We investigated the performance of different approaches to parallelizing bounding volume hierarchy construction for triangle meshes using the surface area heuristic. We used OpenMP to parallelize various sections of the serial algorithm, and compared the runtimes of different implementations.
Halfedge triangle meshes are used in computer graphics to represent surfaces in three dimensions. One technique used to render meshes from a particular viewpoint is raytracing, which produces very physically accurate images by simulating the paths rays of light take through a scene to the camera. Since naively intersecting a ray with a mesh with
N primitives is
O(N), a bounding volume hierarchy is used to reduce the cost of ray-mesh intersection to
O(logN). A BVH is a tree structure in which each node knows which primitives lie inside of it, as well as the axis-aligned bounding box that contains all of its primitives.
A good BVH minimizes the amount of empty space inside of the bounding box of a node. In other words, when choosing a partitioning plane between a collection of primitives, we want to choose the plane that results in the smallest bounding boxes around the primitives on each side of it. The surface area heuristic is commonly used to do this: It assigns a cost to a particular partitioning of primitives based on the probability that a random ray entering a bounding box hits a primitive within the box. Psuedocode for a serial recursive algorithm that takes a list of primitives as input and produces a BVH using the surface area heuristic is shown below:
build_bvh(primitives): if number of primitives is small: return for each axis (x, y, z): for each possible partitioning plane between primitives: generate the bounding boxes for the left and right evaluate surface area heuristic (SAH) pick the partitioning plane and axis with the best SAH partition the primitives into left and right nodes build_bvh(primitives in left node) build_bvh(primitives in right node)There are two main approaches to parallelizing this algorithm:
We started with the serial BVH building algorithm from Aaron's implementation of the 15-462 raytracer. We used OpenMP on machines with 6-core Intel Xeon CPUs running at 3.2GHz. Loops over primitives and recursive calls were parallelized using OpenMP tasks.
We developed 3 implementations to investigate the two approaches discussed above:
We tested the three implementations on two different large triangle meshes and compared the average (over 4 trials) wall clock runtime to build the complete BVH.
Overall, we saw pretty similar speedups between the three implementations. Since the runtime increases mostly linearly up to 6 threads, and then increases more slowly up to 12 threads, we can conclude that the algorithm is mostly, but not completely compute bound. The large compute portion is calculating max and min for expanding bounding boxes, and evaluating the SAH. However, the algorithm then partitions the list of pointers to primitives in memory, which is bandwidth-bound. Since speedup completely tapers off past 12 threads, we can see that the memory bandwidth requirements are not worth the cost of a context switch.
Each iteration of build_bvh
news 2 child nodes. Since the tree structure is so large, this results in many, very frequent calls to
new, which may use internal locks. If we had more time, it would be interesting to experiment with a
malloc implementation that uses thread-private memory or better manages contention.
Memory Bus Traffic
After finding the best partitioning plane, a thread must sort the node's primitives so that its child nodes are handed a list of pointers to primitives that is contiguous in memory. This is done using
std::partition, which performs many swaps of elements that may be far apart in memory.
In addition, since each primitive object is
malloc'd separately, dereferencing each pointer to a primitive to get its bounding box is a random memory access.
Since the partition of the root node may not evenly split the primitives, parallelizing over recursive calls may result in workload imbalance, where some of the threads reach the bottom of the tree much earlier than others. Our work queue implementation solves this problem by pushing recursive calls onto a queue that is consumed by a thread pool. The queue is protected by a mutex, since it maintains internal data structures that allow it to know when the BVH build is complete. In well-balanced meshes like the Dragon, we see that its speedup suffers from the added complexity of the work queue compared to the parallel recursion implementation. However, the Wall-E mesh is very unbalanced, as the partition of the root node separates the base and the character. The work queue implementation has a greater speedup here, as it is able to manage this imbalance.
Equal work was performed by both project members.
maps over the edges and vertices in the mesh, and saw an expected speedup.
We are going to investigate the performance of different implementations of mesh downsampling via quadratic error simplification on multicore CPUs. We will focus on ways to efficiently collapse edges in parallel, while minimizing synchronization necessary to maintain the correctness of the data structure, and producting a high quality downsampling.
Halfedge triangle meshes are used in computer graphics to represent surfaces in three dimensions. When a mesh has more vertices and edges than are necessary to effectively model a shape, downsampling can be used to collapse unnecessary edges. Although it is not possible to determine a completely optimized edge collapse order, the quadric error metric developed by Michael Garland and Paul Heckbert provides a heuristic to determine good edge collapse candidates. The basic algorithm computes a quadric error score for each edge and then uses a priority queue to iteratively collapse the edges with the best score. After each edge is collapsed, other edges may also be removed and the heuristic score for some edges may need to be updated. This introduces many opportunities for parallelism. Trivially, we can parallelize the computation of the quadric error scores for each edge. In addition, we will also collapse edges and update quadric error scores in parallel. This will require synchronization between threads to maintain correctness and provides an opportunity to test different work partitioning schemes.
This problem is challenging since removing edges in a halfedge mesh in parallel requires synchronization so that a thread is not trying to traverse or remove edges that another thread has removed or edited. We will investigate the performance of different synchronization techniques including coarse-grained locks on the entire data structure, fine-grained locks on individual mesh elements, transactional memory operations, as well as lock free solutions that ensure that conflicting updates never occur. We will also investiage different work and spacial partitioning schemes, to ensure that synchronization between threads is relatively infrequent. Finally, traversing the halfedge data structure can have very incoherent memory access.
The starter code that we will be working with will be from assignment 2 from 15-462, MeshEdit. The code contains a serial solution to the probelem that we are trying to tackle and we will be using it as a reference while writing our parallel code. We will start by working on the six-core Xeon e5 in the lateday cluster initially and also the Xeon Phi to see which gives us better performance. We will be looking at the difference in performance obtained when we use different methods of synchronizations across the two machines and as we use different implementations for our code. Then we will investigate the performance of our code when it has been ported to the CUDA language so it can work on NVIDIA GPUs.
Our goals for the project are:
Our problem ends up being IO-bound rather than bound computationally. Since problems that are bound by IO are usually better suited for CPU tasks, we will be implementing our code on the latedays cluster first. We will then later check the performance of our code on the GPUs of the Gates, but this will be a secondary objective in nature.