Antialiasing
What is alias?
Alias

A false signal in telecommunication links from
beats between signal frequency and sampling frequency
(from dictionary.com)
Not just telecommunication, alias is everywhere in
computer graphics because rendering is essentially a
sampling process
Examples:
Jagged edges
False texture patterns
Rendering is a sampling process?
Yes indeed.
Image display example
Rendering is a sampling process?
Another example
Render a smooth surface
Ssignal sampling and filtering
Two important steps when information are
represented digitally:
sampling
and
reconstruction
Alias can be caused by any of the steps
Alias caused by under

sampling
Have
you ever look
ed
at an electric fan and
found it is rotated in a reverse direction?
Sample at only ¼ of the frequency
Incorrect result is seen !!

under

sampling
problem
Alias caused by under

sampling
1D signal sampling example
Actual signal
Sampled signal
How often is enough?
What is the right sampling frequency?
Sampling theorem (or Nyquist limit)

the
sampling frequency has to be at least twice
the
maximum
frequency of the signal to be
sampled
Need two samples in this cycle
www.
microsoft
.ca
Can alias be totally avoided
Given the sampling theorem by Nyquist ?
Not really
–
this is because the
maximum
frequency might be infinite, i.e., not
bandlimited
Most of
graphics scenes are not bandlimited:
sharp edges can never be completely sampled
in a
discrete manner (point sampling)
We will talk about what we can do to correct
alias for real time graphics
Strobe light on dripping water:
Temporal aliasing
Spokes on a rotating wheel:
Temporal aliasing
Moir
é
patterns:
Spatial aliasing
Examples of Aliasing
Antialiasing Lines
Idea:
Make line “fatter”
Fade line out (removes high frequencies)
Now sample the line
Antialiasing Lines
Solution 1
–
Unweighted Area Sampling:
Treat line as a single

pixel wide rectangle
Color pixels according to the percentage of each
pixel that is “covered” by the rectangle
Solution 1: Unweighted Area Sampling
Pixel area is unit square
Constant weighting function
Pixel color is determined by computing the
amount of the pixel covered by the line, then
shading accordingly
Easy to compute, gives reasonable results
Line
One Pixel
Solution 2: Weighted Area Sampling
Treat pixel area as a circle with a radius of one pixel
Use a radially symmetric weighting function
(e.g.,
cone)
:
Areas closer to the pixel center are weighted more heavily
Better results than unweighted, slightly higher cost
Line
One Pixel
Solution 3: Super

sampling
Divide pixel up into “sub

pixels”:
2
2, 3
3, 4
4, etc.
Sub

pixel is colored if inside line
Pixel color is the average of its sub

pixel colors
Easy to implement
(in software and hardware)
Expensive
No antialiasing
Antialiasing
(2
2 super

sampling)
Different Supersampling Schemes
The common
formula:
c(i,x,y): color of the ith sample
for pixel(x,y)
;
Wi: weight
Different super

sampling schemes
Reconstruction
After the (ideal) sampling is done, we still
need to reconstruct back the original
continuous signal
The reconstruction is done by reconstruction
filter
Reconstruction Filters
Common reconstruction filters:
Box filter
Tent filter
Sinc filter = sin(
p
x)/
p
x
Box filter
Very simple, but not very good
Tent Filter
Based on linear interpolation. Better, but still
not smooth
Sinc Filter
Ideal low

pass filter
Anti

aliased texture mapping
The above anti

aliasing methods work better
for removing jagged edges but not texture
aliasing
Two problems to address
–
Magnification
Minification
Re

sampling
Minification and Magnification
–
resample the
signal to a different resolution
Magnification
Minification
(note the minification is done badly here)
Magnification
Simpler to handle, just resample the
reconstructed signal
Reconstructed signal Resample the reconstructed signal
Magnification
Common methods: nearest neighbor (box
filter) or linear interpolation (tent filter)
Nearest neighbor bilinear interpolation
Minification
Harder to handle
The signal’s frequency is too high to avoid
aliasing
A possible solution:
Increase the low pass filter width of the ideal sinc
filter
–
this effectively blur the image
Blur the image first (using any method), and then
sample it
Minification
Several texels cover one pixel (under
sampling happens)
Under sample artifact
one pixel
Solution:
Either increase
sampling rate or
reduce the
texture
Frequency
We will discuss:
1.
Mip

mapping
2.
Rip

mapping
3.
Sum Area Table
Mip

mapping
By Lance Williams
(SIGGRAPH ’83)
Mip
–
many things in a
small place
The original texture is
augmented with a set of low
resolution textures
Which resolution to use
depends on the screen
projection size
–
try to
maintain pixel to texel
ration close to 1
average 2x2
texels
Mipmapping problem
Overblurring!
–
When a pixel covers many u
texels but few v texels, we always choose the
largest pixel coverage to decide the level
Non mipmapping
mipmapping
Ripmapping
Correct the over

blurring problem of
mipmapping
–
separating x and y filtering
•
Scale by half by x
across a row.
•
Scale by half in y
going down a column.
•
The diagonal has the
equivalent mip

map.
•
Four times the amount
of storage is required.
•
Four different texel
values to average
?
Summed Area Table
Another way to perform anisotropic filtering

can be used to compute the average color for
any arbitrary rectangular region in the texture
space
at a constant speed
A two dimensional array that has the same
size as the texture
Every pixel stores the sum of all the texel colors in the
lower left corner
Summed Area Table (SAT)
How to calculate the sum of texels in the area
between A and B?
R
A
B
C
D
R = SAT[B]
–
SAT[C]
–
SAT[D] + SAT[A]
The average value can be computed by
R / num of texels in R
Summed Area Table (SAT)
is used to filter the texture
Compute the rectangular bounding box of the pixel
area in the texture and then
use summed area table to compute the average
color (i.e., filter the texels when
magnification takes place)
pixel
polygon
Texture space
x
y
Summed Area Table
Less blurry than Mipmapping
Still blurry when the texture is viewed along
its diagonal
–
the bounding box of the pixel area is larger than
the actual pixel area
pixel
polygon
Texture space
x
y
not quit fit
Comparison
Non filtering
Mipmapping
Summed area table
Note that only mipmap is implemented by hardware and supported by OpenGL
Motion Blur
–
Temporal Antialias
Tricky
Simplest Solution:
supersampling in time
brute force approach
motion blur increases temporal
resolution
Just as pixel antialiasing increases
spatial resolution
If an object moves 16 pixels in one
frame, then 16 supersample images
(or even less) should be adequate
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