Sampling and Aliasing.ppt

Sampling and Aliasing
Kurt Akeley
CS248 Lecture 3
2 October 2007
http://graphics.stanford.edu/courses/cs248-07/

CS248 Lecture 3 Kurt Akeley, Fall 2007
Aliasing
Aliases are low frequencies in a rendered
image that are due to higher frequencies
in the original image.

Jaggies
Are jaggies due to aliasing? How?
Original:
Rendered:

Demo

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

Why point samples?
Clear and unambiguous semantics
Matches theory well (as we’ll 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 many
small, complex scenes

Fourier theory
“Fourier’s theorem is not only one of the most
beautiful results of modern analysis, but it
may be said to furnish an indispensable
instrument in the treatment of nearly every
recondite question in modern physics.”
-- Lord Kelvin

Reference sources
Marc Levoy’s notes
Ronald N. Bracewell, The Fourier Transform and its
Applications, Second Edition, McGraw-Hill, Inc.,
1978.
Private conversations with Pat Hanrahan
MATLAB

Ground rules
You don’t have to be an engineer to get this
 We’re looking to develop instinct / understanding
 Not to be able to do the mathematics
We’ll 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

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

Fourier series
Any periodic function can be exactly
represented by a (typically infinite) sum
of harmonic sine and cosine functions.
Harmonics are integer multiples of the
fundamental frequency

Fourier series example: sawtooth wave
…
…
( )
( )
( )
1
1
1
2
sin
n
n
f x n x
n
p
p
+
¥
=
-
= å
1
1
-1

Sawtooth wave summation
Harmonics Harmonic sums
1
2
3
n

Sawtooth wave summation (continued)
Harmonics Harmonic sums
5
10
50
n

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 will
continue to be useful

Basic Fourier transform pairs
1 ( )
s
d
f(x) F(s)
( )
cos x
p II(s)

Reciprocal property
Swapped
left/right from
previous slide
f(x) F(s)
1
( )
cos x
p
( )
s
d
II(s)

Scaling theorem
( )
cos 2 x
p
cos
2
x
p
æ ö
÷
ç ÷
ç ÷
ç
è ø
II(2s)
II(s/2)
f(x) F(s)

Band-limited transform pairs
( )
s
Ù
F(s)
f(x)
sinc(x)
( )
sin x
x
p
p
( )
s
Õ
sinc
2
(x)

Finite / infinite extent
If one member of the transform pair is finite, the other
is infinite
 Band-limited  infinite spatial extent
 Finite spatial extent  infinite spectral extent

Convolution
( ) ( ) ( ) ( )
f x g x f u g x u du
¥
- ¥
* = -
ò

Convolution example
*
=

Convolution theorem
Let f and g be the transforms of f and g. 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)

Sampling theory
x
=
*
=
F(s)
f(x)
Spectrum is
replicated an
infinite number of
times

Reconstruction theory
x
=
*
=
F(s)
f(x)
sinc

Sampling at the Nyquist rate
x
=
*
=
F(s)
f(x)

Reconstruction at the Nyquist rate
x
=
*
=
F(s)
f(x)

Sampling below the Nyquist rate
x
=
*
=
F(s)
f(x)

Reconstruction below the Nyquist rate
x
=
*
=
F(s)
f(x)

Reconstruction error
Original Signal
Undersampled
Reconstruction

Reconstruction with a triangle function
x
=
*
=
F(s)
f(x)

Original Signal
Triangle
Reconstruction

Reconstruction with a rectangle function
x
=
*
=
F(s)
f(x)

Original Signal
Rectangle
Reconstruction

Sampling a rectangle
x
=
*
=
F(s)
f(x)

Reconstructing a rectangle (jaggies)
x
=
*
=
F(s)
f(x)

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)

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 correct
reconstruction:
 Signal must be band-limited
 Sampling must be at or above Nyquist rate
 Reconstruction must be done with a sinc function

Assignments
Before Thursday’s class, read
 FvD 3.17 Antialiasing
Project 1:
 Breakout: a simple interactive game
 Demos Wednesday 10 October

End

Sampling and Aliasing.ppt

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Sampling and Aliasing.ppt

Similar to Sampling and Aliasing.ppt (20)

Recently uploaded

Recently uploaded (20)

Sampling and Aliasing.ppt