1April 1, 2003Frank PfenningCarnegie Mellon Universityhttp://www.cs.cmu.edu/~fp/courses/graphics/Hierarchical Bounding VolumesRegular GridsOctreesBSP TreesConstructive Solid Geometry (CSG)[Angel 9.10]Spatial Data Structures15-462 Computer Graphics ILecture 1704/01/2003 15-462 Graphics I 2Ray Tracing Acceleration• Faster intersections– Faster ray-object intersections• Object bounding volume• Efficient intersectors– Fewer ray-object intersections• Hierarchical bounding volumes (boxes, spheres)• Spatial data structures• Directional techniques• Fewer rays– Adaptive tree-depth control– Stochastic sampling• Generalized rays (beams, cones)04/01/2003 15-462 Graphics I 3Spatial Data Structures• Data structures to store geometric information• Sample applications– Collision detection– Location queries– Chemical simulations– Rendering• Spatial data structures for ray tracing– Object-centric data structures (bounding volumes)– Space subdivision (grids, octrees, BSP trees)– Speed-up of 10x, 100x, or more04/01/2003 15-462 Graphics I 4Bounding Volumes• Wrap complex objects in simple ones• Does ray intersect bounding box?– No: does not intersect enclosed objects– Yes: calculate intersection with enclosed objects• Common types– Boxes, axis-aligned– Boxes, oriented– Spheres– Finite intersections or unions of above04/01/2003 15-462 Graphics I 5Selection of Bounding Volumes• Effectiveness depends on:– Probability that ray hits bounding volume, but not enclosed objects (tight fit is better)– Expense to calculate intersections with bounding volume and enclosed objects• Amortize calculation of bounding volumes• Use heuristicsgoodbad04/01/2003 15-462 Graphics I 6Hierarchical Bounding Volumes• With simple bounding volumes, ray casting still has requires O(n) intersection tests• Idea use tree data structure– Larger bounding volumes contain smaller ones etc.– Sometimes naturally available (e.g. human figure)– Sometimes difficult to compute• Often reduces complexity to O(log(n))204/01/2003 15-462 Graphics I 7Ray Intersection Algorithm• Recursively descend tree• If ray misses bounding volume, no intersection• If ray intersects bounding volume, recurse with enclosed volumes and objects• Maintain near and far bounds to prune further• Overall effectiveness depends on model and constructed hierarchy04/01/2003 15-462 Graphics I 8Spatial Subdivision• Bounding volumes enclose objects, recursively• Alternatively, divide space• For each segment of space keep list of intersecting surfaces or objects• Basic techniques– Regular grids– Octrees (axis-aligned, non-uniform partition)– BSP trees (recursive Binary Space Partition, planes)04/01/2003 15-462 Graphics I 9Grids• 3D array of cells (voxels) that tile space• Each cell points to all intersecting surfaces• Intersection algsteps from cellto cell04/01/2003 15-462 Graphics I 10Caching Intersection points• Objects can span multiple cells• For A need to test intersection only once• For B need to cache intersection and check next cell for closer one• If not, C could be missed(yellow ray)ABC04/01/2003 15-462 Graphics I 11Assessment of Grids• Poor choice when world is non-homogeneous• Size of grid– Too small: too many surfaces per cell– Too large: too many empty cells to traverse– Can use alg like Bresenham’s for efficient traversal• Non-uniform spatial subdivision more flexible– Can adjust to objects that are present04/01/2003 15-462 Graphics I 12Outline• Hierarchical Bounding Volumes• Regular Grids• Octrees• BSP Trees• Constructive Solid Geometry (CSG)304/01/2003 15-462 Graphics I 13Quadtrees• Generalization of binary trees in 2D– Node (cell) is a square– Recursively split into 4 equal sub-squares– Stop subdivision based on number of objects• Ray intersection has to traverse quadtree• More difficult to step to next cell04/01/2003 15-462 Graphics I 14Octrees• Generalization of quadtree in 3D• Each cell may be split into 8 equal sub-cells• Internal nodes store pointers to children• Leaf nodes store list of surfaces• Adapts well to non-homogeneous scenes04/01/2003 15-462 Graphics I 15Assessment for Ray Tracing• Grids– Easy to implement– Require a lot of memory– Poor results for non-homogeneous scense• Octrees– Better on most scenes (more adaptive)• Alternative: nested grids• Spatial subdivision expensive for animations• Hierarchical bounding volumes– Natural for hierarchical objects– Better for dynamic scenes04/01/2003 15-462 Graphics I 16Other Spatial Subdivision Techniques• Relax rules for quadtrees and octrees• k-dimensional tree (k-d tree)– Split at arbitrary interior point– Split one dimension at a time• Binary space partitioning tree (BSP tree)– In 2 dimensions, split with any line– In k dims. split with k-1 dimensional hyperplane– Particularly useful for painter’s algorithm– Can also be used for ray tracing [see handout]04/01/2003 15-462 Graphics I 17Outline• Hierarchical Bounding Volumes• Regular Grids• Octrees• BSP Trees• Constructive Solid Geometry (CSG)04/01/2003 15-462 Graphics I 18BSP Trees• Split space with any line (2D) or plane (3D)• Applications– Painters algorithm for hidden surface removal– Ray casting• Inherent spatial ordering given viewpoint– Left subtree: in front, right subtree: behind• Problem: finding good space partitions– Proper ordering for any viewpoint– Balance tree• For details, see http://reality.sgi.com/bspfaq/404/01/2003 15-462 Graphics I 19Building a BSP Tree• Use hidden surface removal as intuition• Using line 1 or line 2 as root is easyLine 2Line 3Line 1Viewpoint1123B A C Da BSP treeusing 2 as rootABDC32the subdivisionof space it implies04/01/2003 15-462 Graphics I 20Splitting of surfaces• Using line 3 as root requires splittingLine 2aLine 3Line 1Viewpoint12a2bLine 2b304/01/2003 15-462 Graphics I 21Building a Good Tree• Naive partitioning of n polygons yields O(n3) polygons (in 3D)• Algorithms with O(n2) increase exist– Try all, use polygon with fewest splits– Do not need to split exactly along polygon planes• Should balance tree– More splits allow easier balancing– Rebalancing?04/01/2003 15-462 Graphics I 22Painter’s Algorithm with BSP Trees• Building the tree– May need to split some polygons– Slow, but done only once• Traverse back-to-front or front-to-back– Order is
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