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Aliasing and Antialiasing
Aliasing and Antialiasing
What is Aliasing?
“Errors and Artifacts arising during rendering, due to the conversionfrom a continuously defined illumination field to a discrete rastergrid of pixels” ITCS 4120/
What is Aliasing?
Aliasing and Antialiasing
What is Aliasing?
ITCS 4120/
What is Aliasing?
Aliasing and Antialiasing
Effects of Aliasing
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Effects of Aliasing
Aliasing and Antialiasing
Effects of Aliasing
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Area Sampling Techniques
Aliasing and Antialiasing
Area Sampling Techniques
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Area Sampling Techniques
Aliasing and Antialiasing
Area Sampling Techniques
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Unweighted Area Sampling
Pixel intensity is varied in proportion to the area of the pixel inter-cepted by the primitive.
Unweighted – equivalent to a box filter of unit height over pixel.
Properties^
Intensity of pixel decreases as the distance between the pixel centerand primitive increases.
A primitive cannot influence a pixel’s intensity if it does not intersectit.
Equal areas (intersected) contribute equal intensity – not a desirableproperty.
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Aliasing and Antialiasing
Weighted Area Sampling
Equal areas can contribute unequally in terms of pixel intensity.
Areas closer to the pixel center contribute more.
Essentially results in filtering with a mask that is centered over thepixel with decreasing radial influence.
Cone filters are a compromise between computational expense andoptimality.
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Postfiltering Techniques
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Aliasing and Antialiasing
Supersampling (Regular Sampling)
Very expensive.
Not very satisfactory.
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Filtering Example
Aliasing and Antialiasing
Filtering Example
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Filtering Example
Aliasing and Antialiasing
Aliasing from a Sampling Theory Viewpoint
Sampling(Spatial Domain) ITCS 4120/
Sampling(Spatial Domain)
Image is a spatial signal
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Aliasing and Antialiasing
Frequency Domain
X axis (position): frequency
Y axis (height): strength of each frequency
Examples:
sine wave:
impulse, square wave:
infinite train of im-
pulses
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How do we get to the Frequency Domain?
Use the Fourier Transform
Let
(x
)^
be a continuous function of a real variable
x
. Then
(x
)^
(x
)e
−j
2 πωx
dx
is the
Fourier Transform
of
(x
), with
j
and,
−^1
(x
)e
j^2
πωx
dω
is the
Inverse Fourier Transform.
◦^
(x
)^
is continuous and integrable
◦^
)^
is integrable
◦^
x
(spatial domain),
(frequency domain)
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Aliasing and Antialiasing
What does the Fourier Transform Do to A
Spatial Signal?
Signal in frequency domain is an integration of individual sinusoids.
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Comb Function
Application:
Used to digitize continuous functions.
Series of impulses (delta functions)
Identity element of convolution: reproduces an indentical copy of thefunction f(x)
FT of a comb function is another comb function
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Aliasing and Antialiasing
Comb Function(contd)
Multiplying
f
(x
)^
with a comb in image space
convolving their
Fourier transforms, resulting in multiple identical copies of
f^
(x
Can result in aliasing if copies overlap
Maximum allowable frequency is the Nyquist Frequency, which ishalf the sampling frequency.
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Aliasing and Antialiasing
Reconstruction Example(Adequate Sampling) ITCS 4120/
Aliasing and Antialiasing
Reconstruction Example(Inadequate Sampling) ITCS 4120/
Aliasing and Antialiasing
Box Filter
Reconstruction filter for nearest neighbor interpolation.
Resampling images/volumes to a higher resolution using nearestneighbor values.
FT of a box filter is the Sinc function (
sinπxπx
Large side lobes continuing at regular intervals will cause aliasing.
Aliasing in images manifests itself as “jaggies”
Aliasing and Antialiasing
Pyramid Filter
Reconstruction filter used in linear interpolation
Computationally more expensive, but more accurate
FT is much better behaved (side lobes much smaller)
Less tendency to produce aliasing
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Aliasing and Antialiasing
Gaussian Filter
The optimal filter in terms of avodiding side lobes
FT of a Gaussian is another Gaussian
Widely used to blur images and the basis for scale space
Aliasing and Antialiasing
