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Sampling and Aliasing
Kurt Akeley
CS248 Lecture 3
2 October 2007
http://graphics.stanford.edu/courses/cs248-07/
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Aliasing
Aliases are low frequencies in a renderedimage that are due to higher frequenciesin the original image.
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Jaggies
Are jaggies due to aliasing? How?
Original:
Rendered:
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Demo
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What is a point sample (aka sample)?
An evaluation
At an infinitesimal point (2-D)
Or along a ray (3-D)
What is evaluated
Inclusion (2-D) or intersection (3-D)
Attributes such as distance and color
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Why point samples?
Clear and unambiguous semantics
Matches theory well (as well see)
Supports image assembly in the framebuffer
will need to resolve visibility based on distance,
and this works well with point samples
Anything else just puts the problem off
Exchange one large, complex scene for manysmall, complex scenes
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Fourier theory
Fouriers theorem is not only one of the most
beautiful results of modern analysis, but itmay be said to furnish an indispensableinstrument in the treatment of nearly everyrecondite question in modern physics.
-- Lord Kelvin
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Reference sources
Marc Levoys notes
Ronald N. Bracewell, The Fourier Transform and itsApplications, Second Edition, McGraw-Hill, Inc.,1978.
Private conversations with Pat HanrahanMATLAB
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Ground rules
You dont have to be an engineer to get this
Were looking to develop instinct / understanding
Not to be able to do the mathematics
Well make minimal use of equations
No integral equations
No complex numbers
Plots will be consistent
Tick marks at unit distances
Signal on left, Fourier transform on the right
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Dimensions
1-D
Audio signal (time)
Generic examples (x)
2-D
Image (x and y)
3-D
Animation (x, y, and time)
All examples in this presentation are 1-D
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Fourier series
Any periodic function can be exactly
represented by a (typically infinite) sumof harmonic sine and cosine functions.
Harmonics are integer multiples of thefundamental frequency
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Fourier series example: sawtooth wave
( )( )
( )
1
1
12sin
n
n
f x n xn
pp
+
=
-=
1
1-1
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Sawtooth wave summation
Harmonics Harmonic sums
1
2
3
n
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Sawtooth wave summation (continued)
Harmonics Harmonic sums
5
10
50
n
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Fourier integral
Any function (that matters in graphics)can be exactly represented by an
integration of sine and cosine functions.
Continuous, not harmonic
But the notion of harmonics willcontinue to be useful
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Basic Fourier transform pairs
1 ( )sd
f(x) F(s)
( )cos xp II(s)
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Reciprocal property
Swappedleft/right fromprevious slide
f(x) F(s)
1
( )cos xp
( )sd
II(s)
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Scaling theorem
( )cos 2 xp
cos
2
xp
II(2s)
II(s/2)
f(x) F(s)
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Band-limited transform pairs
( )s
F(s)f(x)
sinc(x)( )sin x
x
p
p ( )s
sinc2(x)
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Finite / infinite extent
If one member of the transform pair is finite, the otheris infinite
Band-limited infinite spatial extent Finite spatial extent infinite spectral extent
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Convolution
( ) ( ) ( ) ( )f x g x f u g x u du
-
* = -
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Convolution example
*
=
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Convolution theorem
Letfandgbe the transforms offandg. Then
f g f g* =
f g f g = *
f g f g* =
f g f g = *
Something difficult to do in one domain(e.g., convolution) may be easy to do in the
other (e.g., multiplication)
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Sampling theory
x
=
*
=
F(s)f(x)
Spectrum isreplicated an
infinite number oftimes
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Reconstruction theory
x
=
*
=
F(s)f(x)
sinc
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Sampling at the Nyquist rate
x
=
*
=
F(s)f(x)
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Reconstruction at the Nyquist rate
x
=
*
=
F(s)f(x)
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Sampling below the Nyquist rate
x
=
*
=
F(s)f(x)
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Reconstruction below the Nyquist rate
x
=
*
=
F(s)f(x)
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Reconstruction error
Original Signal
Undersampled
Reconstruction
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Reconstruction with a triangle function
x
=
*
=
F(s)f(x)
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Reconstruction error
Original Signal
Triangle
Reconstruction
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Reconstruction with a rectangle function
x
=
*
=
F(s)f(x)
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Reconstruction error
Original Signal
Rectangle
Reconstruction
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Sampling a rectangle
x
=
*
=
F(s)f(x)
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Reconstructing a rectangle (jaggies)
x
=
*
=
F(s)f(x)
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Sampling and reconstruction
Aliasing is caused by
Sampling below the Nyquist rate,
Improper reconstruction, or
Both
We can distinguish between
Aliasing of fundamentals (demo)
Aliasing of harmonics (jaggies)
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Summary
Jaggies matter
Create false cues
Violate rule 1
Sampling is done at points (2-D) or along rays (3-D)
Sufficient for depth sorting
Matches theory
Fourier theory explains jaggies as aliasing. For correctreconstruction:
Signal must be band-limited
Sampling must be at or above Nyquist rate
Reconstruction must be done with a sinc function
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Assignments
Before Thursdays class, read
FvD 3.17 Antialiasing
Project 1:
Breakout: a simple interactive game
Demos Wednesday 10 October
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End
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