Transcript lecture06.ppt

CS 551/651:
Advanced Computer Graphics
Antialiasing Continued:
Prefiltering and Supersampling
David Luebke
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Recap:
The Sampling Theorem

The sampling theorem:
A continuous bandlimited function can be completely
represented by a set of equally spaced samples, if the
samples occur at more than twice the frequency of the
highest frequency component of the function
To capture a function with maximum
frequency F, sample it at frequency N = 2F.
 N is called the Nyquist limit.

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Recap: Fourier Theory
Can decompose any signal I(x) into a spectrum
of sine waves F(u)
 F(u) is complex; we represent it with:
|F(u)| = [Real(u)2 + Imag(u)2]1/2
Ø(u) = tan-1[Real(u)/Imag(u)]
 These are called the amplitude and phase
spectra

 Back
David Luebke
to transparencies…
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Recap: Aliasing
Sampling a function
= multiplying I(x) by sampling fn Sx
= convolving F(u) by sampling fn S1/x
= copying F(u) at regular intervals
 If the copies of F(u) overlap, high frequencies
“fold over”, appearing as lower frequencies
 This is known as aliasing

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Prefiltering
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Eliminate high frequencies before sampling
(Foley & van Dam p. 630)
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Convert I(x) to F(u)
Apply a low-pass filter (e.g., multiply F(u) by a
box function)
Then sample. Result: no aliasing!
So what’s the problem?
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Prefiltering

Eliminate high frequencies before sampling
(Foley & van Dam p. 630)




Convert I(x) to F(u)
Apply a low-pass filter (e.g., multiply F(u) by a
box function)
Then sample. Result: no aliasing!
Problem: most rendering algorithms generate
sampled function directly

e.g., Z-buffer, ray tracing
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Supersampling

The simplest way to reduce aliasing artifacts is
supersampling

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
Increase the resolution of the samples
Average the results down
Sometimes called postfiltering


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Create virtual image at higher resolution than the
final image
Apply a low-pass filter
Resample filtered image
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Supersampling:
Limitations
Q: What practical consideration hampers
supersampling?
 A: Storage goes up quadratically
 Q: What theoretical problem does
supersampling suffer?
 A: Doesn’t eliminate aliasing!
Supersampling simply shifts the Nyquist limit
higher

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Supersampling:
Worst Case
Q: Give a simple scene containing infinite
frequencies
 A: A checkered ground plane receeding into
infinity
 Show Watt & Watt, Figure 4.9

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Supersampling
Despite these limitations, people still use
supersampling (why?)
 So how can we best perform it?

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Supersampling

The process:



Create virtual image at higher resolution than the
final image
Apply a low-pass filter
Resample filtered image
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Supersampling
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Create virtual image at higher resolution than
the final image

This is easy
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Supersampling

Apply a low-pass filter
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Convert to frequency domain
Multiply by a box function
Expensive! Can we simplify this?
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Recall: multiplication in frequency equals
convolution in space, so…
Just convolve initial sampled image with the FT of
a box function
Isn’t this a lot of work?
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Supersampling:
Digital Convolution
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In practice, we combine steps 2 & 3:
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Create virtual image at higher resolution than the
final image
Apply a low-pass filter
Resample filtered image
Idea: only create filtered image at new sample
points

I.e., only convolve filter with image at new points
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Supersampling:
Digital Convolution
Q: What does convolving a filter with an
image entail at each sample point?
 A: Multiplying and summing values
 Example (also F&vD p 642):

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
=
1 2 1
2 4 2
1 2 1
…
1 2 1
2 4 2
1 2 1
=
1 2 1
2 4 2
1 2 1
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Supersampling
Q: What filter should we use in space to effect
multiplying by a box function in frequency?
 A: The sinc function (sin(x) / x)
 Q: Why is this hard?
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Sinc function has infinite support
Sinc function has negative lobes
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Supersampling
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Common filters:
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Truncated sinc
Box
Triangle
Guassian
What do the frequency spectra of each of these
filters look like?
 A: see F&vD, p. 634-635
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Supersampling

Typical supersampling algorithm:
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Compute multiple samples per pixel
Combine sample values for pixel’s value using
simple average
Q: What filter does this equate to?
 A: Box filter -- one of the worst!
 Q: What’s wrong with box filters?

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Passes infinitely high frequencies
Attenuates desired frequencies
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Supersampling
In Practice
Sinc function: ideal but impractical
 One approximation: sinc2 (=what?)
 Another: Guassian falloff
 Q: How wide (what res) should filter be?
 A: As wide as possible (duh)
 In practice: 3x3, 5x5, at most 7x7

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Supersampling:
Summary
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Supersampling improves aliasing artifacts by shifting
the Nyquist limit
It works by calculating a high-res image and filtering
down to final res
“Filtering down” means simultaneous convolution
and resampling
This equates to a weighted average
Wider filterbetter resultsmore work
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Antialiasing:
Coming Up

Stochastic supersampling
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
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Fundamentally the same idea
Works better in practice
A-buffer

Back to the prefiltering idea
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Recap

Prefiltering



Before sampling the image, use a low-pass filter to
eliminate frequencies above the Nyquist limit
This blurs the image…
But ensures that no high frequencies will be
misrepresented as low frequencies
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Recap
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Supersampling
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Sample image at higher resolution than final
image, then “average down”
“Average down” means multiply by low-pass
function in frequency domain
Which means convolving by that function’s FT in
space domain
Which equates to a weighted average of nearby
samples at each pixel
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Recap

Supersampling cons
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Doesn’t eliminate aliasing, just shifts the Nyquist
limit higher
 Can’t

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fix some scenes (e.g., checkerboard)
Badly inflates storage requirements
Supersampling pros
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Relatively easy
Often works all right in practice
Can be added to a standard renderer
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Antialiasing in the
Continuous Domain
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Problem with prefiltering:
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Sampling and image generation inextricably linked
in most renderers
 Z-buffer
algorithm
 Ray tracing


Why?
Still, some approaches try to approximate
effect of convolution in the continuous domain
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Antialiasing in the
Continuous Domain
Pixel
Grid
Filter kernel
Polygons
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Antialiasing in the
Continuous Domain
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The good news
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Exact polygon coverage of the filter kernel can be
evaluated
What does this entail?
 Clipping
 Hidden
David Luebke
surface determination
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Antialiasing in the
Continuous Domain
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The bad news
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Evaluating coverage is very expensive
The intensity variation is too complex to integrate
over the area of the filter
 Q:
Why does intensity make it harder?
 A: Because polygons might not be flat- shaded
 Q: How bad a problem is this?
 A: Intensity varies slowly within a pixel, so shape
changes are more important
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Catmull’s Algorithm
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Find fragment
areas
Multiply by
fragment colors
Sum for final
pixel color
A2
A1
AB
David Luebke
A3
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Catmull’s Algorithm
First real attempt to filter in continuous
domain
 Very expensive
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Clipping polygons to fragments
Sorting polygon fragments by depth (What’s wrong
with this as a hidden surface algorithm?)
Equates to box filter (Is that good?)
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The A-Buffer
Idea: approximate continuous filtering by
subpixel sampling
 Summing areas now becomes simple
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The A-Buffer
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Advantages:
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Incorporating into scanline renderer reduces
storage costs dramatically
Processing per pixel depends only on number of
visible fragments
Can be implemented efficiently using bitwise
logical ops on subpixel masks
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The A-Buffer

Disadvantages

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Still basically a supersampling algorithm
Not a hardware-friendly algorithm
 Lists
of potentially visible polygons can
grow without limit
 Work per-pixel non-deterministic
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The A-Buffer
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Comments
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Watt & Watt claims this is most common algorithm
for high-quality rendering
I’m not so sure, anymore
They give much gory detail
I won’t test you on it
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The End
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