Basic Signal Processing: Sampling, Aliasing, Antialiasing




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Basic Signal Processing: Sampling, Aliasing, Antialiasing 113362_3sampling.pdf CS148: Introduction to Computer Graphics and Imaging

Basic Signal Processing:

Sampling, Aliasing, Antialiasing

No Jaggies

CS148 Lecture 13Pat Hanrahan, Fall 2011

Key Concepts

Frequency space

Filters and convolution

Sampling and the Nyquist frequency

Aliasing and Antialiasing

Frequency Space

sin2 -x

CS148 Lecture 13Pat Hanrahan, Fall 2011

Sines and Cosines

cos2 x

Frequencies

cos4 -x cos2 fx f=1 f=1 T f=2

CS148 Lecture 13Pat Hanrahan, Fall 2011

cos2 πx

Euler's Formula

Odd (-x)

Therefore

Hence, use complex exponentials for sines/cosines

Recall Complex Exponentials

CS148 Lecture 13Pat Hanrahan, Fall 2011

e jx = cosx+jsinx e -jx = cos-x+jsin-x= cosx-jsinx cosx=e jx +e -jx 2 sinx=e jx -e -jx 2 j

CS148 Lecture 13Pat Hanrahan, Fall 2011

Constant

Spatial DomainFrequency Domain

sin(2-/32)x

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

Frequency = 1/32; 32 pixels per cycle

sin(2?/16)x

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

sin(2/16)y

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

sin(2?/32)x×sin(2?/16)y

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

e r 2 / 16 2

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

e ?r 2 / 32
2

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

e -x 2 / 32
2 ×e -y 2 / 16 2

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

Rotate 45

CS148 Lecture 13Pat Hanrahan, Fall 2011

Spatial DomainFrequency Domain

e -x 2 / 32
2 ×e -y 2 / 16 2

Filtering

CS148 Lecture 13Pat Hanrahan, Fall 2011

My Humble Frequencies

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Remove Low Frequencies (Edges)

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Remove High Frequencies (Blur)

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Remove Low and High Frequencies

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Remove Low and High Frequencies

Spatial DomainFrequency Domain

Filters = Convolution

CS148 Lecture 13Pat Hanrahan, Fall 2011

Convolution

130421

12 1 * 1 + 3 * 2 = 7

CS148 Lecture 13Pat Hanrahan, Fall 2011

Convolution

7

130421

12

CS148 Lecture 13Pat Hanrahan, Fall 2011

Convolution

3 * 1 + 0 * 2 = 3

130421

12 73

CS148 Lecture 13Pat Hanrahan, Fall 2011

Convolution

0 * 1 + 4 * 2 = 8

130421

12 738

CS148 Lecture 13Pat Hanrahan, Fall 2011

Convolution Theorem

A ? lter can be implemented in the spatial domain using convolution A ? lter can also be implemented in the frequency domain

Convert image to frequency domain

Convert

? lter to frequency domain

Multiply

? lter times image in frequency domain

Convert result to the spatial domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Box Filter

11 11

CS148 Lecture 13Pat Hanrahan, Fall 2011

Box Filter = Low-Pass Filter

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Wider Filters, Lower Frequencies

Spatial DomainFrequency Domain

CS148 Lecture 13Pat Hanrahan, Fall 2011

Size of Filter

As a ? lter is localized in space, it spreads out in frequency

Conversely, as a

? lter is localized in frequency, it spreads out in space

A box

? lter is very localized in space; it has in ? nite extent in frequency space

CS148 Lecture 13Pat Hanrahan, Fall 2011

Ef ? ciency?

When would it be faster to apply the

? lter in the spatial domain?

When would it be faster to apply the

? lter in the frequency domain?

Sampling

CS148 Lecture 13Pat Hanrahan, Fall 2011

Image Generation = Sampling

Evaluating a function at a point is sampling

for( int x = 0; x < xmax; x++ ) for( int y = 0; y < ymax; y++ ) Image[x][y] = f(x,y);

Rasterization is equivalent to evaluating the

function inside(triangle,x,y)

CS148 Lecture 13Pat Hanrahan, Fall 2011

Sampling Causes Jaggies

Retort, by Don Mitchell

Staircase pattern or jaggies

CS148 Lecture 13Pat Hanrahan, Fall 2011

Sampling in Computer Graphics

Artifacts due to sampling -

Aliasing

Jaggies - sampling in space

Wagon wheel effect - sampling in time

Temporal strobing - sampling in space-time

Moire - sampling texture coordinates

Sparkling highlights - sampling normals

Preventing these artifacts -

Antialiasing

Aliasing

Wagon Wheel Effect

http://www.michaelbach.de/ot/mot_wagonWheel/

CS148 Lecture 13Pat Hanrahan, Fall 2011

"Aliases"

These two sine waves are indistinguishable

Indistinguishable frequencies are called "aliases"

CS148 Lecture 13Pat Hanrahan, Fall 2011

Nyquist Frequency

De ? nition: The Nyquist frequency is ½ the sampling frequency (1/Ts)

Frequencies above the Nyquist frequency appear

as aliases

No aliases appear if the function being sampled

has no frequencies above the Nyquist frequency

Antialiasing

CS148 Lecture 13Pat Hanrahan, Fall 2011

Antialiasing

Simple idea:

Remove frequencies above the Nyquist

frequency before sampling

How? Filtering before sampling

CS148 Lecture 13Pat Hanrahan, Fall 2011

Pre ? ltering by Computing Coverage

A 1 pixel box

? lter removes frequencies whose period is less than or equal to 1 pixel

Original

Filtered

CS148 Lecture 13Pat Hanrahan, Fall 2011

Point- vs. Area-Sampled

PointArea

Checkerboard sequence by Tom Duff

CS148 Lecture 13Pat Hanrahan, Fall 2011

Antialiasing

JaggiesPre?lter

CS148 Lecture 13Pat Hanrahan, Fall 2011

Antialiasing vs. Blurred Aliases

Blurred JaggiesPre?lter

CS148 Lecture 13Pat Hanrahan, Fall 2011

Things to Remember

Signal processing

Frequency domain vs. spatial domain

Filters in the frequency domain

Filters in the spatial domain = convolution

Sampling and aliasing

Image generation involves sampling

May also sample geometry, motion, ...

Nyquist frequency is

½ the sampling rate

Frequencies above the Nyquist frequency

appear as other frequencies - aliases

Antialiasing - Filter before sampling

Extra Slides

Supersampling

CS148 Lecture 13Pat Hanrahan, Fall 2011

Approximate a box

? lter by taking more samples and averaging them together

Supersampling

4 x 4 supersampling

CS148 Lecture 13Pat Hanrahan, Fall 2011

Point-sampling vs. Super-sampling

Point4x4 Super-sampled

Checkerboard sequence by Tom Duff

CS148 Lecture 13Pat Hanrahan, Fall 2011

Area-Sampling vs. Super-sampling

Exact Area4x4 Super-sampled


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