Anti-aliased and accelerated ray tracing

Reading Required: Further reading: Watt, sections 12.5.3 – 12.5.4, 14.7 Further reading: A. Glassner. An Introduction to Ray Tracing. Academic Press, 1989. [In the lab.] University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 2

Aliasing in rendering Sketch the triangle that would create the given image -- lower right corner coverage. Called the jaggies. Point to frame 1, 4, 7, 10, 13 One of the most common rendering artifacts is the “jaggies”. Consider rendering a white polygon against a black background: We would instead like to get a smoother transition: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 3

Anti-aliasing Q: How do we avoid aliasing artifacts? Sampling: 1. Increase sampling rate -- not practical for fixed resolution display. 2. Smooth out high frequences analytically. Requires an analytic function. 3. Supersample and average down. Q: How do we avoid aliasing artifacts? Sampling: Pre-filtering: Combination: Example - polygon: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4

Polygon anti-aliasing Show anti-aliased image with ray sample. A: Motion blur Draw a dot streaking across the screen, first as jumps then as smear. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 5

Antialiasing in a ray tracer Put a box around the “support of a pixel”. Show ray-traced output We would like to compute the average intensity in the neighborhood of each pixel. When casting one ray per pixel, we are likely to have aliasing artifacts. To improve matters, we can cast more than one ray per pixel and average the result. A.k.a., super-sampling and averaging down. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 6

Speeding it up Vanilla ray tracing is really slow! Put a box around the “support of a pixel”. Show ray-traced output Vanilla ray tracing is really slow! Consider: m x m pixels, k x k supersampling, and n primitives, average ray path length of d, with 2 rays cast recursively per intersection. Complexity = For m=1,000,000, k = 5, n = 100,000, d=8…very expensive!! In practice, some acceleration technique is almost always used. We’ve already looked at reducing d with adaptive ray termination. Now we look at reducing the effect of the k and n terms. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 7

Antialiasing by adaptive sampling |I1 - I2| > thresh I1/I2 or I2/I1 > 1 + thresh -- better since it’s based on relative values, which is the way the eye perceives. Show how ray sample doesn’t take 16X longer to trace. Casting many rays per pixel can be unnecessarily costly. For example, if there are no rapid changes in intensity at the pixel, maybe only a few samples are needed. Solution: adaptive sampling. Q: When do we decide to cast more rays in a particular area? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 8

Faster ray-polyhedron intersection Let’s say you were intersecting a ray with a polyhedron: Straightforward method intersect the ray with each triangle return the intersection with the smallest t-value. Q: How might you speed this up? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 9

Ray Tracing Acceleration Techniques 1 N Faster Intersection Fewer Rays Generalized Approaches Tighter bounds Faster intersector Uniform grids Spatial hierarchies k-d, oct-tree, bsp hierarchical grids Hierarchical bounding volumes (HBV) Early ray termination Adaptive sampling Beam tracing Cone tracing Pencil tracing Readings Arvo and Kirk Arvo, Metahierarchies Havran and “Best Efficiency Scheme” Beam tracing techniques fail for two reasons: Intersection calculations are too complicated; imagine cone-algebraic surface Beams don’t remain closed under reflection and refraction off curved surfaces Bounds First test bounds, then do detailed intersection calculation Trade-off on how tight the bounds are; the tighter the bound, the higher the Probability of a real intersection given an intersection with the bound, but the higher the cost of rejection. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 10

Uniform spatial subdivision Another approach is uniform spatial subdivision. Idea: Partition space into cells (voxels) Associate each primitive with the cells it overlaps Trace ray through voxel array using fast incremental arithmetic to step from cell to cell University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 11

Uniform Grids Preprocess scene Find bounding box Two reasons for using a grid. Cull out objects not near the ray. Only consider those objects in the voxels along the ray’s path Early termination. Stop when you find the point of closest intersection. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 12

Uniform Grids Preprocess scene Find bounding box Determine resolution Setting the resolution is tricky. If the resolution is too high, then you step through too many voxels. If the resolution is too low, then you perform too many intersections. Add Tim’s study of voxel density. Roughly set so that the average time spent walking voxels is equal to the average time spent test for intersection. D=3 is typical. Woo describes an algorithm for non-uniformly allocating voxels in each direction. D is the voxel density, and was introduced by Cleary and Wyvill? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 13

Uniform Grids Preprocess scene Find bounding box Determine resolution Place object in cell, if object overlaps cell University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 14

Uniform Grids Preprocess scene Find bounding box Determine resolution Place object in cell, if object overlaps cell Check that object intersects cell Could remove interior voxels as well. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 15

Uniform Grids Preprocess scene Traverse grid 3D line – 3D-DDA 6-connected line University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 16

Caveat: Overlap Optimize for objects that overlap multiple cells Traverse until tmin(cell) > tmax(ray) Problem: Redundant intersection tests: Solution: Mailboxes Assign each ray an increasing number Primitive intersection cache (mailbox) Store last ray number tested in mailbox Only intersect if ray number is greater Mailboxes first proposed by Amanaidas and Woo Each ray is assigned a unique number. Latter rays have higher numbers than earlier rays. Store the ray number with each primitive. Don’t test rays if their ray number is less than or equal to the primitive’s ray number. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 17

Non-uniform spatial subdivision Still another approach is non-uniform spatial subdivision. Other variants include k-d trees and BSP trees. Various combinations of these ray intersections techniques are also possible. See Glassner and pointers at bottom of project web page for more. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 18

Non-uniform spatial subdivision Best approach - k-d trees or perhaps BSP trees More adaptive to actual scene structure BSP vs. k-d tradeoff between speed from simplicity and better adaptability 1 2 3 4 5 6 7 8 University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 19

Spatial Hierarchies Letters correspond to planes (A) As questions? Kay-Kajiya Properties of hierarchical bounding volumes … A Letters correspond to planes (A) Point Location by recursive search University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 20

Spatial Hierarchies Letters correspond to planes (A, B) As questions? Kay-Kajiya Properties of hierarchical bounding volumes … Letters correspond to planes (A, B) Point Location by recursive search University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 21

Spatial Hierarchies Letters correspond to planes (A, B, C, D) As questions? Kay-Kajiya Properties of hierarchical bounding volumes … Letters correspond to planes (A, B, C, D) Point Location by recursive search University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 22

Variations kd-tree oct-tree bsp-tree Kd-tree where you set the position of the splitting plane. Oct-tree where you set the position of the center point. kd-tree oct-tree bsp-tree University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 23

Ray Traversal Algorithms Recursive inorder traversal [Kaplan, Arvo, Jansen] Intersect(L,tmin,tmax) Intersect(L,tmin,t*) Intersect(R,t*,tmax) Intersect(R,tmin,tmax) University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 24

Build Hierarchy Top-Down ? What exactly is the surface-area heuristic Choose splitting plane Midpoint Median cut Surface area heuristic University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 25

Surface Area and Rays Number of rays in a given direction that hit an object is proportional to its projected area The total number of rays hitting an object is Crofton’s Theorem: For a convex body For example: sphere University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 26

Surface Area and Rays The probability of a ray hitting a convex shape that is completely inside a convex cell equals Question after class. How is this used? Need to show in detail how this information Is used to find the optimal hierarchy. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 27

Surface Area Heuristic Intersection time Traversal time What exactly is the surface-area heuristic a b University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 28

Surface Area Heuristic b 2n splits What exactly is the surface-area heuristic University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 29

Hierarchical bounding volumes We can generalize the idea of bounding volume acceleration with hierarchical bounding volumes. Key: build balanced trees with tight bounding volumes. Many different kinds of bounding volumes. Note that bounding volumes can overlap. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 30