2007.10.02 CS248-03 Sampling and Aliasing

7/30/2019 2007.10.02 CS248-03 Sampling and Aliasing

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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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CS248 Lecture 3 Kurt Akeley, Fall 2007

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


7/40CS248 Lecture 3 Kurt Akeley, Fall 2007

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



Reference sources

Marc Levoys notes

Ronald N. Bracewell, The Fourier Transform and itsApplications, Second Edition, McGraw-Hill, Inc.,1978.

Private conversations with Pat HanrahanMATLAB



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



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) sumof harmonic sine and cosine functions.

Harmonics are integer multiples of thefundamental frequency



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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Original Signal

Triangle

Reconstruction


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Reconstruction with a rectangle function

x

=

*

=

F(s)f(x)


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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

2007.10.02 CS248-03 Sampling and Aliasing

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