two probabilities as well as the set of primitives on each side of the split.

In pbrt, we will make the simplifying assumption that is the same for all the primitives; this assumption is probably not too far from reality, and any error that it introduces does not seem to affect the performance of accelerators very much. Another possibility would be to add a method to Primitive that returned an estimate of the number of processing cycles that its intersection test requires.

The probabilities and can be computed using ideas from geometric probability. It can be shown that for a convex volume contained in another convex volume , the conditional probability that a uniformly distributed random ray passing through will also pass through is the ratio of their surface areas, and :

Because we are interested in the cost for rays passing through the node, we can use this result directly. Thus, if we are considering refining a region of space such that there are two new subregions with bounds and (Figure 7.6), the probability that a ray passing through will also pass through either of the subregions is easily computed.

When splitMethod has the value SplitMethod::SAH, the SAH is used for building the BVH; a partition of the primitives along the chosen axis that gives a minimal SAH cost estimate is found by considering a number of candidate partitions. (This is the default SplitMethod, and it creates the most efficient hierarchies of the partitioning options.) However, once it has refined down to two primitives, the implementation switches over to directly partitioning them in half. The incremental computational cost for applying the SAH at that point is not beneficial.

<<Partition primitives using approximate SAH>>=
if (bvhPrimitives.size() <= 2) { <<Partition primitives into equally sized subsets>>
mid = bvhPrimitives.size() / 2; std::nth_element(bvhPrimitives.begin(), bvhPrimitives.begin() + mid, bvhPrimitives.end(), [dim](const BVHPrimitive &a, const BVHPrimitive &b) { return a.Centroid()[dim] < b.Centroid()[dim]; });
} else { <<Allocate BVHSplitBucket for SAH partition buckets>>
constexpr int nBuckets = 12; BVHSplitBucket buckets[nBuckets];
<<Initialize BVHSplitBucket for SAH partition buckets>>
for (const auto &prim : bvhPrimitives) { int b = nBuckets * centroidBounds.Offset(prim.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; buckets[b].count++; buckets[b].bounds = Union(buckets[b].bounds, prim.bounds); }
<<Compute costs for splitting after each bucket>>
constexpr int nSplits = nBuckets - 1; Float costs[nSplits] = {}; <<Partially initialize costs using a forward scan over splits>>
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }
<<Finish initializing costs using a backward scan over splits>>
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }
<<Find bucket to split at that minimizes SAH metric>>
int minCostSplitBucket = -1; Float minCost = Infinity; for (int i = 0; i < nSplits; ++i) { <<Compute cost for candidate split and update minimum if necessary>>
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }
} <<Compute leaf cost and SAH split cost for chosen split>>
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();
<<Either create leaf or split primitives at selected SAH bucket>>
if (bvhPrimitives.size() > maxPrimsInNode || minCost < leafCost) { auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [=](const BVHPrimitive &bp) { int b = nBuckets * centroidBounds.Offset(bp.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; return b <= minCostSplitBucket; }); mid = midIter - bvhPrimitives.begin(); } else { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
}
}

Rather than exhaustively considering all possible partitions along the axis, computing the SAH for each to select the best, the implementation here instead divides the range along the axis into a small number of buckets of equal extent. It then only considers partitions at bucket boundaries. This approach is more efficient than considering all partitions while usually still producing partitions that are nearly as effective. This idea is illustrated in Figure 7.7.

<<BVHSplitBucket Definition>>=
struct BVHSplitBucket { int count = 0; Bounds3f bounds; };

We have found that 12 buckets usually work well in practice. An improvement may be to increase this value when there are many primitives and to decrease it when there are few.

<<Allocate BVHSplitBucket for SAH partition buckets>>=
constexpr int nBuckets = 12; BVHSplitBucket buckets[nBuckets];

For each primitive, the following fragment determines the bucket that its centroid lies in and updates the bucket’s bounds to include the primitive’s bounds.

<<Initialize BVHSplitBucket for SAH partition buckets>>=
for (const auto &prim : bvhPrimitives) { int b = nBuckets * centroidBounds.Offset(prim.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; buckets[b].count++; buckets[b].bounds = Union(buckets[b].bounds, prim.bounds); }

For each bucket, we now have a count of the number of primitives and the bounds of all of their respective bounding boxes. We want to use the SAH to estimate the cost of splitting at each of the bucket boundaries. The fragment below loops over all the buckets and initializes the cost[i] array to store the estimated SAH cost for splitting after the ith bucket. (It does not consider a split after the last bucket, which by definition would not split the primitives.)

We arbitrarily set the estimated intersection cost to 1, and then set the estimated traversal cost to . (One of the two of them can always be set to 1 since it is the relative, rather than absolute, magnitudes of the estimated traversal and intersection costs that determine their effect.) However, not only is the absolute amount of computation necessary for node traversal—a ray–bounding box intersection—much less than the amount of computation needed to intersect a ray with a shape, the full cost of a shape intersection test is even higher. It includes the overhead of at least two instances of dynamic dispatch (one or more via Primitives and one via a Shape), the cost of computing all the geometric information needed to initialize a SurfaceInteraction if an intersection is found, and any resulting costs from possibly applying additional transformations and interpolating animated transformations.

We have intentionally underestimated the performance ratio between these two costs because the raw amount of computation each performs does not measure their full expense. With a lower traversal cost, the resulting BVHs would be deeper and require more nodes. For complex scenes, this additional memory use may be undesirable. Even for simpler scenes, visiting more nodes when a ray is traced will generally incur the cost of cache misses, which not only may reduce performance for that ray, but may harm future performance from displacing other useful data from the cache. We have found the ratio that we have used here to make a reasonable trade-off between all of these issues.

In order to be able to choose a split in linear time, the implementation first performs a forward scan over the buckets and then a backward scan over the buckets that incrementally compute each bucket’s cost. There is one fewer candidate split than the number of buckets, since all splits are between pairs of buckets.

<<Compute costs for splitting after each bucket>>=
constexpr int nSplits = nBuckets - 1; Float costs[nSplits] = {}; <<Partially initialize costs using a forward scan over splits>>
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }
<<Finish initializing costs using a backward scan over splits>>
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }

The loop invariant is that countBelow stores the number of primitives that are below the corresponding candidate split, and boundsBelow stores their bounds. With these values in hand, the value of the first sum in Equation (7.1) can be evaluated for each split.

<<Partially initialize costs using a forward scan over splits>>=
int countBelow = 0; Bounds3f boundBelow; for (int i = 0; i < nSplits; ++i) { boundBelow = Union(boundBelow, buckets[i].bounds); countBelow += buckets[i].count; costs[i] += countBelow * boundBelow.SurfaceArea(); }

A similar backward scan over the buckets finishes initializing the costs array.

<<Finish initializing costs using a backward scan over splits>>=
int countAbove = 0; Bounds3f boundAbove; for (int i = nSplits; i >= 1; --i) { boundAbove = Union(boundAbove, buckets[i].bounds); countAbove += buckets[i].count; costs[i - 1] += countAbove * boundAbove.SurfaceArea(); }

Given all the costs, a linear search over the potential splits finds the partition with minimum cost.

<<Find bucket to split at that minimizes SAH metric>>=
int minCostSplitBucket = -1; Float minCost = Infinity; for (int i = 0; i < nSplits; ++i) { <<Compute cost for candidate split and update minimum if necessary>>
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }
} <<Compute leaf cost and SAH split cost for chosen split>>
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();

To find the best split, we evaluate a simplified version of Equation (7.1), neglecting the traversal cost and the division by the surface area of the bounding box of all the primitives to compute the probabilities and ; these have no effect on the choice of the best split. That cost is precisely what is stored in costs, so the split with minimum cost is easily found.

<<Compute cost for candidate split and update minimum if necessary>>=
if (costs[i] < minCost) { minCost = costs[i]; minCostSplitBucket = i; }

To compute the final SAH cost for a split, we need to divide by the surface area of the overall bounding box to compute the probabilities and before adding the estimated traversal cost, . Because we set the estimated intersection cost to 1 previously, the estimated cost for just creating a leaf node is equal to the number of primitives.

<<Compute leaf cost and SAH split cost for chosen split>>=
Float leafCost = bvhPrimitives.size(); minCost = 1.f / 2.f + minCost / bounds.SurfaceArea();

If the chosen bucket boundary for partitioning has a lower estimated cost than building a node with the existing primitives or if more than the maximum number of primitives allowed in a node is present, the std::partition() function is used to do the work of reordering nodes in the bvhPrimitives array. Recall from its use earlier that it ensures that all elements of the array that return true from the given predicate appear before those that return false and that it returns a pointer to the first element where the predicate returns false.

<<Either create leaf or split primitives at selected SAH bucket>>=
if (bvhPrimitives.size() > maxPrimsInNode || minCost < leafCost) { auto midIter = std::partition(bvhPrimitives.begin(), bvhPrimitives.end(), [=](const BVHPrimitive &bp) { int b = nBuckets * centroidBounds.Offset(bp.Centroid())[dim]; if (b == nBuckets) b = nBuckets - 1; return b <= minCostSplitBucket; }); mid = midIter - bvhPrimitives.begin(); } else { <<Create leaf BVHBuildNode>>
int firstPrimOffset = orderedPrimsOffset->fetch_add(bvhPrimitives.size()); for (size_t i = 0; i < bvhPrimitives.size(); ++i) { int index = bvhPrimitives[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[index]; } node->InitLeaf(firstPrimOffset, bvhPrimitives.size(), bounds); return node;
}

### 7.3.3 Linear Bounding Volume Hierarchies

While building bounding volume hierarchies using the surface area heuristic gives very good results, that approach does have two disadvantages: first, many passes are taken over the scene primitives to compute the SAH costs at all the levels of the tree. Second, top-down BVH construction is difficult to parallelize well: the approach used in buildRecursive()—performing parallel construction of independent subtrees—suffers from limited independent work until the top few levels of the tree have been built, which in turn inhibits parallel scalability. (This second issue is particularly an issue on GPUs, which perform poorly if massive parallelism is not available.)

Linear bounding volume hierarchies (LBVHs) were developed to address these issues. With LBVHs, the tree is built with a small number of lightweight passes over the primitives; tree construction time is linear in the number of primitives. Further, the algorithm quickly partitions the primitives into clusters that can be processed independently. This processing can be fairly easily parallelized and is well suited to GPU implementation.

The key idea behind LBVHs is to turn BVH construction into a sorting problem. Because there is no single ordering function for sorting multidimensional data, LBVHs are based on Morton codes, which map nearby points in dimensions to nearby points along the 1D line, where there is an obvious ordering function. After the primitives have been sorted, spatially nearby clusters of primitives are in contiguous segments of the sorted array.

Morton codes are based on a simple transformation: given -dimensional integer coordinate values, their Morton-coded representation is found by interleaving the bits of the coordinates in base 2. For example, consider a 2D coordinate where the bits of and are denoted by and . The corresponding Morton-coded value is

Figure 7.8 shows a plot of the 2D points in Morton order—note that they are visited along a path that follows a reversed “z” shape. (The Morton path is sometimes called “z-order” for this reason.) We can see that points with coordinates that are close together in 2D are generally close together along the Morton curve.

A Morton-encoded value also encodes useful information about the position of the point that it represents. Consider the case of 4-bit coordinate values in 2D: the and coordinates are integers in and the Morton code has 8 bits: . Many interesting properties follow from the encoding; a few examples include:

• For a Morton-encoded 8-bit value where the high bit is set, we then know that the high bit of its underlying coordinate is set and thus (Figure 7.9(a)).
• The next bit value, , splits the axis in the middle (Figure 7.9(b)). If is set and is off, for example, then the corresponding point must lie in the shaded area of Figure 7.9(c). In general, points with a number of matching high bits lie in a power-of-two sized and axis-aligned region of space determined by the matching bit values.
• The value of splits the axis into four regions (Figure 7.9(d)).

Another way to interpret these bit-based properties is in terms of Morton-coded values. For example, Figure 7.9(a) corresponds to the index being in the range , and Figure 7.9(c) corresponds to . Thus, given a set of sorted Morton indices, we could find the range of points corresponding to an area like Figure 7.9(c) by performing a binary search to find each endpoint in the array.

LBVHs are BVHs built by partitioning primitives using splitting planes that are at the midpoint of each region of space (i.e., equivalent to the SplitMethod::Middle path defined earlier). Partitioning is extremely efficient, as it takes advantage of properties of the Morton encoding described above.

Just reimplementing Middle in a different manner is not particularly interesting, so in the implementation here, we will build a hierarchical linear bounding volume hierarchy (HLBVH). With this approach, Morton-curve-based clustering is used to first build trees for the lower levels of the hierarchy (referred to as “treelets” in the following), and the top levels of the tree are then created using the surface area heuristic. The buildHLBVH() method implements this approach and returns the root node of the resulting tree.

<<BVHAggregate Method Definitions>>+=
BVHBuildNode *BVHAggregate::buildHLBVH( Allocator alloc, const std::vector<BVHPrimitive> &bvhPrimitives, std::atomic<int> *totalNodes, std::vector<Primitive> &orderedPrims) { <<Compute bounding box of all primitive centroids>>
Bounds3f bounds; for (const BVHPrimitive &prim : bvhPrimitives) bounds = Union(bounds, prim.Centroid());
<<Compute Morton indices of primitives>>
std::vector<MortonPrimitive> mortonPrims(bvhPrimitives.size()); ParallelFor(0, bvhPrimitives.size(), [&](int64_t i) { <<Initialize mortonPrims[i] for ith primitive>>
constexpr int mortonBits = 10; constexpr int mortonScale = 1 << mortonBits; mortonPrims[i].primitiveIndex = bvhPrimitives[i].primitiveIndex; Vector3f centroidOffset = bounds.Offset(bvhPrimitives[i].Centroid()); Vector3f offset = centroidOffset * mortonScale; mortonPrims[i].mortonCode = EncodeMorton3(offset.x, offset.y, offset.z);
});
<<Create LBVH treelets at bottom of BVH>>
<<Find intervals of primitives for each treelet>>
std::vector<LBVHTreelet> treeletsToBuild; for (size_t start = 0, end = 1; end <= mortonPrims.size(); ++end) { uint32_t mask = 0b00111111111111000000000000000000; if (end == (int)mortonPrims.size() || ((mortonPrims[start].mortonCode & mask) != (mortonPrims[end].mortonCode & mask))) { <<Add entry to treeletsToBuild for this treelet>>
size_t nPrimitives = end - start; int maxBVHNodes = 2 * nPrimitives - 1; BVHBuildNode *nodes = alloc.allocate_object<BVHBuildNode>(maxBVHNodes); treeletsToBuild.push_back({start, nPrimitives, nodes});
start = end; } }
<<Create LBVHs for treelets in parallel>>
std::atomic<int> orderedPrimsOffset(0); ParallelFor(0, treeletsToBuild.size(), [&](int i) { <<Generate ith LBVH treelet>>
int nodesCreated = 0; const int firstBitIndex = 29 - 12; LBVHTreelet &tr = treeletsToBuild[i]; tr.buildNodes = emitLBVH(tr.buildNodes, bvhPrimitives, &mortonPrims[tr.startIndex], tr.nPrimitives, &nodesCreated, orderedPrims, &orderedPrimsOffset, firstBitIndex); *totalNodes += nodesCreated;
});
<<Create and return SAH BVH from LBVH treelets>>
std::vector<BVHBuildNode *> finishedTreelets; for (LBVHTreelet &treelet : treeletsToBuild) finishedTreelets.push_back(treelet.buildNodes); return buildUpperSAH(alloc, finishedTreelets, 0, finishedTreelets.size(), totalNodes);
}

The BVH is built using only the centroids of primitive bounding boxes to sort them—it does not account for the actual spatial extent of each primitive. This simplification is critical to the performance that HLBVHs offer, but it also means that for scenes with primitives that span a wide range of sizes, the tree that is built will not account for this variation as an SAH-based tree would.

Because the Morton encoding operates on integer coordinates, we first need to bound the centroids of all the primitives so that we can quantize centroid positions with respect to the overall bounds.

<<Compute bounding box of all primitive centroids>>=
Bounds3f bounds; for (const BVHPrimitive &prim : bvhPrimitives) bounds = Union(bounds, prim.Centroid());

Given the overall bounds, we can now compute the Morton code for each primitive. This is a fairly lightweight calculation, but given that there may be millions of primitives, it is worth parallelizing.

<<Compute Morton indices of primitives>>=
std::vector<MortonPrimitive> mortonPrims(bvhPrimitives.size()); ParallelFor(0, bvhPrimitives.size(), [&](int64_t i) { <<Initialize mortonPrims[i] for ith primitive>>
constexpr int mortonBits = 10; constexpr int mortonScale = 1 << mortonBits; mortonPrims[i].primitiveIndex = bvhPrimitives[i].primitiveIndex; Vector3f centroidOffset = bounds.Offset(bvhPrimitives[i].Centroid()); Vector3f offset = centroidOffset * mortonScale; mortonPrims[i].mortonCode = EncodeMorton3(offset.x, offset.y, offset.z);
});

A MortonPrimitive instance is created for each primitive; it stores the index of the primitive, as well as its Morton code, in the bvhPrimitives array.

<<MortonPrimitive Definition>>=
struct MortonPrimitive { int primitiveIndex; uint32_t mortonCode; };

We use 10 bits for each of the , , and dimensions, giving a total of 30 bits for the Morton code. This granularity allows the values to fit into a single 32-bit variable. Floating-point centroid offsets inside the bounding box are in , so we scale them by to get integer coordinates that fit in 10 bits. The EncodeMorton3() function, which is defined with other bitwise utility functions in Section B.2.7, returns the 3D Morton code for the given integer values.

<<Initialize mortonPrims[i] for ith primitive>>=
constexpr int mortonBits = 10; constexpr int mortonScale = 1 << mortonBits; mortonPrims[i].primitiveIndex = bvhPrimitives[i].primitiveIndex; Vector3f centroidOffset = bounds.Offset(bvhPrimitives[i].Centroid()); Vector3f offset = centroidOffset * mortonScale; mortonPrims[i].mortonCode = EncodeMorton3(offset.x, offset.y, offset.z);

Once the Morton indices have been computed, we will sort the mortonPrims by Morton index value using a radix sort. We have found that for BVH construction, our radix sort implementation is noticeably faster than using std::sort() from our system’s standard library (which is a mixture of a quicksort and an insertion sort).

Recall that a radix sort differs from most sorting algorithms in that it is not based on comparing pairs of values but rather is based on bucketing items based on some key. Radix sort can be used to sort integer values by sorting them one digit at a time, going from the rightmost digit to the leftmost. Especially with binary values, it is worth sorting multiple digits at a time; doing so reduces the total number of passes taken over the data. In the implementation here, bitsPerPass sets the number of bits processed per pass; with the value 6, we have 5 passes to sort the 30 bits.

<<BVHAggregate Utility Functions>>=
static void RadixSort(std::vector<MortonPrimitive> *v) { std::vector<MortonPrimitive> tempVector(v->size()); constexpr int bitsPerPass = 6; constexpr int nBits = 30; constexpr int nPasses = nBits / bitsPerPass; for (int pass = 0; pass < nPasses; ++pass) { <<Perform one pass of radix sort, sorting bitsPerPass bits>>
int lowBit = pass * bitsPerPass; <<Set in and out vector references for radix sort pass>>
std::vector<MortonPrimitive> &in = (pass & 1) ? tempVector : *v; std::vector<MortonPrimitive> &out = (pass & 1) ? *v : tempVector;
<<Count number of zero bits in array for current radix sort bit>>
constexpr int nBuckets = 1 << bitsPerPass; int bucketCount[nBuckets] = { 0 }; constexpr int bitMask = (1 << bitsPerPass) - 1; for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; ++bucketCount[bucket]; }
<<Compute starting index in output array for each bucket>>
int outIndex[nBuckets]; outIndex[0] = 0; for (int i = 1; i < nBuckets; ++i) outIndex[i] = outIndex[i - 1] + bucketCount[i - 1];
<<Store sorted values in output array>>
for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; out[outIndex[bucket]++] = mp; }
} <<Copy final result from tempVector, if needed>>
if (nPasses & 1) std::swap(*v, tempVector);
}

Each pass sorts bitsPerPass bits, starting at lowBit.

<<Perform one pass of radix sort, sorting bitsPerPass bits>>=
int lowBit = pass * bitsPerPass; <<Set in and out vector references for radix sort pass>>
std::vector<MortonPrimitive> &in = (pass & 1) ? tempVector : *v; std::vector<MortonPrimitive> &out = (pass & 1) ? *v : tempVector;
<<Count number of zero bits in array for current radix sort bit>>
constexpr int nBuckets = 1 << bitsPerPass; int bucketCount[nBuckets] = { 0 }; constexpr int bitMask = (1 << bitsPerPass) - 1; for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; ++bucketCount[bucket]; }
<<Compute starting index in output array for each bucket>>
int outIndex[nBuckets]; outIndex[0] = 0; for (int i = 1; i < nBuckets; ++i) outIndex[i] = outIndex[i - 1] + bucketCount[i - 1];
<<Store sorted values in output array>>
for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; out[outIndex[bucket]++] = mp; }

The in and out references correspond to the vector to be sorted and the vector to store the sorted values in, respectively. Each pass through the loop alternates between the input vector *v and the temporary vector for each of them.

<<Set in and out vector references for radix sort pass>>=
std::vector<MortonPrimitive> &in = (pass & 1) ? tempVector : *v; std::vector<MortonPrimitive> &out = (pass & 1) ? *v : tempVector;

If we are sorting bits per pass, then there are buckets that each value may land in. We first count how many values will land in each bucket; this will let us determine where to store sorted values in the output array. To compute the bucket index for the current value, the implementation shifts the index so that the bit at index lowBit is at bit 0 and then masks off the low bitsPerPass bits.

<<Count number of zero bits in array for current radix sort bit>>=
constexpr int nBuckets = 1 << bitsPerPass; int bucketCount[nBuckets] = { 0 }; constexpr int bitMask = (1 << bitsPerPass) - 1; for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; ++bucketCount[bucket]; }

Given the count of how many values land in each bucket, we can compute the offset in the output array where each bucket’s values start; this is just the sum of how many values land in the preceding buckets.

<<Compute starting index in output array for each bucket>>=
int outIndex[nBuckets]; outIndex[0] = 0; for (int i = 1; i < nBuckets; ++i) outIndex[i] = outIndex[i - 1] + bucketCount[i - 1];

Now that we know where to start storing values for each bucket, we can take another pass over the primitives to recompute the bucket that each one lands in and to store their MortonPrimitives in the output array. This completes the sorting pass for the current group of bits.

<<Store sorted values in output array>>=
for (const MortonPrimitive &mp : in) { int bucket = (mp.mortonCode >> lowBit) & bitMask; out[outIndex[bucket]++] = mp; }

When sorting is done, if an odd number of radix sort passes were performed, then the final sorted values need to be copied from the temporary vector to the output vector that was originally passed to RadixSort().

<<Copy final result from tempVector, if needed>>=
if (nPasses & 1) std::swap(*v, tempVector);

Given the sorted array of primitives, we can now find clusters of primitives with nearby centroids and then create an LBVH over the primitives in each cluster. This step is a good one to parallelize as there are generally many clusters and each cluster can be processed independently.

<<Create LBVH treelets at bottom of BVH>>=
<<Find intervals of primitives for each treelet>>
std::vector<LBVHTreelet> treeletsToBuild; for (size_t start = 0, end = 1; end <= mortonPrims.size(); ++end) { uint32_t mask = 0b00111111111111000000000000000000; if (end == (int)mortonPrims.size() || ((mortonPrims[start].mortonCode & mask) != (mortonPrims[end].mortonCode & mask))) { <<Add entry to treeletsToBuild for this treelet>>
size_t nPrimitives = end - start; int maxBVHNodes = 2 * nPrimitives - 1; BVHBuildNode *nodes = alloc.allocate_object<BVHBuildNode>(maxBVHNodes); treeletsToBuild.push_back({start, nPrimitives, nodes});
start = end; } }
<<Create LBVHs for treelets in parallel>>
std::atomic<int> orderedPrimsOffset(0); ParallelFor(0, treeletsToBuild.size(), [&](int i) { <<Generate ith LBVH treelet>>
int nodesCreated = 0; const int firstBitIndex = 29 - 12; LBVHTreelet &tr = treeletsToBuild[i]; tr.buildNodes = emitLBVH(tr.buildNodes, bvhPrimitives, &mortonPrims[tr.startIndex], tr.nPrimitives, &nodesCreated, orderedPrims, &orderedPrimsOffset, firstBitIndex); *totalNodes += nodesCreated;
});

Each primitive cluster is represented by an LBVHTreelet. It encodes the index in the mortonPrims array of the first primitive in the cluster as well as the number of following primitives. (See Figure 7.10.)

<<LBVHTreelet Definition>>=
struct LBVHTreelet { size_t startIndex, nPrimitives; BVHBuildNode *buildNodes; };

Recall from Figure 7.9 that a set of points with Morton codes that match in their high bit values lie in a power-of-two aligned and sized subset of the original volume. Because we have already sorted the mortonPrims array by Morton-coded value, primitives with matching high bit values are already together in contiguous sections of the array.

Here we will find sets of primitives that have the same values for the high 12 bits of their 30-bit Morton codes. Clusters are found by taking a linear pass through the mortonPrims array and finding the offsets where any of the high 12 bits changes. This corresponds to clustering primitives in a regular grid of total grid cells with cells in each dimension. In practice, many of the grid cells will be empty, though we will still expect to find many independent clusters here.

<<Find intervals of primitives for each treelet>>=
std::vector<LBVHTreelet> treeletsToBuild; for (size_t start = 0, end = 1; end <= mortonPrims.size(); ++end) { uint32_t mask = 0b00111111111111000000000000000000; if (end == (int)mortonPrims.size() || ((mortonPrims[start].mortonCode & mask) != (mortonPrims[end].mortonCode & mask))) { <<Add entry to treeletsToBuild for this treelet>>
size_t nPrimitives = end - start; int maxBVHNodes = 2 * nPrimitives - 1; BVHBuildNode *nodes = alloc.allocate_object<BVHBuildNode>(maxBVHNodes); treeletsToBuild.push_back({start, nPrimitives, nodes});
start = end; } }

When a cluster of primitives has been found for a treelet, BVHBuildNodes are immediately allocated for it. (Recall that the number of nodes in a BVH is bounded by twice the number of leaf nodes, which in turn is bounded by the number of primitives.) It is simpler to preallocate this memory now in a serial phase of execution than during parallel construction of LBVHs.

<<Add entry to treeletsToBuild for this treelet>>=
size_t nPrimitives = end - start; int maxBVHNodes = 2 * nPrimitives - 1; BVHBuildNode *nodes = alloc.allocate_object<BVHBuildNode>(maxBVHNodes); treeletsToBuild.push_back({start, nPrimitives, nodes});

Once the primitives for each treelet have been identified, we can create LBVHs for them in parallel. When construction is finished, the buildNodes pointer for each LBVHTreelet will point to the root of the corresponding LBVH.

There are two places where the worker threads building LBVHs must coordinate with each other. First, the total number of nodes in all the LBVHs needs to be computed and returned via the totalNodes pointer passed to buildHLBVH(). Second, when leaf nodes are created for the LBVHs, a contiguous segment of the orderedPrims array is needed to record the indices of the primitives in the leaf node. Our implementation uses atomic variables for both.

<<Create LBVHs for treelets in parallel>>=
std::atomic<int> orderedPrimsOffset(0); ParallelFor(0, treeletsToBuild.size(), [&](int i) { <<Generate ith LBVH treelet>>
int nodesCreated = 0; const int firstBitIndex = 29 - 12; LBVHTreelet &tr = treeletsToBuild[i]; tr.buildNodes = emitLBVH(tr.buildNodes, bvhPrimitives, &mortonPrims[tr.startIndex], tr.nPrimitives, &nodesCreated, orderedPrims, &orderedPrimsOffset, firstBitIndex); *totalNodes += nodesCreated;
});

The work of building the treelet is performed by emitLBVH(), which takes primitives with centroids in some region of space and successively partitions them with splitting planes that divide the current region of space into two halves along the center of the region along one of the three axes.

Note that instead of taking a pointer to the atomic variable totalNodes to count the number of nodes created, emitLBVH() updates a non-atomic local variable. The fragment here then only updates totalNodes once per treelet when each treelet is done. This approach gives measurably better performance than the alternative—having the worker threads frequently modify totalNodes over the course of their execution. (To understand why this is so, see the discussion of the overhead of multi-core memory coherence models in Appendix B.6.3.)

<<Generate ith LBVH treelet>>=
int nodesCreated = 0; const int firstBitIndex = 29 - 12; LBVHTreelet &tr = treeletsToBuild[i]; tr.buildNodes = emitLBVH(tr.buildNodes, bvhPrimitives, &mortonPrims[tr.startIndex], tr.nPrimitives, &nodesCreated, orderedPrims, &orderedPrimsOffset, firstBitIndex); *totalNodes += nodesCreated;

Thanks to the Morton encoding, the current region of space does not need to be explicitly represented in emitLBVH(): the sorted MortonPrims passed in have some number of matching high bits, which in turn corresponds to a spatial bound. For each of the remaining bits in the Morton codes, this function tries to split the primitives along the plane corresponding to the bitIndex bit (recall Figure 7.9(d)) and then calls itself recursively. The index of the next bit to try splitting with is passed as the last argument to the function: initially it is , since is the index of the th bit with zero-based indexing, and we previously used the high bits of the Morton-coded value to cluster the primitives; thus, we know that those bits must all match for the cluster.

<<BVHAggregate Method Definitions>>+=
BVHBuildNode *BVHAggregate::emitLBVH(BVHBuildNode *&buildNodes, const std::vector<BVHPrimitive> &bvhPrimitives, MortonPrimitive *mortonPrims, int nPrimitives, int *totalNodes, std::vector<Primitive> &orderedPrims, std::atomic<int> *orderedPrimsOffset, int bitIndex) { if (bitIndex == -1 || nPrimitives < maxPrimsInNode) { <<Create and return leaf node of LBVH treelet>>
++*totalNodes; BVHBuildNode *node = buildNodes++; Bounds3f bounds; int firstPrimOffset = orderedPrimsOffset->fetch_add(nPrimitives); for (int i = 0; i < nPrimitives; ++i) { int primitiveIndex = mortonPrims[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[primitiveIndex]; bounds = Union(bounds, bvhPrimitives[primitiveIndex].bounds); } node->InitLeaf(firstPrimOffset, nPrimitives, bounds); return node;
} else { int mask = 1 << bitIndex; <<Advance to next subtree level if there is no LBVH split for this bit>>
if ((mortonPrims[0].mortonCode & mask) == (mortonPrims[nPrimitives - 1].mortonCode & mask)) return emitLBVH(buildNodes, bvhPrimitives, mortonPrims, nPrimitives, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1);
<<Find LBVH split point for this dimension>>
int splitOffset = FindInterval(nPrimitives, [&](int index) { return ((mortonPrims[0].mortonCode & mask) == (mortonPrims[index].mortonCode & mask)); }); ++splitOffset;
<<Create and return interior LBVH node>>
(*totalNodes)++; BVHBuildNode *node = buildNodes++; BVHBuildNode *lbvh[2] = { emitLBVH(buildNodes, bvhPrimitives, mortonPrims, splitOffset, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1), emitLBVH(buildNodes, bvhPrimitives, &mortonPrims[splitOffset], nPrimitives - splitOffset, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1) }; int axis = bitIndex % 3; node->InitInterior(axis, lbvh[0], lbvh[1]); return node;
} }

After emitLBVH() has partitioned the primitives with the final low bit, no more splitting is possible and a leaf node is created. Alternatively, it also stops and makes a leaf node if it is down to a small number of primitives.

Recall that orderedPrimsOffset is the offset to the next available element in the orderedPrims array. Here, the call to fetch_add() atomically adds the value of nPrimitives to orderedPrimsOffset and returns its old value before the addition. Given space in the array, leaf construction is similar to the approach implemented earlier in <<Create leaf BVHBuildNode>>.

<<Create and return leaf node of LBVH treelet>>=
++*totalNodes; BVHBuildNode *node = buildNodes++; Bounds3f bounds; int firstPrimOffset = orderedPrimsOffset->fetch_add(nPrimitives); for (int i = 0; i < nPrimitives; ++i) { int primitiveIndex = mortonPrims[i].primitiveIndex; orderedPrims[firstPrimOffset + i] = primitives[primitiveIndex]; bounds = Union(bounds, bvhPrimitives[primitiveIndex].bounds); } node->InitLeaf(firstPrimOffset, nPrimitives, bounds); return node;

It may be the case that all the primitives lie on the same side of the splitting plane; since the primitives are sorted by their Morton index, this case can be efficiently checked by seeing if the first and last primitive in the range both have the same bit value for this plane. In this case, emitLBVH() proceeds to the next bit without unnecessarily creating a node.

<<Advance to next subtree level if there is no LBVH split for this bit>>=
if ((mortonPrims[0].mortonCode & mask) == (mortonPrims[nPrimitives - 1].mortonCode & mask)) return emitLBVH(buildNodes, bvhPrimitives, mortonPrims, nPrimitives, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1);

If there are primitives on both sides of the splitting plane, then a binary search efficiently finds the dividing point where the bitIndexth bit goes from 0 to 1 in the current set of primitives.

<<Find LBVH split point for this dimension>>=
int splitOffset = FindInterval(nPrimitives, [&](int index) { return ((mortonPrims[0].mortonCode & mask) == (mortonPrims[index].mortonCode & mask)); }); ++splitOffset;

Given the split offset, the method can now claim a node to use as an interior node and recursively build LBVHs for both partitioned sets of primitives. Note a further efficiency benefit from Morton encoding: entries in the mortonPrims array do not need to be copied or reordered for the partition: because they are all sorted by their Morton code value and because it is processing bits from high to low, the two spans of primitives are already on the correct sides of the partition plane.

<<Create and return interior LBVH node>>=
(*totalNodes)++; BVHBuildNode *node = buildNodes++; BVHBuildNode *lbvh[2] = { emitLBVH(buildNodes, bvhPrimitives, mortonPrims, splitOffset, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1), emitLBVH(buildNodes, bvhPrimitives, &mortonPrims[splitOffset], nPrimitives - splitOffset, totalNodes, orderedPrims, orderedPrimsOffset, bitIndex - 1) }; int axis = bitIndex % 3; node->InitInterior(axis, lbvh[0], lbvh[1]); return node;

Once all the LBVH treelets have been created, buildUpperSAH() creates a BVH of all the treelets. Since there are generally tens or hundreds of them (and in any case, no more than 4096), this step takes very little time.

<<Create and return SAH BVH from LBVH treelets>>=
std::vector<BVHBuildNode *> finishedTreelets; for (LBVHTreelet &treelet : treeletsToBuild) finishedTreelets.push_back(treelet.buildNodes); return buildUpperSAH(alloc, finishedTreelets, 0, finishedTreelets.size(), totalNodes);

The implementation of buildUpperSAH() is not included here, as it follows the same approach as fully SAH-based BVH construction, just over treelet root nodes rather than scene primitives.

### 7.3.4 Compact BVH for Traversal

Once the BVH is built, the last step is to convert it into a compact representation—doing so improves cache, memory, and thus overall system performance. The final BVH is stored in a linear array in memory. The nodes of the original tree are laid out in depth-first order, which means that the first child of each interior node is immediately after the node in memory. In this case, only the offset to the second child of each interior node must be stored explicitly. See Figure 7.11 for an illustration of the relationship between tree topology and node order in memory.

The LinearBVHNode structure stores the information needed to traverse the BVH. In addition to the bounding box for each node, for leaf nodes it stores the offset and primitive count for the primitives in the node. For interior nodes, it stores the offset to the second child as well as which of the coordinate axes the primitives were partitioned along when the hierarchy was built; this information is used in the traversal routine below to try to visit nodes in front-to-back order along the ray.

The structure is declared to require 32-byte alignment in memory. It could otherwise be allocated at an alignment that was sufficient to satisfy the first member variable, which would be 4 bytes for the Float-valued Bounds3f::pMin::x member variable. Because modern processor caches are organized into cache lines of a size that is a multiple of 32, a more stringent alignment constraint ensures that no LinearBVHNode straddles two cache lines. In turn, no more than a single cache miss will be incurred when one is accessed, which improves performance.

<<LinearBVHNode Definition>>=
struct alignas(32) LinearBVHNode { Bounds3f bounds; union { int primitivesOffset; // leaf int secondChildOffset; // interior }; uint16_t nPrimitives; // 0 -> interior node uint8_t axis; // interior node: xyz };

The built tree is transformed to the LinearBVHNode representation by the flattenBVH() method, which performs a depth-first traversal and stores the nodes in memory in linear order. It is helpful to release the memory in the bvhPrimitives array before doing so, since that may be a significant amount of storage for complex scenes and is no longer needed at this point. This is handled by the resize(0) call.

<<Convert BVH into compact representation in nodes array>>=
bvhPrimitives.resize(0); nodes = new LinearBVHNode[totalNodes]; int offset = 0; flattenBVH(root, &offset);

The pointer to the array of LinearBVHNodes is stored as a BVHAggregate member variable.

<<BVHAggregate Private Members>>+=
LinearBVHNode *nodes = nullptr;

Flattening the tree to the linear representation is straightforward; the *offset parameter tracks the current offset into the BVHAggregate::nodes array. Note that the current node is added to the array before any recursive calls to process its children.

<<BVHAggregate Method Definitions>>+=
int BVHAggregate::flattenBVH(BVHBuildNode *node, int *offset) { LinearBVHNode *linearNode = &nodes[*offset]; linearNode->bounds = node->bounds; int nodeOffset = (*offset)++; if (node->nPrimitives > 0) { linearNode->primitivesOffset = node->firstPrimOffset; linearNode->nPrimitives = node->nPrimitives; } else { <<Create interior flattened BVH node>>
linearNode->axis = node->splitAxis; linearNode->nPrimitives = 0; flattenBVH(node->children[0], offset); linearNode->secondChildOffset = flattenBVH(node->children[1], offset);
} return nodeOffset; }

At interior nodes, recursive calls are made to flatten the two subtrees. The first one ends up immediately after the current node in the array, as desired, and the offset of the second one, returned by its recursive flattenBVH() call, is stored in this node’s secondChildOffset member.

<<Create interior flattened BVH node>>=
linearNode->axis = node->splitAxis; linearNode->nPrimitives = 0; flattenBVH(node->children[0], offset); linearNode->secondChildOffset = flattenBVH(node->children[1], offset);

### 7.3.5 Bounding and Intersection Tests

Given a built BVH, the implementation of the Bounds() method is easy: by definition, the root node’s bounds are the bounds of all the primitives in the tree, so those can be returned directly.

<<BVHAggregate Method Definitions>>+=
Bounds3f BVHAggregate::Bounds() const { return nodes[0].bounds; }

The BVH traversal code is quite simple—there are no recursive function calls and a small amount of data to maintain about the current state of the traversal. The Intersect() method starts by precomputing a few values related to the ray that will be used repeatedly.

<<BVHAggregate Method Definitions>>+=
pstd::optional<ShapeIntersection> BVHAggregate::Intersect(const Ray &ray, Float tMax) const { pstd::optional<ShapeIntersection> si; Vector3f invDir(1 / ray.d.x, 1 / ray.d.y, 1 / ray.d.z); int dirIsNeg[3] = {int(invDir.x < 0), int(invDir.y < 0), int(invDir.z < 0)}; <<Follow ray through BVH nodes to find primitive intersections>>
int toVisitOffset = 0, currentNodeIndex = 0; int nodesToVisit[64]; while (true) { const LinearBVHNode *node = &nodes[currentNodeIndex]; <<Check ray against BVH node>>
if (node->bounds.IntersectP(ray.o, ray.d, tMax, invDir, dirIsNeg)) { if (node->nPrimitives > 0) { <<Intersect ray with primitives in leaf BVH node>>
for (int i = 0; i < node->nPrimitives; ++i) { <<Check for intersection with primitive in BVH node>>
pstd::optional<ShapeIntersection> primSi = primitives[node->primitivesOffset + i].Intersect(ray, tMax); if (primSi) { si = primSi; tMax = si->tHit; }
} if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset];
} else { <<Put far BVH node on nodesToVisit stack, advance to near node>>
if (dirIsNeg[node->axis]) { nodesToVisit[toVisitOffset++] = currentNodeIndex + 1; currentNodeIndex = node->secondChildOffset; } else { nodesToVisit[toVisitOffset++] = node->secondChildOffset; currentNodeIndex = currentNodeIndex + 1; }
} } else { if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset]; }
}
return si; }

Each time the following while loop starts an iteration, currentNodeIndex holds the offset into the nodes array of the node to be visited. It starts with a value of 0, representing the root of the tree. The nodes that still need to be visited are stored in the nodesToVisit[] array, which acts as a stack; toVisitOffset holds the offset to the next free element in the stack. With the following traversal algorithm, the number of nodes in the stack is never more than the maximum tree depth. A statically allocated stack of 64 entries is sufficient in practice.

<<Follow ray through BVH nodes to find primitive intersections>>=
int toVisitOffset = 0, currentNodeIndex = 0; int nodesToVisit[64]; while (true) { const LinearBVHNode *node = &nodes[currentNodeIndex]; <<Check ray against BVH node>>
if (node->bounds.IntersectP(ray.o, ray.d, tMax, invDir, dirIsNeg)) { if (node->nPrimitives > 0) { <<Intersect ray with primitives in leaf BVH node>>
for (int i = 0; i < node->nPrimitives; ++i) { <<Check for intersection with primitive in BVH node>>
pstd::optional<ShapeIntersection> primSi = primitives[node->primitivesOffset + i].Intersect(ray, tMax); if (primSi) { si = primSi; tMax = si->tHit; }
} if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset];
} else { <<Put far BVH node on nodesToVisit stack, advance to near node>>
if (dirIsNeg[node->axis]) { nodesToVisit[toVisitOffset++] = currentNodeIndex + 1; currentNodeIndex = node->secondChildOffset; } else { nodesToVisit[toVisitOffset++] = node->secondChildOffset; currentNodeIndex = currentNodeIndex + 1; }
} } else { if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset]; }
}

At each node, the first step is to check if the ray intersects the node’s bounding box (or starts inside of it). The node is visited if so, with its primitives tested for intersection if it is a leaf node or its children are visited if it is an interior node. If no intersection is found, then the offset of the next node to be visited is retrieved from nodesToVisit[] (or traversal is complete if the stack is empty). See Figures 7.12 and 7.13 for visualizations of how many nodes are visited and how many intersection tests are performed at each pixel for two complex scenes.

<<Check ray against BVH node>>=
if (node->bounds.IntersectP(ray.o, ray.d, tMax, invDir, dirIsNeg)) { if (node->nPrimitives > 0) { <<Intersect ray with primitives in leaf BVH node>>
for (int i = 0; i < node->nPrimitives; ++i) { <<Check for intersection with primitive in BVH node>>
pstd::optional<ShapeIntersection> primSi = primitives[node->primitivesOffset + i].Intersect(ray, tMax); if (primSi) { si = primSi; tMax = si->tHit; }
} if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset];
} else { <<Put far BVH node on nodesToVisit stack, advance to near node>>
if (dirIsNeg[node->axis]) { nodesToVisit[toVisitOffset++] = currentNodeIndex + 1; currentNodeIndex = node->secondChildOffset; } else { nodesToVisit[toVisitOffset++] = node->secondChildOffset; currentNodeIndex = currentNodeIndex + 1; }
} } else { if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset]; }

If the current node is a leaf, then the ray must be tested for intersection with the primitives inside it. The next node to visit is then found from the nodesToVisit stack; even if an intersection is found in the current node, the remaining nodes must be visited in case one of them yields a closer intersection.

<<Intersect ray with primitives in leaf BVH node>>=
for (int i = 0; i < node->nPrimitives; ++i) { <<Check for intersection with primitive in BVH node>>
pstd::optional<ShapeIntersection> primSi = primitives[node->primitivesOffset + i].Intersect(ray, tMax); if (primSi) { si = primSi; tMax = si->tHit; }
} if (toVisitOffset == 0) break; currentNodeIndex = nodesToVisit[--toVisitOffset];

If an intersection is found, the tMax value can be updated to the intersection’s parametric distance along the ray; this makes it possible to efficiently discard any remaining nodes that are farther away than the intersection.

<<Check for intersection with primitive in BVH node>>=
pstd::optional<ShapeIntersection> primSi = primitives[node->primitivesOffset + i].Intersect(ray, tMax); if (primSi) { si = primSi; tMax = si->tHit; }

For an interior node that the ray hits, it is necessary to visit both of its children. As described above, it is desirable to visit the first child that the ray passes through before visiting the second one in case the ray intersects a primitive in the first one. If so, the ray’s tMax value can be updated, thus reducing the ray’s extent and thus the number of node bounding boxes it intersects.

An efficient way to perform a front-to-back traversal without incurring the expense of intersecting the ray with both child nodes and comparing the distances is to use the sign of the ray’s direction vector for the coordinate axis along which primitives were partitioned for the current node: if the sign is negative, we should visit the second child before the first child, since the primitives that went into the second child’s subtree were on the upper side of the partition point. (And conversely for a positive-signed direction.) Doing this is straightforward: the offset for the node to be visited first is copied to currentNodeIndex, and the offset for the other node is added to the nodesToVisit stack. (Recall that the first child is immediately after the current node due to the depth-first layout of nodes in memory.)

<<Put far BVH node on nodesToVisit stack, advance to near node>>=
if (dirIsNeg[node->axis]) { nodesToVisit[toVisitOffset++] = currentNodeIndex + 1; currentNodeIndex = node->secondChildOffset; } else { nodesToVisit[toVisitOffset++] = node->secondChildOffset; currentNodeIndex = currentNodeIndex + 1; }

The BVHAggregate::IntersectP() method is essentially the same as the regular intersection method, with the two differences that Primitive’s IntersectP() methods are called rather than Intersect(), and traversal stops immediately when any intersection is found. It is thus not included here.