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Sampling, Aliasing & Antialiasing
Slides courtesy: Durand, Cutler and Pat Hanrahan
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Today What is a Pixel? Examples of aliasing Sampling & Reconstruction Filters in Computer Graphics
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What is a Pixel? A pixel is not: a box a disk a teeny tiny little light
A pixel “looks different” ondifferent display devices
A pixel is a sample it has no dimension it occupies no area it cannot be seen it has a coordinate it has a value
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More on Samples Most things in the real world are continuous,
yet everything in a computer is discrete The process of mapping a continuous function to a discrete
one is called sampling The process of mapping a continuous variable to a discrete
one is called quantization To represent or render an image using a computer,
we must both sample and quantize
discrete position
discretevalue
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An Image is a 2D Function An ideal image is a continuous function I(x,y) of intensities. It can be plotted as a height field. In general an image
cannot be represented as a continuous, analytic function.
Instead we represent images as tabulated functions.
How do we fill this table?
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Dirac Function Defined by its behavior when integrated
Zero almost everywhere Except at zero, where its infinite
)()()( 00 xfdxxxxf
x
(x)
0
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Sampling Grid We can generate the table values by multiplying the
continuous image function by a sampling grid of Kroneckerdelta functions.
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Sampling an Image The result is a set of point samples, or pixels.
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Today What is a Pixel? Examples of Aliasing Sampling & Reconstruction Filters in Computer Graphics
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Examples of Aliasing
Aliasing occurs because of sampling and reconstruction
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Examples of Aliasing
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Examples of Aliasing
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Examples of Aliasing
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Examples of AliasingTexture Errors
point sampling
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Examples of Aliasing
Image courtesy: http://www.siggraph.org/education/materials/HyperGraph/aliasing/alias2a.htm
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Today What is a Pixel? Examples of Aliasing Sampling & Reconstruction Sampling Density Fourier Analysis & Convolution
Filters in Computer Graphics
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Sampling Density How densely must we sample an image in order to
capture its essence?
If we under-sample the signal, we won't be able to accurately reconstruct it...
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Sampling Density If we insufficiently sample the signal, it may be
mistaken for something simpler during reconstruction (that's aliasing!)
Image from Robert L. Cook, "Stochastic Sampling and Distributed Ray Tracing",
An Introduction to Ray Tracing, Andrew Glassner, ed.,
Academic Press Limited, 1989.
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Sampling Theorem When sampling a signal at discrete intervals, the sampling
frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version (Shannon, Nyquist)
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Sampling Applet http://www.jhu.edu/~signals/sampling/index.html Amazing Sampling and Reconstruction Applet:
http://www.cs.brown.edu/exploratories/freeSoftware/repository/edu/brown/cs/exploratories/applets/nyquist/nyquist_limit_java_browser.html
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Sampling Density
Aliasing in 2D because of insufficient sampling density
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Images from http://axion.physics.ubc.ca/341-02/fourier/fourier.html
Remember Fourier Analysis? All periodic signals
can be represented as a summation of sinusoidal waves.
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Fourier Analysis Applet http://www.jhu.edu/~signals/fourier2/index.html
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Remember Fourier Analysis? Every periodic signal in the spatial domain has a dual in the
frequency domain.
This particular signal is band-limited, meaning it has no frequencies above some threshold
frequency domainspatial domain
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Remember Fourier Analysis? We can transform from one domain to the other using
the Fourier Transform.
spatial domainfrequency domain
Fourier Transform
Inverse Fourier
Transform
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Fourier Transform of an Image
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Fourier Transform pairs
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Convolution
Example:
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Convolution
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Convolution Some operations that are difficult to compute in the
spatial domain can be simplified by transforming to its dual representation in the frequency domain.
For example, convolution in the spatial domain is the same as multiplication in the frequency domain.
And, convolution in the frequency domain is the same as multiplication in the spatial domain
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Sampling in the Frequency Domain
(convolution)(multiplication)
originalsignal
samplinggrid
sampledsignal
Fourier Transform
Fourier Transform
Fourier Transform
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Reconstruction If we can extract a copy of the original signal from the
frequency domain of the sampled signal, we can reconstruct the original signal!
But there may be overlap between the copies.
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Guaranteeing Proper Reconstruction Separate by removing high
frequencies from the original signal (low pass pre-filtering)
Separate by increasing the sampling density
If we can't separate the copies, we will have overlapping frequency spectrum during reconstruction → aliasing.
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Sampling Theorem When sampling a signal at discrete intervals, the sampling
frequency must be greater than twice the highest frequency of the input signal in order to be able to reconstruct the original perfectly from the sampled version (Shannon, Nyquist)
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Today What is a Pixel? Examples of Aliasing Sampling & Reconstruction Filters in Computer Graphics Pre-Filtering, Post-Filtering Ideal, Gaussian, Box, Bilinear, Bicubic
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Filters Weighting function or a
convolution kernel Area of influence often
bigger than "pixel" Sum of weights = 1 Each sample contributes
the same total to image Constant brightness as object
moves across the screen.
No negative weights/colors (optional)
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Filters Filters are used to reconstruct a continuous signal from a sampled signal
(reconstruction filters) band-limit continuous signals to avoid aliasing during sampling
(low-pass filters)
Desired frequency domain properties are the same for both types of filters
Often, the same filters are used as reconstruction and low-pass filters
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Low-Pass Filters
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High-Pass Filter
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Pre-Filtering Filter continuous primitives Treat a pixel as an area Compute weighted
amount of object overlap What weighting function
should we use?
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Post-Filtering Filter samples Compute the weighted average of many samples Regular or jittered sampling (better)
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The Ideal Filter Unfortunately it has
infinite spatial extent Every sample contributes
to every interpolated point
Expensive/impossible to compute
spatial
frequency
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Problems with Practical Filters Many visible artifacts in re-sampled images are caused
by poor reconstruction filters Excessive pass-band attenuation results in blurry
images Excessive high-frequency leakage
causes "ringing" and can accentuate the sampling grid (anisotropy)
frequency
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Gaussian Filter This is what a CRT
does for free!
spatial
frequency
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Box Filter / Nearest Neighbor Pretending pixels
are little squares.
spatial
frequency
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Tent Filter / Bi-Linear Interpolation Simple to implement Reasonably smooth
spatial
frequency
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Bi-Cubic Interpolation Begins to approximate
the ideal spatial filter, the sinc function
spatial
frequency
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Why is the Box filter bad? (Why is it bad to think of pixels as squares)
Down-sampled with a 5x5 box filter
(uniform weights)
Original high-resolution image
Down-sampled with a 5x5 Gaussian filter
(non-uniform weights)
notice the ugly horizontal banding
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Trivia Real Time Antialiasing Techniques Sample the scene more than once and average the value
(supersampling) AntiAliasing in the GeForce 6800 (NV40) has changed from
that used in previous Nvidia cards. The method has changed from ordered grid multi- sampling to
rotated grid multi sampling. Only performed at the edge pixels – GPU has more
intelligence (the main difference from supersampling)
ATI SmoothVisionTM uses a jittered sampling patternOGSS
RGSS