1. Antialiasing Miniaturized Textures
CSS 552 – Prof. Sung
Daniel R. Lewis
June 3, 2013

2. Texels and Pixels (1) Textures are composed of discrete texels:
Pixels are projected onto texture:
Textured
Geometry
Eye
Image
Plane
- Projection is both scene- and view-dependent.
- Two grids: the texel grid, which is composed of squares containing a color; and the pixel grid, composed of irregular quadrilaterals which need to be assigned a color. Our task is to map between these grids.

3. Texels and Pixels (2)
Image credit: Heckbert

4. Texture Magnification
Image resolution higher than texture resolution
Pixel grid “denser” than texel grid
- Magnification is a.k.a. upsampling

5. Texture Minification Image resolution lower than texture resolution
Texel grid “denser” than pixel grid
Pixel Footprint
Texel
Grid
- Minification is a.k.a. downsampling

6. Point Sampling
Color comes from texel that happens to be at the intersection position
Poor representation of pixel area
Temporal aliasing e.g. “blinking”
At right: How would the pixel coloration change as the camera or geometry moved?
Improved by supersampling to a degree—becomes expensive as the texel:pixel ratio increases
- a.k.a. nearest neighbor
- this is the approach used by our MPs

7. Aliasing from point sampling
Image credit:
Angel

8. Prefiltered Texel Averaging
(1b)
camera moves back
How many?
What shape?
Reminder: pixel coverage is view- and scene-dependent (1a, 1b)
Pixels cover various numbers of texels even within the same frame (2)
Cannot know a priori what to average
All methods prefilter the texture to generate approximations of all possible pixel footprints
(2)

9. Mipmapping Invented by Lance Williams, seminal paper published in 1983
MIP an acronym for multum in parvo, meaning “many things in a small place”
Well-supported in hardware
point sampling
mipmapping
detail
Image credit: Williams

10. Mipmapping: How It Works (1)
Start with 2n×2n texture
Create n downsampled texture maps, where each map Mi is 25% of the size of map Mi-1
Average four pixels into one
How should pixels be “averaged”? Hold that thought...
Mipmaps require only 33.3% of the memory of the original texture
Image credit: Wikipedia(a)
- 21 / 64 = 32.8

11. Mipmapping: How It Works (2)
Pixel color derived from (u, v, d) triple, where d is the level of detail
Image credit: Akenine-Moller et al.

12. Finding d (a) d ∝ || longest_pixel_edge || (b)
lpe = || longest_pixel_edge ||
d = 1 - ( lpe / texture_width )
(b)
dx = sqrt((Δu / Δx)2 + (Δv / Δx)2)
dy = sqrt((Δu / Δy)2 + (Δv / Δy)2)
d ∝ max(dx, dy)
e.g.:
d = (1 / sqrt(2)) × max(dx, dy, sqrt(2))
Challenge: in ray tracing, all we have is a point–not enough to reconstruct the red quadrilateral or find dx/dy

13. Using d Derive two maps from d
Sample both at (u, v) and interpolate between the samples
Common to add/subtract a level of detail bias from d to tweak results. Specified by user.

14. Filtering Algorithms
When downsampling, want to eliminate high frequency information that cannot be reproduced, before it causes aliasing
Low-pass filter (LPF)
Pixel averaging, a.k.a. box filtering, is a bad LPF
Ideal LPF is sinc
sin(πx)/(πx)
Infinitely wide
Approximate sinc by multiplying it by a windowing function and cutting it off
Purple: Ideal filter
Yellow: Box filter
Green: Kaiser filter
Blue: Point sampling
Image credit: Blow(a)

15. Brightness and Gamma Correction (1)
Human eye is more sensitive to darker shades than brighter ones
Linear encoding of color wastes space on bright shades that are indistinguishable
Hence, images store color non-linearly, which is then scaled exponentially by γ to become linear
Clinear = (Cstore)γ
Typically, 1.8 ≤ γ ≤ 2.8

16. Brightness and Gamma Correction (2)
Image credit:
Blow(b)
Filtering should not reduce brightness (ideally)
However:
A = tc1γ + tc2γ + tc3γ + tc4γ
B = ((tc1 + tc2 + tc3 + tc4) / 4)γ
A > B
Fixed by converting texel colors to linear space (decode), filter that, and then convert back (encode)

17. Brightness and Gamma Correction (3)
Detail view of mipmapped textures
direct filtering
linear filtering
Image credit: Blow(b)

18. Achilles’ Heel of Mipmapping
Mipmapping weakness: all pixel projections that do not vaguely approximate squares
Below: color of pixel (red outline) comes from all shaded texels—mostly error!

19. Aliasing from mipmapping
Image credit:
Angel

20. Ripmapping Extension of mipmapping generating rectangular maps
Requires 300% more memory than original texture
Rarely used today dv du du dv
Image credit: Wikipedia(b)

21. Summed Area Tables
Invented by Frank Crow, seminal paper published in 1984
Based on rectangles like ripmapping, but with a dissimilar algorithm
Limited hardware support
Top: Point sampling
Middle: Mipmapping
Bottom: Summed area tables
Image credit: Akenine-Moller et al.

22. Generating the SAT n×m SAT for n×m texture
(0,0)
(i, j) m n
n×m SAT for n×m texture
Each slot (i, j) holds the sum of all texels in rectangle between origin (0, 0) and (i, j)
Requires more bits per channel (16–32) than original texture color (8)
Efficiently computed in a single pass:
I(i,j) = tc(i,j) + I(i−1,j) + I(i,j−1) − I(i−1,j−1)

23. Finding Rectangle Color
Rs = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)
Ra = Rs / ((ir − il)(jb − jt))
(0,0) il ir jt jb

24. Finding Rectangle Color
Rs = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)
Ra = Rs / ((ir − il)(jb − jt))
(0,0) il ir jt jb

25. Finding Rectangle Color
Rs = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)
Ra = Rs / ((ir − il)(jb − jt))
(0,0) il ir jt jb

26. Finding Rectangle Color
Rs = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)
Ra = Rs / ((ir − il)(jb − jt))
(0,0) il ir jt jb

27. Finding Rectangle Color
Rs = I(il,jb) − I(ir,jt) − I(il,jb) + I(il,jt)
Ra = Rs / ((ir − il)(jb − jt))
(0,0) il ir jt jb

28. Finding the Bounding Box
(ua, va)
(ub, vb)
(uc, vc)
(ud, vd) v u
umin = min(ua, ub, uc, ud)
umax = max(ua, ub, uc, ud)
vmin = min(va, vb, vc, vd)
vmax = max(va, vb, vc, vd) v u v0 v1 v3 v2
Rectangle:
Vertice[0] = (umin, vmin)
Vertice[1] = (umin, vmax)
Vertice[2] = (umax, vmax)
Vertice[3] = (umax, vmin)

29. Interpolating SAT Rectangles
Instead of rounding bounding box vertices to integers, allow the corners to fall between texel boundaries
Find color of all boxes at neighboring texels and interpolate between them based on distance

30. Achilles’ Heel of Ripmapping and SATs
Weakness of rectangle-based methods is pixel projections that cut diagonally across the texture
Below: color of pixel (red outline) comes from the texels in the shaded bounding box—mostly error!

31. Unconstrained Anisotropic Filtering
Built on top of mipmapping
Find (approximate) the major axis of the pixel footprint
Sample the mipmap (trilinear interpolation) at least twice along the axis, more if the footprint is long
Unlike normal mipmapping, use the length of the shorter side to compute the d value
Supported by PC graphics cards since early 2000s

32. Not Herein Considered...
Non-color textures: most of the antialiasing techniques can be extended for textures of other types (e.g., bump maps)
Curved surfaces
Literal edge cases

33. Implementation Plan (1)
Focus on implementing basic mipmapping
“Color” textures only
Time allowing, will implement more
Use RayTracer framework, with MP4 as the starting point
Risk: do not have enough information to compute the size or shape of the pixel footprint
Substantial modifications required

34. Implementation Plan (2)
Must have additional intersections (either world space or uv space) down within GetTexile()
Deep stack; might be easiest to stuff extra information into the IntersectionRecord
ComputeImage()
ComputeShading()
GetDiffuse()
AttributeLookup()
TextureLookup()
GetTexile()
Naïve implementation: ray trace four extra intersections at the pixel corners
Better: shifting “box” of four intersection records
No more expensive, except edges

35. Image Credits
Akenine-Moller, Tomas; Eric Haines; Naty Hoffman. Real-Time Rendering (3rd ed.).
Angel, Edward. Interactive Computer Graphics.
Found at
Blow, Jonathan (a). “Mipmapping, Part 1.” Game Developer Magazine Dec
Blow, Jonathan (b). “Mipmapping, Part 2.” Game Developer Magazine Jan
Heckbert, Paul. “Fundamentals of Texture Mapping and Image Warping.” Thesis. University of California, Berkeley, 1989.
Wikipedia (a). <
Wikipedia (b). <
Williams, Lance. “Pyramidal Parametrics.” Computer Graphics 13.3 (1983).

36. Questions?