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Aliasing10/4/2010

CGGC1

(Anti)Aliasing

and

Image Manipulation

for (y = 0; y < Size; y++) {

for (x = 0; x < Size; x++) {

Image[x][y] = 127 + 128 *

sin((SQR(x – Size2) +

SQR(y – Size2)) /

32.0);

}

} // Size2 = Size / 2; Aliasing - Center for Graphics and Geometric Computing, Technion 2

Aliasing

Can’t draw smooth lines on

discrete raster device – get

staircased lines (“jaggies”):

Images with too fine

details create artifacts

(Moire patterns)

Why? How to solve?

Aliasing - Center for Graphics and Geometric Computing, Technion 3

Antialiasing


Antialiasing


Sampling

2D raster image = discrete sampling of

continuous image (signal)

Consider 1D signal sampling


Sampling




Aliasing10/4/2010

CGGC2


Sampling





Sampling





Sampling Theory in a Nutshell

Terminology

Cycle – a repeated sequence over infinite signal

Frequency – number of cycles in a unit length

Example – sine wave

Cycle = 2

Frequency = 1/ 2


Sampling Theory

Theorem: Any periodic signal can be represented as

(infinite) sum of sine waves, at different frequencies

Can represent signal using coefficients of the sine

waves - alternative basis used by Fourier Analysis

Fourier Transform- Tool to transform to/from sine

basis, to/from the frequency domain


Example A square wave is represented by infinite sum

First few components:

...5

)5sin(

3

)3sin()sin()(

ttttSqrWave


Shannon Sampling Theorem

Shannon Theorem: Any signal that is band-limited to

frequency fmax and sampled at equally spaced intervals

smaller than ts = 1/(2fmax) can be fully reconstructed

The minimal sampling frequency of ts = 1/(2fmax) is

called the Nyquist frequency of the signal

That this condition is necessary can be trivially shown:

Question: What is the result of this sampling? Aliasing!

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CGGC3


More Aliasing Examples

Sampling high frequency wave below Nyquist frequency

yields results that “look” like a low frequency wave

The appearance of high frequencies as low frequencies,

due to sampling, is called Aliasing:


Even More Aliasing ExamplesConsider two sine waves: -sin(3t) and sin(5t):

-sin(3t) = sin(5t) is satisfied for t = n / 4

Sampling sin(5t) at ti = i / 4 looks like

sampling sin(3t)


Aliasing

Aliasing – the appearance of high frequencies

as low frequencies

To avoid

Sample at a higher frequency - only if band-limited

to begin with

A-priori remove high frequencies – Low pass

filtering before sampling

For images – the discrete image is a

sampling set of a continuous image (band

limits?)


The Fourier Transform

Takes us from the spatial domain to the

frequency domain:

The inverse Fourier transform equals

We will see some examples shortly

dxexftF itx

)()(

dxexFtf itx

)(

2

1)(


Convolution

The convolution, denoted by *, is defined as:

Question: What is the intuition behind the

convolution operation?

Theorem: Convolution in the spatial domain

is the same as a multiplication in the

frequency domain and vice-versa

dttxgtfxgf

)()())(*(


The Fourier Transform (cont’d)

Question: What is the Fourier transform of a hat?

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CGGC4


Convolution

Consider an fmax band-limited signal.

Sampling is the same as multiplying the signal

with equally spaced impulse functions (x):

By the theorem, this operation is equivalent to a convolution in the frequency domain:


Convolution (cont’d)

During the reconstruction process, we apply a low pass

filter that removes all frequencies above fmax

This ideal filter is a box filter in the frequency domain:

Samples that are less dense in the spatial domain will

become more dense in the frequency domain

Question: What will happen if we sample below the

Nyquist limit


Convolution (cont’d)We will get aliasing:

The different convolved elements, in the

frequency domain, overlap.

We have high frequencies intermixed with low

frequencies and they are inseparable.

The only way to prevent aliasing is pre-filtering


Back to Computer Graphics

Question: how all this sampling theory relates

to computer graphics?

When a discrete image Id(x,y) is created, we

are effectively generating a sample set of a

(possibly imaginary) continuous image Ic(x,y)

Extending sampling theory to two dimensions:

Ic(x,y) should be band-limited for it to be

reconstructable

Sampling should be above the Nyquist limit for Ic(x,y)

to be reconstructable

The box filter in the frequency domain is the ideal

low-pass filter (what is it in the spatial domain?)


Back to Computer Graphics (cont’d)

Question: is Ic(x,y) band-limited?

Question: can we pre-filter Ic(x,y)?

One can reduce aliasing by super-sampling

Question: can we

post-filter the data using

convolution with a sinc

filter in the spatial domain?

22

22sin

yx

yx


Sinc/Truncated sinc

Box

Triangle (Hat)

Guassian

22

22 )sin(

yx

yx

otherwise,0

1and1,1 yx

otherwise,0

1,1 2222 yxyx

232

343

232

24

1

121

242

121

16

1

111

111

111

9

1

121

252

121

17

1

21012

14641

06960

14641

21012

33

1

12321

24642

36963

24642

12321

81

1

11111

12421

141041

12421

11111

50

1

Common Filters

2

22

2

1yx

e

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Common Filters (cont’d)

All filters are normalized so they sum to one.

Why?

Due to obvious reasons we cannot apply the

infinite filters and must compromise on the

truncated versions

Note the sinc filter can assume negative values

which is a potential for minor problems. Why?

The difference between different filters are quite

minor and most filters of size 44 offers similar

improvements


Filter results

Gaussian

Box

Unfiltered


Supersampling Example

None 3x3 5x5


A Word on Temporal Aliasing

So far we dealt with Spatial Aliasing

In animation, aliasing could also occur between one frame and the

next

This aliasing is known as Temporal Aliasing

Recall that in a movie each

frame represents a finite

time range of 1/25 or 1/30

of a second.

Hence, it would serve better

to accumulate the influence

of moving objects over this time


Motion Blur

The technique of

Motion Blur

accumulates the affect

of each object over the

time step of the frame

The result will look like

the image on the right

Playing animation with

motion blur is more

natural to the eye


Image ManipulationImage operations

Scaling

Rotation

Composition

Scientific American,

Feb 1994

Aliasing10/4/2010

CGGC6


Consider Linear Transformation L of image I:

What is wrong with this algorithm?

Image Scaling & Rotation

);,()),((

to0 from For

to0 from For

color background toclear

),ans(ImageLinTr

yxIyxLI

YMax(I)y

XMax(I)x

I

LI


Guarantee coverage – iterate over target

What can be wrong with this algorithm? Any

reason to limit L to a linear transformations?

Correct Image Scaling & Rotation

));,((),(

to0 from For

to0 from For


),ans(ImageLinTr

1 yxLIyxI

YMax(I)y

XMax(I)x

I

LI


Image Transformations

Each pixel in the target image is computed from a single source pixel – prone to Aliasing

Solution in filtering

A pixel in target image is

some affine combination

of several source pixels

Coefficients based on area coverage


Rotation By ShearConvenient for images as it

involved only intra-row or

intra-column operations

Rotation can be performed

by two shears – can be lossy

for large angles

Image size might be reduced in intermediate stage

Rotation by three shears is lossless up to

integer precision.

y

x

y

x

cossin

sincos

12

tan

01

10

sin1

12

tan

01

cossin

sincos


Image Composition

+

=


CompositionBasic Method

Will work until portion of foreground has

background color

Unfortunately we are facing more problems

);,(),(

else

);,(),(

color background),( if

to0 from For

to0 from For


),ompose(ImageOverC

1

2

1

21

yxIyxI

yxIyxI

yxI

YMax(I)y

XMax(I)x

I

II

Aliasing10/4/2010

CGGC7


Composition (cont’d)Question: This is the result of applying the

basic algorithm to teapot-over-white-

background. What is going on?


Composition (cont’d)

The reason for this behavior could be found in

the filtering (anti-aliasing) conducted during

the rendering process

On the boundary there exist pixels that are

only partially covered by the teapot

Rendered with black background,

these boundary pixels are darker

Yet, as dark as they are, they are

not black and hence they are

being copied verbatim

Question: Any remedy?



A background pixel has zero coverage while a

pixel totally contained in the teapot’s projection

will have a coverage of one.

This coverage information is also known as

the Alpha channel that accompanies the RGB

Question: How can we compute

the Alpha channel?

Question: How can we use the

Alpha channel to resolve this

boundary problem?



T – Teapot, S - Sand

Out: S(1 - T)

In: ST

Over: TT + S(1 - T)


Composition

(cont’d)

Scientific American,

Feb 1994


Special Effects – “Chroma Key”

Aliasing10/4/2010

CGGC8


Special Effects – “Chroma Key II”


Special Effects – “No Blue Background”

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