Aliasing and Anti-Aliasing Copyright Zachary Wartell, University of North Carolina at Charlotte, All...

Aliasing and Anti-Aliasing

Copyright Zachary Wartell, University of North Carolina at Charlotte, All Rights Reserved

Revision: 10/24/2007 3:38:00 AM

©Zachary Wartell - 1/26/2005

Describe what you see!

(Hint)

Example A

Example B


Reconstructed Signal (Low Frequency)


Sampling (Lower Frequency)

Signal (High Frequency)

Reconstruction


Example A

Example B

ReconstructionReconstructed Signal (Low freq.)







Reconstruction


Example A

Aliasing = a high frequency signal masquarading as a low frequency signal due to a poor sampling

frequency.


position

inte

nsity

Recall: frequency, cycle/wavelength

Δxcycle

sin 2 /

is amplitude

is wavelength

(ignoring y-offset and phase shift)

1/ frequency

cycle

cycle

cycle

y A x x

A

x

f x

A

What is a “poor sampling frequency”?

Nyquist Sampling Frequency:

fs = 2 fmax

Δxs = Δxcycle / 2

fs : Nyquist Sampling Frequencyfmax : maximum frequency component in signalΔxs : Nyquist Sampling IntervalΔxcycle : Cycle interval

“To avoid losing information we need to set sampling frequency to at least twice the highest frequency component of signal.”©Zachary Wartell - 1/26/2005

Δx, f = 1/ΔxΔt, f = 1/Δt

space or time

ampl

itude

Δx, f = 1/ΔxΔt, f = 1/Δt

space or time

ampl

itude

TADA!highest freq.comp. of S(x)

Example of highest frequency component

● What is highest frequency component of this signal, S(x)?

● What if I tell you the equation is:

Sin[x (2 Pi /10)] + Sin[x (2 Pi / 1)]*0.25

= +

f= 1/10, a=1

Sin[x (2 Pi / 1)]*0.25Sin[x (2 Pi /10)]

f= 1, a=0.25

Highest frequency component of general signal

Any signal represented as sum of sinusoids:Amplitude vs Space

Input Signal S

=+

+

+

+ …

∑


Fourier Transform

Amplitude vs Frequency

Input Signal S

=+

++

+ …∑

Insensity vs Position

=

©Zachary Wartell - 1/26/2005 f

af -how much does frequency f contribute to the original signal S

sinusoid a frequency f of amplitude af

How does this relate to image capture?

-”real” image is a 2D signal of infinite resolution- the sampled image is of finite resolution (finite # of pixels)-the original image contains details that are very much smaller than the pixel sampling rate, result is jagged lines/boundaries

- high detail means high frequency components- jagged lines/boundaries is aliasing

“real” image - 2D Signal (“Mia”)


sampled image - (“Eye”)

Sampling “Mia” signal at too low frequency

©Zachary Wartell

→

“1D” Slice

Signal

Sample

Reconstruction ()


http://xkcd.com/26/

Anti-Aliasing Options (Lines)

increase sampling rate with more pixels in framebuffer & displayUse same framebuffer resolution but enable multiple pixels based on pixel coverage pre-filter image - remove high frequenciesbefore sampling

area sampling – calculate area of overlap of each pixel with displayed primitives

supersampling (post-filter) – use more framebufferpixels and “average” intensities of groups of super-pixels results to the lower resolution actual pixels


-Bresenham's line algorithm can be modified to account for pixel coverage [Gupta-Sproull]. We treat line as 1 pixel thick rectangle which can intersect up to 3 pixels and use a measure of the distance (D?) from line to the upper (Pk+1

u), standard (Pk+1) and lower pixel (Pk+1

l) to compute the weight each pixels intensity.

Use Pixel Coverage

Pαu

Pαl

Mk

Pk Pα

Du

Dl

D

Pαu

Pαu

Pαl

Pαl

Mk

Pk Pα

Du

Dl

D

Figure 13: Assume case Pk+1=Pα. Distances Du,D,Dl are functions of decision parameter (pk). Each distance used to get a intensity weighting from a pre-computed lookup table


Supersampling Straight-Line Segments

supersample subpixel positions of zero-width line

2 3 4

2

3

4

Intensity of pixel Pis number ofsub-pixels turned onin P


Supersampling Straight-Line Segments

©Zachary Wartell

supersample subpixel positions of finite-width line

2 3 4

2

3

4

-Intensity of pixel Pis number ofsub-pixels turned onin P

-Use lower left corner rule

-Also “pick-up” color ofbackground


Subpixel Weighting Masks (Discrete Array)

2 3 4

2

3

4

1

1

1

1

2

2

2 24

-Mask (sum to 16)


Area Sampling Straight-Line Segments

©Zachary Wartell©Zachary Wartell

Compute pixel to rectangle intersection area (trapezoid)

2 3 4

2

3

4


Filtering Techniques (Continuous Function)

Box Cone Gaussian

-Integrate over volume-Highest potential accuracy (= slow)

Sinc(sin θ/θ)

∞


Line-Intensity and Line Angle

√2 * L

L


Anti-aliasing boundaries: Pixel Coverage

-adjust intensity on boundary

x x+1

y

y+1


Anti-aliasing boundaries: Supersampling

2 3 4

2

3

4


Anti-aliasing boundaries: Pitteway & Watkinson

-variant of Bresenham Midpoint decision parameter p (assume m in [0,1])

xk

yk

yk+1

xk+1

Pβ

Pα

Mk

L: y=mx+b

Pk

yk+1/2



xk

yk

yk+1

xk+1

Pβ

Pα

Mk

L: y=mx+b

Pk

yk+1/2

xk

yk

yk+1

xk+1

Pβ

Pα

Mk

L: y=mx+b

Pk

yk+1/2

add 1 – m, p [0,1]

p = [m(xk+1) + b] – (yk + ½) + (1-m)

Pα ↔ p < 1 – m and Pβ ↔ p > 1 – m



Insight: p = [m(xk+1) + b] – (yk + ½) + (1-m) =coverage area = m xk + b – yk + ½

xk- 1/2 xk+ 1/2xk

yk+1/2

yk-1/2

yk

y = m (xk+ ½) + b

y = m (xk- ½) + b

Atrap=B H0 + ½ B (H1 - H0 )


Anti-Aliasing: Nvidia

-GeForce 4 has Accuview “multi-sampling”:

“1. Subpixels are rendered in parallel (thanks to multisampling technology) to a back buffer. This back buffer is a factor that is larger than the final display resolution.2. The image is filtered and written out to a front frame buffer.3. The frame buffer is sent to the display.”

-GeForce 8800 uses “coverage samples”…, and is Coverage Sampling Antialiasing (CSAA).

“CSAA provides higher levels of AA than with current 4x or 8x multisampling AA. CSAA modes include 8x, 16x and 16xQ.

CSAA produces antialiased images that rival the quality of 8x or 16x MSAA, while introducing only a minimal performance hit over standard (typically 4x) MSAA. ”


Revisions

-1.2 – update with 8800 GPU commentary-to tedious s to keep this updated see SVN logs

Aliasing and Anti-Aliasing Copyright Zachary Wartell, University of North Carolina at Charlotte, All...

Documents

Transcript of Aliasing and Anti-Aliasing Copyright Zachary Wartell, University of North Carolina at Charlotte, All...