Aliasing - Lecture Slides - Computer Graphics | CAP 4720, Study notes of Computer Graphics

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10/21/2008 1

Juraj Obert [email protected] http://graphics.cs.ucf.edu/cap4720/fall2008/

Aliasing

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What is Sampling?

Most things in the real world are

continuous

Everything in a computer is discrete

The process of mapping a continuous

function to a discrete one is called

sampling.

When we represent or render an image

using a computer we must sample.

A Sampling & Reconstruction System

Analog Information

Sampling System

Digital Operation

Reconstruction System

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Sampling an Image

From Leonard McMillan, CS Department, MIT

Sampling Grid

y

x

1

1

We create a digital representation of this continuous function by multiplying the function by a sampling grid. A sampling grid is composed of periodically spaced Kronecker delta functions.

Kronecker delta is :

And a 2-D sampling grid:

−

=

−

=

1

0

1

0

h

j

w

i

S x y δ x i y j

xy otherwise

xy iff x y ( , ) 0

( , ) 1 ( , ) ( 0 , 0 )

= =

δ

δ

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Sampling a Sine Wave

I ( x )=sin( 2 π fx )

Sample Points

Function to be Sampled

Sampling interval

X

Y

Sampling a Sine Wave

X

Y

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Sampling a Sine Wave

Sampling frequency < 2 x function frequency ( f )

X

Y

Aliased Sine wave

Sampling a Sine Wave

X

Y

Aliased Sine wave

Sampling frequency < 2 x function frequency ( f )

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Aliasing

Aliasing is the artifact caused when

high-frequency information appears as

low-frequency information.

Nyquist Theorem

For accurate reconstruction, the sampling Frequency, f s, must be at least twice the maximum Frequency, f max, of the function.

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Fourier Analysis

Every function defined in the spatial domain, has a dual representation in the frequency domain.

= (^) ∫

∞

− ∞

F ( u ) f ( x ) e^ −^ i^2 π uxdx

= (^) ∫

∞

− ∞

f ( x ) F ( u ) ei^2 π ux du

Equation to transform from spatial domain to frequency domain is:

Equation to transform from frequency domain to spatial domain is:

Convolution

Convolution describes how a system with impulse response, h(x), reacts to a signal, f(x).

This integral evaluation is equivalent to multiplication in the frequency domain

The inverse is also true

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Sampling Functions

Sampling takes measurements of a continuous function at discrete points
Equivalent to product of continuous function and sampling function
Uses a sampling function s ( x )
Sampling function is a collection of spikes
Frequency of spikes corresponds to their resolution
Frequency is inversely proportional to the distance between spikes
Fourier domain also spikes
Distance between spikes is the frequency

p

1/p

s ( x )

S ( u )

Spatial Domain

Frequency Domain

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Sampling

p (^) 1/p

I ( x )

s ( x )

( Is )( x )

( Is ’)( x )

s ’( x )

F ( u )

S ( u )

( F*S )( u )

S ’( u )

( F*S ’)( u ) Aliasing, can’t retrieve original signal

( Is ’)( x )

s ’( x ) S ’( u )

( F*S ’)( u )

I ’ = ( I *sinc)

( I’s ’)( x ) ( F’*S ’)( u )

F ’ = ( F box)

(sinc)( I’s ’)( x )^ (box)( _F’S_ ’)( u )

Prefiltering