Experiences with Streaming Construction of SAH KD Trees Stefan Popov, Johannes Günther, Hans-Peter Seidel, Philipp Slusallek

Streaming Construction of KD TreesStefan Popov Motivation  Large speed-up of ray tracing lately  Better algorithms (packet tracing [Wald04, Reshetov05] )  Optimized spatial index structures  Best known: KD trees [Havran00]  Faster hardware  Research concentrated mainly on static scenes  Dynamic scenes  Building – slow for SAH based KD trees  Done in a pre-processing step

Streaming Construction of KD TreesStefan Popov Dynamic Scenes Approaches  Embed dynamics in the index structure  Use a two level approach [Wald 03 ]  Fuzzy KD trees [Günther06]  Update index structure  Grids, BVHs and KD tree hybrids  Faster build/update  Lower traversal performance  No efficient approach for KD trees  Rebuild entire KD tree  Need to make it fast  Lazy build

Streaming Construction of KD TreesStefan Popov SAH Algorithm  Extract & sort events in advance  Abstract objects with AABBs  Events given by AABB boundaries  Recursive top-down construction  Find split plane using SAH  Compute minimum cost  Distribute objects to children  By distributing the events  Keep them sorted

Streaming Construction of KD TreesStefan Popov SAH Cost Function   Piecewise linear  Discontinuities at object boundaries  Evaluate only before opening and after closing event

Streaming Construction of KD TreesStefan Popov Distribution Along the Split Axis  Given: event list & split position  Sweep event list and classify  Open event  Before split  label object “both”  After split  label object “right”  Close event  Before split  re-label object “left”  Copy event to corresponding child’s list  Might have to insert new events  Random memory access

Streaming Construction of KD TreesStefan Popov Distribution Along the Other Axes  Sweep event lists. Copy event to  Left, if corresponding object labeled “left” or “both”  Right, if corresponding object labeled “right” or “both”  Look up in object array  Random memory access

Streaming Construction of KD TreesStefan Popov Problems of KD Tree Construction  Random memory accesses  Expensive cost function evaluation  Initial sorting – inefficient for lazy builds

Streaming Construction of KD TreesStefan Popov Streaming Algorithm Overview  Work with unsorted lists of AABBs  Avoid initial sorting  Sweep list once to locate initial split plane  In a single sweep  Distribute objects (straightforward)  Determine split positions of children  Once data fits in caches, switch to conventional build

Streaming Construction of KD TreesStefan Popov SAH Cost Estimation  Cost function typically varies only slowly  No need to evaluate SAH at every event  Use sampling!  Naïve approach  For every event: check all samples  O(kN)  How to sample efficiently?

Streaming Construction of KD TreesStefan Popov Efficient Sampling  Two step approach  #Objects to left of sample = #Opening events to its left  #Objects to right of sample = #Closing events to its right  Count opening/closing events between samples  Regular sampling  index computation in O(1)  Reconstruct left/right object counts at samples  Using two partial sums from left and right  O(k+N)

Streaming Construction of KD TreesStefan Popov Refining of Samples  SAH – sum of two monotone functions – C l and C r  Cost between two samples a < b is bounded from below  C  C min = min(C l ) + min(C r ) = C l (a) + C r (b)  Resample areas where C min < current minimum  Typically only few intervals need to be re-sampled (< 5%)

Streaming Construction of KD TreesStefan Popov Algorithm properties  Streaming memory accesses  SAH cost function estimated by sampling  No initial sorting required  Refining of Samples

Streaming Construction of KD TreesStefan Popov Improvements  Conventional Algorithm  Use radix sort – O(N)  Fastest algorithm if data set fits into caches  No need to order events at same position  Count opening/closing events instead  Removes one radix sort pass  Multiple cores  parallelize build  Most time spent in the lower tree levels  One sub-tree  one core

Streaming Construction of KD TreesStefan Popov Results  Speed-up up to 50%  Only effective in the upper levels  Limited by copying of object/events  The larger the scene, the higher the speedup  Performance independent of triangle order  Small decrease in traversal performance (< 2%)  With 1024 samples  Multi-threading  4 cores (no local memory management)

Streaming Construction of KD TreesStefan Popov Future Work  Fully multi-threaded implementation  Carefully memory management on NUMA architectures  Extend to other spatial index structures  BVHs, BKD trees, SKD trees, …

Streaming Construction of KD TreesStefan Popov Conclusion  Streaming construction algorithm  50% speedup  Cost function sampling  Very low quality degradation  Refining of samples

Streaming Construction of KD TreesStefan Popov Thank you!

Streaming Construction of KD TreesStefan Popov Advantages  Sequential memory access in the upper levels  Small data foot print in conventional build  Fits in caches  Radix sort is efficient  Less computations needed for split plane position estimation  But, what about the tree cost?

Streaming Construction of KD TreesStefan Popov Memory Managment  Use two arrays and alternate them

Streaming Construction of KD TreesStefan Popov SAH tree cost  Optimal KD tree for ray tracing  SAH based  Minimize average expected traversal cost of an arbitrary ray  

Streaming Construction of KD TreesStefan Popov SAH computation  Efficient computation – extract & sort events in advance  Compute incrementally. Keep track of objects on left/right  Evaluate after close, before an open events

Streaming Construction of KD TreesStefan Popov Alternative Multi-Threading  required on NUMA architectures)  Sub-tree  core not suitable for the first log(#cores) levels  Also unsuitable for some architecture (Cell)  Alternative  Bring data to cores from sequential pages  Gather event counts in bins at each core  Merge counts before actual cost evaluation

Streaming Construction of KD TreesStefan Popov Extension: Multi-Threading  Multiple cores  parallelize build  Most time spent in the lower tree levels  One sub-tree  one core