Nima Kalantari

CSCE 441 - Computer Graphics

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Antialiasing

Some slides from Ren Ng

Rasterization

This is Displayed

Compare: The Continuous Triangle Function

What’s Wrong With This Picture?

Jaggies!

Jaggies (Staircase Pattern)

Is this the best we can do?

Outline

• Sampling & aliasing
• Frequency domain
• Antialiasing

Sampling

• Ubiquitous in Computer Graphics and Imaging

Rasterization = Sample 2D Positions

Photograph = Sample Image Sensor Plane

Video = Sample Time

Harold Edgerton Archive, MIT

Sampling Artifacts in Graphics and Imaging

Jaggies (Staircase Pattern)

This is also an example of “aliasing” – a sampling error

Moiré Patterns in Imaging

lystit.com

Read every sensor pixel

Skip odd rows and columns

Wagon Wheel Illusion (False Motion)

Created by Jesse Mason, https://www.youtube.com/watch?v=QOwzkND_ooU

Aliasing in video

Slide by Steve Seitz

Aliasing in video

Sampling Artifacts in Computer Graphics

• Artifacts due to sampling - “Aliasing”
• Jaggies – sampling in space
• Moire – undersampling images (and textures)
• Wagon wheel effect – sampling in time
• [Many more] …

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• Aliasing
• Fast-changing signals (high frequency) sampled too slowly

Antialiasing

Video: Point vs Antialiased Sampling

Point in Time

Motion Blurred

Video: Point Sampling in Time

30 fps video. 1/800 second exposure is sharp in time, causes time aliasing.

Credit: Aris & cams youtube, https://youtu.be/NoWwxTktoFs

Rasterization: Point Sampling in Space

Note jaggies in rasterized triangle �where pixel values are pure red or white

Rasterization: Antialiased Sampling

Pre-Filter

Sample

Note antialiased edges in rasterized triangle�where pixel values take intermediate values

Point Sampling

Antialiased

Aliasing and Antialiasing

• Why undersampling results in aliasing?
• Why pre-filtering reduces aliasing?

Outline

• Sampling & aliasing
• Frequency domain
• Antialiasing

Sines and Cosines

Frequencies

Fourier Transform

• Represent a function as a weighted sum

of sines and cosines

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Joseph Fourier 1768 - 1830

f₀ +

Extension to 2D

Image as a sum of basis images

=

Constant

Frequency Domain

(0,0)

Spatial Domain

— frequency 1/32; 32 pixels per cycle

(0,0)

Frequency Domain

Spatial Domain

— frequency 1/16; 16 pixels per cycle

(0,0)

Frequency Domain

Spatial Domain

Frequency Domain

Spatial Domain

2D Frequency Space

Note: Frequency domain also known as frequency space, Fourier domain, spectrum, …

Frequency Domain

Spatial Domain

Sampling

• Simple example: a sine wave

© 2006 Steve Marschner

Undersampling

• What if we “missed” things between the samples?
• Simple example: undersampling a sine wave
• Unsurprising result: information is lost

© 2006 Steve Marschner

Undersampling

• What if we “missed” things between the samples?
• Simple example: undersampling a sine wave
• Unsurprising result: information is lost
• Surprising result: indistinguishable from lower frequency
• Aliasing: high frequencies traveling in disguise as low frequencies

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© 2006 Steve Marschner

Moiré Patterns in Imaging

lystit.com

Read every sensor pixel

Skip odd rows and columns

Higher Frequencies Need Faster Sampling

x

Nyquist Theorem

• Sampling rate should be equal to or greater than twice the highest frequency in the analog signal

Signal Interval

Maximum Acceptable

Sampling Interval

Nyquist Theorem

• Sampling rate should be equal to or greater than twice the highest frequency in the analog signal

Nyquist Theorem

• Sampling rate should be equal to or greater than twice the highest frequency in the analog signal

Nyquist Theorem

• Sampling rate should be equal to or greater than twice the highest frequency in the analog signal

Antialiasing

• What can we do about aliasing?
• Sample more often
• Higher res displays, sensors, framebuffers
• Only shifts the problem to higher frequencies
• Make the signal less “wiggly” (filtering)
• Get rid of some high frequencies
• Will lose information
• But it’s better than aliasing

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Outline

• Sampling & aliasing
• Frequency domain
• Antialiasing

Rasterization: Point Sampling in Space

Note jaggies in rasterized triangle �where pixel values are pure red or white

Rasterization: Antialiased Sampling

Pre-Filter

Sample

Note antialiased edges in rasterized triangle�where pixel values take intermediate values

A Practical Pre-Filter

• One pixel-width box filter will attenuate frequencies whose period is less than or equal to 1 pixel-width

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Spatial Domain

Antialiasing By Averaging Values in Pixel Area

• The following are the same:�
• Option 1:
• Convolve scene by a 1-pixel box-blur
• Then sample at every pixel
• �Option 2:
• Compute the average value of scene in the pixel

Antialiasing by Computing Average Pixel Value

• In rasterizing one triangle, the average value inside a pixel area of f(x,y) = inside(triangle, x, y) is equal to the area of the pixel covered by the triangle.

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• Original

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• Filtered

1 pixel width

Super Sampling Antialiasing (SSAA)

Supersampling

• We can approximate the effect of the 1-pixel box filter by sampling multiple locations within a pixel and averaging their values:

4x4 supersampling

Point Sampling: One Sample Per Pixel

Supersampling: Step 1

• Take NxN samples in each pixel

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2x2 supersampling

Supersampling: Step 2

• Average the NxN samples “inside” each pixel

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2x2 supersampling

Supersampling: Step 2

• Average the NxN samples “inside” each pixel

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2x2 supersampling

Supersampling: Step 2

• Average the NxN samples “inside” each pixel

Supersampling: Result

This is the corresponding signal emitted by the display

75%

75%

100%

50%

50%

50%

50%

25%

Point Sampling

• One sample per pixel

4x4 Supersampling

• Pixel value is average of 4x4 samples per pixel

Sample Locations

(0,h)

(w,h)

(0,0)

(w,0)

Regular sampling: sample location for pixel (i,j)

(i+1/2,j+1/2)

Sample Locations

(i+1/4,j+3/4)

(i+1/4,j+1/4)

(i+3/4,j+1/4)

(i+3/4,j+3/4)

2x2 supersampling: locations for pixel (i,j)

(0,h)

(w,h)

(0,0)

(w,0)