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Summed Area Tables using Graphics Hardware

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In Computer Graphics 18, 3 (July 1984), pp. 207-212. ... Summed-Area Tables for Texture Mapping. Proceedings of SIGGRAPH `84. In Computer Graphics 18, 3 ... – PowerPoint PPT presentation

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Title: Summed Area Tables using Graphics Hardware


1
Summed Area Tables using Graphics Hardware
  • Simon Green, NVIDIA

2
What are Summed Area Tables (SATs)?
  • Pre-integrated texture representation
  • Invented by Frank Crow, 1984
  • Each texel is the sum of all texels below and to
    the left of it
  • Allows rapid box filtering (average) over any
    rectangle for a fixed cost
  • Based on the algebraic identity
  • (xa)(yb) (xa)y x(yb) xy ab

3
Summed Area Table
Summed area table
Regular image
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
4 8 12 16
3 6 9 12
2 4 6 8
1 2 3 4
4
Summed Area Table
a
(x, yb)
(xa, yb)
(0, yb)
b
(x, y)
(0, y)
(xa, y)
(x, 0)
(xa, 0)
(0, 0)
ab Sxa, yb Sxa, y Sx, yb Sx, y
5
Original Image
6
Summed Area Table (scaled)
7
SATs on Graphics Hardware
  • Floating point textures have sufficient precision
    to represent SATs
  • Can decode SAT using a fragment program
  • Enables texture lookup with any width box filter
    for constant cost (4 texture lookups math)
  • Amount of blur can vary from pixel to pixel
    (spatially variant)

8
SAT Texture Decode
// look up into summed area table texture//
returns average of pixels in rectangle with
corners (x,y) - (xa,yb)float4
texRECT_summedArea(uniform texobjRECT tex, float2
xy, float2 ab) float4 r r
f4texRECT(tex, xy ab) //
Sxa, yb r r - f4texRECT(tex, xy
vec2(ab0, 0) ) // Sxa, y r r -
f4texRECT(tex, xy vec2(0, ab1) ) // Sx,
yb r r f4texRECT(tex, xy)
// Sx, y return r / (ab0 ab1)
  • Bilinear lookup requires 4 bilinear lookups into
    float texture, followed by SAT decode
  • May also require clamping texture coordinates to
    avoid problems at edges

9
Building Summed Area Tables using Hardware
  • For dynamic applications (e.g. depth of field),
    we need to be able to create SAT textures
    on-the-fly
  • Possible using render-to-texture
  • Sum columns first, and then rows
  • Each row or column is rendered as a line
    primitive
  • Fragment program adds value of current texel with
    texel to the left or below
  • Can be done using render to active texture, but
    warning - results undefined!

10
Building Summed Area Table
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
4 8 12 16
3 6 9 12
2 4 6 8
1 2 3 4
Original image
Sum columns
Sum rows
  • For n x m image, requires rendering2 x n x m
    pixels, each of which performs two texture
    lookups
  • Other algorithms may be possible

11
Applications of SATs
  • Anisotropic texture filtering
  • Can be better than mip-mapping
  • Rectangle may not match projection of pixel
    exactly
  • Depth of field
  • Blur image of scene based on distance
  • Higher quality blur than using mipmaps
  • Soft shadows
  • Blur image of shadow based on distance from
    occluder

12
Depth of Field Demo
13
References
  • Crow, Frank. Summed-Area Tables for Texture
    Mapping. Proceedings of SIGGRAPH 84. In Computer
    Graphics 18, 3 (July 1984), pp. 207-212.
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