Rasterization

1
Rasterization

• Anselmo Lastra
• COMP290
• Fall 2003

2
Administrative

• When to have the final project presentations?

Last Class
Candidate days circled. Could go as late as 17th.
Last Day of Exams
3
Next Time

• Hakura and Gupta, "The design and analysis of a
  cache architecture for texture mapping", ISCA
  1997.
• Homan Igehy, Matthew Eldridge , Kekoa Proudfoot,
  "Prefetching in a texture cache architecture",
  Graphics Hardware 98 (HWWS).
• Links on web page

4
Rasterization

• Identify pixels/samples covered by a primitive
• Generate fragment parameters
• Z
• Color
• Normals
• Texture coordinates
• etc.

5
Outline

• Historical order
• Skip scan line rasterizer
• Pixel-Planes
• Pineda

6
Rasterization on Pixel-Planes

• Processing built into frame buffer, a processor
  per pixel.
• Each processor used linear expression, Ax By
  C, for each edge
• 0 on edge, positive/negative indicate which side
• Look at all edges to decide whether processor on
  or off

7
Enable All Processors
8
Edge 1
_

Ax By C
9
Edge 2
Ax By C

_
10
Edge 3
_

Ax By C
11
Depth and Color Interpolation
z F(x, y) and color F(x, y) for each of
red, green, and blue
12
Tree

• Each pixel has a one-bit ALU
• Henry came up with a clever way to parallelize
  the computation of Ax By C

ALUs
Memory
A, B, C
13
C3
C2
C1
C0
A1
A0
0
A3
0
A2
C0
C1
C0
C1A0
14
More Details

• Actually 2 trees
• one for A, the other for B
• Each chip had a register to select starting x,y
  screen position
• Supertree on each chip to compute what upper
  levels of tree would have done

15
Pinedas Approach

• Extended edge equation approach to systems that
  did not compute all pixels in parallel
• Incremental computation
• Traversal

16
Different Formulation

• Explicit formulas
• E(x,y) (x-X) dY (y-Y) dX
• Can compute incrementally
• E(x 1, y) E(x,y) dY
• E(x, y 1) E(x,y) dX
• Test Ei for all edges i

17
Traversing the Polygon

• Simplest way test pixels in BBox
• Wasteful

18
More Efficient

• Find top vertex
• Begin to left of edge
• How to find edge?
• Has problems
• Edge search next

19
Edge Search

• Could step back into triangle when incrementing
• Then need to search

20
Bidirectional Search

• Must save context so can return to center line

21
Parallel Implementation

• E is linear, so can compute incrementally for any
  offset
• E(xL,y) E(x) L dY
• for some horizontal offset L
• Test block of pixels

22
Olano and Greer

• First look at interpolation
• Then pixel inclusion test
• Caps (like X) non-homogeneous
• Lower case (x) homogeneous

23
Interpolation

• Parameter varies linearly
• u a X b Y c Z (1)
• In homogeneous coordinates
• u a x b y c w (2)
• Perspective-correct interpolation
• u/w a x/w b y/w c a X b Y c
• a, b, and c same in all eqns

24
Their Observation

• From eqn (2)
• u a x b y c w (2)
• From u at each vertex, we can solve for a, b, and
  c

25
Implications

• The matrix that needs to be inverted just the
  vertex coords
• Then just one vector-matrix multiply for a, b, c
  for each param
• No division by w!
• So no problem when 0 or negative

26
Computational Cost

• Perspective-correct params computed (u/w)
• Need to also compute 1/w at each pixel
• Then take reciprocal and multiply
• (u/w) ? w

27
Scan Conversion

• Compute pseudo-parameter
• One thats positive on one side of vertex,
  negative on the other
• Parameter thats zero at 2 verts of an edge, one
  at the third vert
• So on side of triangle
• Again test for 3 s

0

0
1
28
Matrix Inverse?

• What if its not invertible?
• Either triangle is edge-on, or very small
• Didnt want to render it anyway

29
Precision

• Black line is edge
• Rounded to orange
• Pixel centers sampled differently
• Pixels that should have been on left side now on
  right
• Can minimize with fractional bits

30
Frac Bits

• Easy on Pixel-Planes tree
• Just add cycles to computation
• Configurable
• On a pixel-parallel design would have to decide
  ahead of time

31
Snap to Grid

• On Pixel-Planes we snapped vertices to a subpixel
  grid
• 1/64th of a pixel
• Choose fractional bits to guarantee numerical
  results

32
Another Issue

• What to do about edges?
• Edges that fall right on pixels
• Rules to keep from rendering twice

33
Tiled Traversal

• Texture or frame buffer caches that are
  rectangular (tiled) make sense
• How can you traverse triangles in tiled order to
  make use of cache?
• GH2000 paper by McCormack and McNamara

34
Stamps vs. Tiles

• They first talk about stamps a rectangular
  block of candidate pixels tested at a time
• Then they look at larger tiles (multiple of stamp
  size)

35
Stamp, here 2 x 4

• Thin lines show pixel grid
• X pixel sample point candidate fragment

• Circle called probes sites tested for
  traversal (next)

36
Traversal

• Test whether any part of stamp edge is in
  triangle
• First for stamp edge, are one or both probes
  inside all triangle edges?
• A single probe does not have to be in triangle
  (right)
• Case where this is true, but triangle does not
  cross edge
• Edge RT-RB above right

37
Second Test

• Is stamp edge in bounding box?
• Example at right RT and RB (together) inside
  all tri edges
• Stamp edge crosses shadow of triangle

38
Note

• Note that these stamp tests do not have to be
  correct, just conservative
• False stamp movement just inefficient

39
Centerline Traversal
Circle RightSave Octagon - DownSave

• Start at topmost vertex
• Is right position valid?
• If so, store context
• Context edge functions, interpolated values at
  RT
• Traverse left until not valid

• If right was valid, restore context and traverse
• Always looking for valid down position
• If one found, save in down context
• Go down after traversing whole line
• Context not saved for invalid positions (L of 16)

40
Alternate Starting Position

• They point out that starting at vertex on corner
  of BBox would eliminate need for rightSave
• Only one sideways direction to move
• Would then need to be able to traverse
  top-to-bottom or bottom-to-top

41
Alternate Traversal

• Step to tiled traversal
• Keep DownSave (octagonal) and UpSave (diamond)
• When no more UpSave, pop DownSave

Diamond UpSave Octagon - DownSave
42
Tiling Traversal

• Stamps visited by rows
• Tiles visited by columns
• Now need RightTileSave context in addition to
  UpSave and DownSave
• Right to end of tile
• Pop DownSave
• Pop UpSave
• Pop RightTileSave

Diamond UpSave Octagon DownSave Triangle
RightTileSave
43
Tiling Traversal 2

• Phase 0
• Right to end of tile, save next valid position in
  RightTileSave
• Repeat phase 0 while DownSave is in tile
• Phase 1
• Trace right again, respecting right boundary
• Now go up, but ignore up tile boundaries
• Phase 2
• Like phase 1, but go down
• When reach bottom, go to Phase 0

Diamond UpSave Octagon DownSave Triangle
RightTileSave
White Phase 0 Rose Phase 1 Lavender Phase 2
44
Frame Buffer Tiling

• FB divided into pages
• Two bank checkerboard
• Allows precharge other bank
• Serpentine version of algorithm increases
  frequency of bank changes

45
Texture Cache Tiling

• Texture cache a little different than frame
  buffer pages because uses texels from adjacent
  blocks
• Paper also discusses metatiling, multiple levels
  of tiles
• To stay in FB pages
• And also increase use of texture cache