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CSCI-6962 Advanced Computer Graphics Cutler
Sampling,
Aliasing,
& Mipmaps
CSCI-6962 Advanced Computer Graphics Cutler
Last Time?
CSCI-6962 Advanced Computer Graphics Cutler
(a) Chosen uniformly
within the cone of
directions subtended
by each light source
4 samples / pixel
(b) Chosen with
probability
proportional
to the BRDF
4 samples / pixel
(c) Weighted
combination of
(a) & (b)
8 samples / pixel
“Optimally combining
sampling techniques for
Monte Carlo rendering”
Veach & Guibas SIGGRAPH 1995
Importance Sampling
CSCI-6962 Advanced Computer Graphics Cutler
Today
• What is a Pixel?
• Examples of Aliasing
• Sampling & Reconstruction
• Filters in Computer Graphics
• Anti-Aliasing for Texture Maps
CSCI-6962 Advanced Computer Graphics Cutler
What is a Pixel?
• A pixel is not:
- a box
- a disk
- a teeny tiny little light
• A pixel “looks different” on
different 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
CSCI-6962 Advanced Computer Graphics Cutler
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
discrete
value
CSCI-6962 Advanced Computer Graphics Cutler
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.
images as tabulated
functions.
this table?
CSCI-6962 Advanced Computer Graphics Cutler
Sampling Grid
- We can generate the table values by multiplying the continuous
image function by a sampling grid of Kronecker delta functions.
CSCI-6962 Advanced Computer Graphics Cutler
Sampling an Image
- The result is a set of point samples, or pixels.
CSCI-6962 Advanced Computer Graphics Cutler
Questions?
CSCI-6962 Advanced Computer Graphics Cutler
Today
- What is a Pixel?
- Examples of Aliasing
- Sampling & Reconstruction
- Filters in Computer Graphics
- Anti-Aliasing for Texture Maps
CSCI-6962 Advanced Computer Graphics Cutler
Examples of Aliasing
- Aliasing occurs because of sampling and
reconstruction
CSCI-6962 Advanced Computer Graphics Cutler
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.
CSCI-6962 Advanced Computer Graphics Cutler
Sampling Density
- Aliasing in 2D because of insufficient
sampling density
CSCI-6962 Advanced Computer Graphics Cutler
Images from
http://axion.physics.ubc.ca/341-02/fourier/fourier.html
Remember Fourier Analysis?
can be represented
as a summation of
sinusoidal waves.
CSCI-6962 Advanced Computer Graphics Cutler
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
spatial domain frequency domain
CSCI-6962 Advanced Computer Graphics Cutler
Remember Fourier Analysis?
- We can transform from one domain to the other
using the Fourier Transform.
frequency domain spatial domain
Fourier
Transform
Inverse
Fourier
Transform
CSCI-6962 Advanced Computer Graphics Cutler
Remember Convolution?
Images from Mark Meyer
http://www.gg.caltech.edu/~cs174ta/
CSCI-6962 Advanced Computer Graphics Cutler
Remember 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
CSCI-6962 Advanced Computer Graphics Cutler
Sampling in the Frequency Domain
(multiplication) (convolution)
original
signal
sampling
grid
sampled
signal
Fourier Transform
Fourier Transform
Fourier Transform
CSCI-6962 Advanced Computer Graphics Cutler
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!
overlap between
the copies.
CSCI-6962 Advanced Computer Graphics Cutler
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.
CSCI-6962 Advanced Computer Graphics Cutler
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)
CSCI-6962 Advanced Computer Graphics Cutler
Questions?
CSCI-6962 Advanced Computer Graphics Cutler
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
CSCI-6962 Advanced Computer Graphics Cutler
Gaussian Filter
does for free!
spatial
frequency
CSCI-6962 Advanced Computer Graphics Cutler
Box Filter / Nearest Neighbor
are little squares.
spatial
frequency
CSCI-6962 Advanced Computer Graphics Cutler
Tent Filter / Bi-Linear Interpolation
- Simple to implement
- Reasonably smooth
spatial
frequency
CSCI-6962 Advanced Computer Graphics Cutler
Bi-Cubic Interpolation
the ideal spatial filter,
the sinc function
spatial
frequency
CSCI-6962 Advanced Computer Graphics Cutler
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
CSCI-6962 Advanced Computer Graphics Cutler
Questions?
CSCI-6962 Advanced Computer Graphics Cutler
Today
- What is a Pixel?
- Examples of Aliasing
- Sampling & Reconstruction
- Filters in Computer Graphics
- Anti-Aliasing for Texture Maps
- Magnification & Minification
- Mipmaps
- Anisotropic Mipmaps
CSCI-6962 Advanced Computer Graphics Cutler
Sampling Texture Maps
- When texture mapping it is rare that the screen-space
sampling density matches the sampling density of the
texture.
Original Texture Magnification for Display Minification for Display
for which we must use a reconstruction filter
64x64 pixels
CSCI-6962 Advanced Computer Graphics Cutler
Linear Interpolation
- Tell OpenGL to use a tent filter instead of a box filter.
- Magnification looks better, but blurry
- (texture is under-sampled for this resolution)
box filter tent filter
CSCI-6962 Advanced Computer Graphics Cutler
Spatial Filtering
- Remove the high frequencies
which cause artifacts in texture
minification.
- Compute a spatial integration
over the extent of the pixel
- This is equivalent to convolving
the texture with a filter kernel
centered at the sample (i.e.,
pixel center)!
rasterization, but an
approximation it can be
precomputed
projected texture in image plane
box filter in texture plane
CSCI-6962 Advanced Computer Graphics Cutler
MIP Mapping
of images that are
pre-filtered and
re-sampled at
1/2, 1/4, 1/8, etc.,
of the original
image's sampling
we compute the index of the decimated image that is sampled at
a rate closest to the density of our desired sampling rate
- MIP stands for multum in parvo which means
many in a small place
CSCI-6962 Advanced Computer Graphics Cutler
Anisotropic MIP-Mapping
different
mipmaps for
the 2 directions
extensions can
handle non
axis-aligned
views
Images from http://www.sgi.com/software/opengl/advanced98/notes/node37.html
CSCI-6962 Advanced Computer Graphics Cutler
Questions?
