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
• Objects 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)
Lets see a grubby scanline algorithm do that!
14
A Tiling Traversal Algorithm

• 0. Proceed right as long as valid position and
  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