Aliasing & Antialiasing

Aliasing & AntialiasingAaron Bloomfield

CS 445: Introduction to GraphicsFall 2006

(Slide set originally by David Luebke)

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Overview Introduction Signal Processing Sampling Theorem Prefiltering Supersampling Continuous Antialiasing Catmull's Algorithm The A-Buffer Stochastic Sampling

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Antialiasing Aliasing: signal processing term with very specific

meaning Aliasing: computer graphics term for any unwanted

visual artifact Antialiasing: computer graphics term for avoiding

unwanted artifacts We’ll tackle these in order

4No anti-aliasing

55 sample anti-aliasing

616 sample anti-aliasing

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Signal Processing Raster display: regular sampling of a continuous

function (Really?) Think about sampling a 1-D function:

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Signal Processing Sampling a 1-D function:

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What do you notice?

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Signal Processing Sampling a 1-D function: what do you notice?

Jagged, not smooth

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Jagged, not smooth Loses information!

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Jagged, not smooth Loses information!

What can we do about these? Use higher-order reconstruction Use more samples How many more samples?

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The Sampling Theorem Obviously, the more samples we take the better

those samples approximate the original function The Nyquist 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

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The Sampling Theorem In other words, to adequately capture a function

with maximum frequency F, we need to sample it at frequency N = 2F.

N is called the Nyquist limit.

The Nyquist sampling theorem applied to CDs Most humans can hear to 20 kHz

Some (like me!) to 22 kHz CDs are sampled at 44.1 kHz

Although not twice the “highest frequency component”, it is twice the “highest frequency component” that can be (generally) heard

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The Sampling Theorem An example: sinusoids

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The Sampling Theorem An example: sinusoids

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Fourier Theory All our examples have been sinusoids Does this help with real world signals? Why? Fourier theory lets us decompose any signal into

the sum of (a possibly infinite number of) sine waves

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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

A low-pass filter allows low frequencies through, but attenuates (or reduces) high frequencies

Then sample. Result: no aliasing!

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Prefiltering

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Prefiltering

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Prefiltering So what’s the problem? 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 Increase the resolution of the samples Average the results down

For now, we’ll assume that the samples are evenly spaced In the “Stochastic Sampling” section, we’ll see other

ways of doing it Sometimes called postfiltering

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 super-

sampling? 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 receding into infinity See next slide…

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Supersampling Despite these limitations, people still use super-

sampling (why?) So how can we best perform it?

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Supersampling The process:

1. Create virtual image at higher resolution than the final image

2. Apply a low-pass filter3. Resample filtered image

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Supersampling: Step 1 Create virtual image at higher resolution than the

final image This is easy

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Supersampling: Step 2 Apply a low-pass filter

Convert to frequency domain

Multiply by a box function

Expensive! Can we simplify this? 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

In practice, we combine steps 2 & 3: 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:

1 2 12 4 21 2 1

1 2 12 4 21 2 1=1 2 12 4 21 2 1=…

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Supersampling Typical supersampling algorithm:

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?

Passes infinitely high frequencies Attenuates desired frequencies

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Supersampling Common filters:

Truncated sinc sinc(x) = sin(x)/x

Box Triangle Guassian

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Supersampling In Practice Sinc function: ideal but impractical One approximation: sinc2

Another: Gaussian falloff Q: How wide (what res) should filter be? A: As wide as possible (duh) In practice: 3x3, 5x5, at most 7x7

In other words, the filter is larger than the final pixel So we are using the values of neighboring pixels

This creates a better visual effect

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Supersampling: Summary 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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Summary So Far 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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Summary So Far Supersampling

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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Summary So Far Supersampling cons

Doesn’t eliminate aliasing, just shifts the Nyquist limit higher

Can’t fix some scenes (e.g., checkerboard) Badly inflates storage requirements

Supersampling pros Relatively easy Often works all right in practice Can be added to a standard renderer

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Antialiasing in the Continuous Domain

Problem with prefiltering: 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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PixelGrid

Polygons Filter kernel

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The good news Exact polygon coverage of the filter kernel can be

evaluated What does this entail?

Clipping Hidden surface determination

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The bad news 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

AB

A1A2

A3

Find fragment areas

Multiply by fragment colors

Sum for final pixel color

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Catmull’s Algorithm First real attempt to filter in continuous domain Very expensive

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 Advantages:

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

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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Recap: Antialiasing Strategies Supersampling: sample at

higher resolution, then filter down

Pros: Conceptually simple Easy to retrofit existing

renderers Works well most of the time

Cons: High storage costs Doesn’t eliminate aliasing,

just shifts Nyquist limit upwards

A-Buffer: approximate pre-filtering of continuous signal by sampling

Pros: Integrating with scan-line

renderer keeps storage costs low

Can be efficiently implemented with clever bitwise operations

Cons: Still basically a super-

sampling approach Doesn’t integrate with ray-

tracing

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Stochastic Sampling Stochastic: involving or containing a random

variable Sampling theory tells us that with a regular

sampling grid, frequencies higher than the Nyquist limit will alias

Q: What about irregular sampling? A: High frequencies appear as noise, not aliases This turns out to bother our visual system less!

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Stochastic Sampling An intuitive argument:

In stochastic sampling, every region of the image has a finite probability of being sampled

Thus small features that fall between uniform sample points tend to be detected by non-uniform samples

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Stochastic Sampling Integrating with different renderers:

Ray tracing: It is just as easy to fire a ray one direction as another

Z-buffer: hard, but possible Notable example: REYES system (?) Using image jittering is easier (more later)

A-buffer: nope Totally built around square pixel filter and primitive-to-sample

coherence

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Stochastic Sampling Idea: randomizing distribution of samples scatters

aliases into noise Problem: what type of random distribution

to adopt? Reason: type of randomness used affects spectral

characteristics of noise into which high frequencies are converted

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Stochastic Sampling Problem: given a pixel, how to distribute points

(samples) within it?

Grid Random Poisson Disc Jitter

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Stochastic Sampling Poisson distribution:

Completely random Add points at random until area is full. Uniform distribution: some neighboring

samples close together, some distant

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Stochastic Sampling Poisson disc distribution:

Poisson distribution, with minimum-distance constraint between samples

Add points at random, removing again if they are too close to any previous points

Very even-looking distribution

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Stochastic Sampling Jittered distribution

Start with regular grid of samples Perturb each sample slightly in a

random direction More “clumpy” or granular in appearance

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Stochastic Sampling Spectral characteristics of these distributions:

Poisson: completely uniform (white noise). High and low frequencies equally present

Poisson disc: Pulse at origin (DC component of image), surrounded by empty ring (no low frequencies), surrounded by white noise

Jitter: Approximates Poisson disc spectrum, but with a smaller empty disc.

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Stochastic Sampling Watt & Watt, p. 134 See Foley & van Dam, p 644-645 Should have images next time

Aliasing & Antialiasing

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