Tiled Polygon Traversal Using Half-Plane Edge Functions

1. Tiled Polygon Traversal Using Half-Plane Edge Functions Joel McCormack & Bob McNamara

2. What Do You Want To Do? • Partition screen into rectangular tiles • Visit all locations in triangle in one tile before any in the next tile

3. Why Do You Want To Do That? • Improve frame buffer memory access patterns • DRAM access to open page faster than non-open page • Staying on open page improves average access time... • …and improves prefetching of pages into other banks • Improve texture cache access patterns • Staying within limited area increases cache hits by exploiting 2D spatial coherency of texel accesses

4. Outline • Half-Plane Edge Functions • Fragment Stamp and Movement • A Simple Traversal Algorithm • A Tiling Traversal Algoriththm • Uses of Tiling and Even Metatiling • Conclusions

5. Half-Plane Edge Functions • Many rasterizers traverse polygons by surrounding them with edge functions going from vertex to vertex • E01(x, y) = (x – x0)(y1 – y0) – (y – y0)(x1 – x0) splits 2D plane into two halves: • Non-negative on or to right of edge • Negative to left of edge • If all edge functions at (x,y) have same sign, then (x, y) is within polygon

6. Triangle Surrounded By Edge Functions • Non-negative half-planes have “shadow” lines • All three edges non-negative in shaded yellow area

7. Zen and the Art of Polygon Traversal • Scanline traversal like being in the military: • Start exactly right here • Go exactly to there, then stop • Repeat • Half-plane traversal more like Zen wandering to enlightenment: • Where am I? • Where can I go? Have I reached my limits? • Okay, how about that away? (For now.)

8. Fragment Stamp: Fragment Sample Points and Probe Points •  marks sample point: is fragment inside object? •  marks probe point: combine to determine if more object to left, right, above, below?

9. Fragment Stamp: Which Adjacent Stamp Positions Are Valid? • Does a stamp segment, e.g. (RB, LB), intersect the object (and thus indicate valid adjacent position)? • Probably, if: • Each edge contains at least one stamp segment endpoint • Object’s bounding box contains at least one stamp segment endpoint

10. Non-Tiling Traversal Example 0. Traverse first stampline 1. Traverse stamplines above 2. Traverse stamplines below Zen, yeah right. Looks like goofy scanline traversal

11. A Non-Tiling Traversal Algorithm Starting at leftmost vertex... 0. Proceed right as long as valid position • Save first valid up position • Save first valid down position 1. Jump to saved up, proceed right as long as valid • Save first valid up, repeat 1 until no more up 2. Jump to saved down, proceed right as long as valid • Save first valid down, repeat 2 until no more down

12. Tiling Traversal Observations • Easy to obey top and bottom tile boundaries • Go down from initial scanline until hit tile bottom • Then do original algorithm starting at Step 1 (go up) • Need new saved state to obey right tile boundary • Original algorithm, but stop at right tile boundary, and save first right position in next tile over • Traverse all of object in first tile column • Then move to saved right position and restart algorithm • Combine both to obey all tile boundaries

13. Tiling Traversal Example 0. Traverse first stampline in tile ½. Traverse stamplines below in same tile 1. Traverse stamplines above in tile column 2. Traverse stamplines below in tile column (then next column) Let’s see a grubby scanline algorithm do that!

14. A Tiling Traversal Algorithm 0. Proceed right as long as valid positionand in same tile • Save first valid up and down positions ½. Jump to saved down, proceed right in same tile • Save first valid down, repeat ½ while down in same tile 1. Jump to saved up, proceed right in same tile • Save first valid up, repeat 1 until no more up 2. Jump to saved down, proceed right in same tile • Save first valid down, repeat 2 until no more down 3. Jump to saved right (in new tile), go back to Step 0

15. Obvious Uses for Tiling • Optimize frame buffer access patterns • Tile size & dimensions match 2D page in frame buffer • All positions in object & page visited before new page • Increases available time to prefetch next page • Serpentine variation reduces non-prefetchable crossings • Reduce texture cache miss rate • Tile size related to texture cache size • Tile dimensions long and skinny to minimize texture fetches that will suffer capacity misses

16. Subset Metatiling • Visit all location in tile before next tile, and all tiles in metatile before next metatile • Requires three additional save states, though

17. Subset Metatiling Uses • Optimize hierarchical frame buffer caching • Mitsubishi 3D-RAM has two cache levels • Optimize hierarchical texture caches • Or optimize frame buffer access and texture cache access simultaneously

18. Non-Subset Metatiling • Tiles not contained by metatiles • Visit all locations in tile also in metatile before new tile

19. Conclusions • Tiling is easy to add to a half-plane based rasterizer • Tiling increases frame buffer memory efficiency • Tiling increases effectiveness of texture cache • Scanline rasterizers should die a quiet death