Today. Acceleration Data Structures for Ray Tracing. Cool results from Assignment 2. Last Week: Questions? Schedule
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1 Today Acceleration Data Structures for Ray Tracing Cool results from Assignment 2 Last Week: koi seantek Ray Tracing Shadows Reflection Refraction Local Illumination Bidirectional Reflectance Distribution Function (BRDF) Phong Model θ r θ i φ i φ r Schedule Questions? Wednesday October 1 st : Assignment 3 (Ray Tracing & Phong Materials) due Sunday October 5 th, 5-7 PM, Room TBA: Review Session for Quiz 1 Tuesday October 7 th : Quiz 1: In class Wednesday October 15th: Assignment 4 (Grid Acceleration) due
2 Today Extra rays needed for these effects: Distribution Ray Tracing Soft shadows Anti- aliasing (getting rid of jaggies) Glossy reflection Motion blur Depth of field (focus) Shadows one shadow ray per intersection per point light source no shadow rays Soft Shadows multiple shadow rays to sample area light source one shadow ray one shadow ray lots of shadow rays Antialiasing Supersampling Reflection multiple rays per pixel jaggies w/ antialiasing one reflection ray per intersection point light area light θ θ perfect mirror
3 Glossy Reflection multiple reflection rays Motion Blur Sample objects temporally Justin Legakis θ θ polished surface Rob Cook film Depth of Field multiple rays per pixel focal length Justin Legakis Algorithm Analysis Ray casting Lots of primitives Recursive Distributed Ray Tracing Effects Soft shadows Anti- aliasing Glossy reflection Motion blur Depth of field cost height * width * num primitives * intersection cost * num shadow rays * supersampling * num glossy rays * num temporal samples * max recursion depth *... can we reduce this? Questions? Today of each primitive of groups of ed primitives
4 Acceleration of Ray Casting Goal: Reduce the number of ray/primitive intersections Conservative Bounding Region First check for an intersection with a conservative bounding region Early reject Conservative Bounding Regions Intersection with Axis-Aligned Box tight avoid false positives fast to intersect non-aligned bounding box bounding sphere axis-aligned bounding box From Lecture 3, Ray Casting II t 1y t near t 1x t far x=x1 t 2y t 2x x=x2 y=y2 y=y1 For all 3 axes, calculate the intersection distances t 1 and t 2 t near = max (t 1x, t 1y, t 1z ) t far = min (t 2x, t 2y, t 2z ) If t near > t far, box is missed If t far < t min, box is behind If box survived tests, report intersection at t near arbitrary convex region (bounding half-spaces) Bounding Box of a Triangle Bounding Box of a Sphere (x 1, y 1, z 1 ) (x 0, y 0, z 0 ) = (max(x 0,x 1,x 2 ), max(y 0,y 1,y 2 ), max(z 0,z 1,z 2 )) (x, y, z) r = (x+r, y+r, z+r) = (min(x 0,x 1,x 2 ), min(y 0,y 1,y 2 ), min(z 0,z 1,z 2 )) (x 2, y 2, z 2 ) = (x-r, y-r, z-r)
5 Bounding Box of a Plane Bounding Box of a Group = (+, +, + )* (x max_a, y max_a, z max_a ) (x max_b, y max_b, z max_b ) = (max(x max_a,x max_b ), max(y max_a,y max_b ), max(z max_a,z max_b )) n = (a, b, c) ax + by + cz = d (x min_b, y min_b, z min_b ) (x min_a, y min_a, z min_a ) = (-, -, - )* * unless n is exactly perpendicular to an axis = (min(x min_a,x min_b ), min(y min_a,y min_b ), min(z min_a,z min_b )) Bounding Box of a Transform Special Case: Transformed Triangle (x' max, y' max, z' max ) = (max(x 0,x 1,x 2,x 3,x 4,x 5,x 6,x 7 ), max(y 0,y 1,y 2,y 3,y 4,x 5,x 6,x 7 ), max(z 0,z 1,z 2,z 3,z 4,x 5,x 6,x 7 )) Can we do better? M (x 3,y 3,z 3 ) = M (x max,y max,z min ) M (x 2,y 2,z 2 ) = M (x min,y max,z min ) (x 1,y 1,z 1 ) = M (x max,y min,z min ) (x' min, y' min, z' min ) = (min(x 0,x 1,x 2,x 3,x 4,x 5,x 6,x 7 ), min(y 0,y 1,y 2,y 3,y 4,x 5,x 6,x 7 ), min(z 0,z 1,z 2,z 3,z 4,x 5,x 6,x 7 )) (x 0,y 0,z 0 ) = M (x min,y min,z min ) Special Case: Transformed Triangle Questions? (x 1, y 1, z 1 ) (x 0, y 0, z 0 ) (x 2, y 2, z 2 ) M = (min(x' 0,x' 1,x' 2 ), min(y' 0,y' 1,y' 2 ), min(z' 0,z' 1,z' 2 )) = (max(x' 0,x' 1,x' 2 ), max(y' 0,y' 1,y' 2 ), max(z' 0,z' 1,z' 2 )) (x' 0,y' 0,z' 0 ) = M (x 0,y 0,z 0 ) (x' 1,y' 1,z' 1 ) = M (x 1,y 1,z 1 ) (x' 2,y' 2,z' 2 ) = M (x 2,y 2,z 2 )
6 Today Regular Grid Regular Grid Adaptive Grids Hierarchical Bounding Volumes Create grid Find bounding box of scene Choose grid spacing grid x need not = grid y Cell (i, j) Insert primitives into grid Primitives that overlap multiple cells? Insert into multiple cells (use pointers) grid y grid x For each cell along a ray Does the cell contain an intersection? Yes: return closest intersection No: continue Preventing repeated computation Perform the computation once, "mark" the object Don't re- intersect marked objects
7 Don't return distant intersections Where do we start? If intersection t is not within the cell range, continue (there may be something closer) Intersect ray with scene bounding box Ray origin may be inside the scene bounding box Cell (i, j) t next_h t next_v t min t min t next_v t next_h Is there a pattern to cell crossings? What's the next cell? Yes, the horizontal and vertical crossings have regular spacing dt v = grid y / dir y if t next_v < t next_h i += sign x t min = t next_v t next_v += dt v else j += sign y t min = t next_h Cell (i+1, j) Cell (i, j) t next_v t min t next_h t next_h += dt h (dir x, dir y ) dt h = grid x / dir x grid y (dir x, dir y ) dt h dt v grid x if (dir x > 0) sign x = 1 else sign x = -1 if (dir y > 0) sign y = 1 else sign y = -1 What's the next cell? 3DDDA Three Dimensional Digital Difference Analyzer We'll see this again later, for line rasterization Pseudo-code create grid insert primitives into grid for each ray r find initial cell c(i,j), t min, t next_v & t next_h compute dt v, dt h, sign x and sign y while c!= NULL for each primitive p in c intersect r with p if intersection in range found return c = find next cell
8 Regular Grid Discussion Questions? Advantages? easy to construct easy to traverse Disadvantages? may be only sparsely filled geometry may still be clumped Today Regular Grid Adaptive Grids Hierarchical Bounding Volumes Adaptive Grids Subdivide until each cell contains no more than n elements, or maximum depth d is reached Nested Grids Octree/(Quadtree) Primitives in an Adaptive Grid Can live at intermediate levels, or be pushed to lowest level of grid Adaptive Grid Discussion Advantages? grid complexity matches geometric density Disadvantages? more expensive to traverse (especially octree) Octree/(Quadtree)
9 Bounding Volume Hierarchy Find bounding box of objects Split objects into two groups Recurse Bounding Volume Hierarchy Find bounding box of objects Split objects into two groups Recurse Bounding Volume Hierarchy Find bounding box of objects Split objects into two groups Recurse Bounding Volume Hierarchy Find bounding box of objects Split objects into two groups Recurse Bounding Volume Hierarchy Find bounding box of objects Split objects into two groups Recurse Where to split objects? At midpoint OR Sort, and put half of the objects on each side OR Use modeling hierarchy
10 Intersection with BVH Check subvolume with closer intersection first Intersection with BVH Don't return intersection immediately if the other subvolume may have a closer intersection Bounding Volume Hierarchy Discussion Advantages easy to construct easy to traverse binary Disadvantages may be difficult to choose a good split for a node poor split may result in minimal spatial pruning Today Transformation Hierarchy Questions? group Group & Transformation hierarchy may not be a good spatial hierarchy A B C Flatten group D E A B group C D C E
11 Assignment 4 (due Oct 15 th ) Bounding boxes for primitives Regular grid acceleration data structure Flatten the ation hierarchy Collect statistics Average # of rays per pixel Average # of ray/primitive intersections per pixel Extra Credit: Distribution Ray Tracing (and anything else from past weeks) Next Time: Curves & Surfaces
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