CSE 681 Aliasing and Antialiasing

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CSE 681Aliasing and Anti-aliasing
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Aliasing

• Functions are continuous, but the computer only
  allows for a discrete representation
• Signal processing is very concerned about
  discrete representations of functions
• Sampling a function is the process of determining
  which function values to take and store for
  discrete representation
• Reconstruction is the process of determining the
  function value at non-sampled locations

What is Aliasing? If you insufficiently sample
the signal, it may be mistaken for something
simpler during reconstruction
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Sampling And Reconstruction

• Assume samples are taken at a regular spacing

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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 (see right)
• How do we determine the function values in a 2D
  array table?

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Sampling an Image

• The result is a set of point samples, or pixels

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Sources of Aliasing

• Sampling The process of mapping a continuous
  function to a discrete one
• For example, take samples along the x-axis
• Quantization The process of mapping a continuous
  variable to a discrete one
• For example, take samples on the y-axis
• Convert color intensities to 8-bits per channel
  (framebuffer). We do this a lot!

discretevalue
discrete position
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An Aliasing Example
Another source of aliasing
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Reconstruction Another Source of Aliasing

• The choice of a reconstruction filter affects
  aliasing
• For example, texture mapping

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Nearest Neighbor Reconstruction

• Acts like the round() function
• OK approximation, but not so great

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Nearest Neighbor Reconstruction
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Bilinear Interpolation

• Blend the four texel values surrounding the
  sample by weighted distance

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Bilinear Interpolation

• Blending (or averaging) eliminates abrupt color
  changes
• ? Averaging (or smoothing) is a common way to
  reduce aliasing artifacts
• (i.e., anti-aliasing)

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Nearest Neighbor vs. Bilinear
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Nearest Neighbor vs. Bilinear
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Where Do We Find Aliasing?

• Aliasing occurs through out the rendering
  pipeline
• Point, normal, depth, and color quantization
• Full float precision is a recent advancement in
  PC graphics hardware pipelines
• Rasterization of geometric primitives
• Texture mapping
• Display hardware

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Line Rasterization
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Curve Rasterization
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Low Screen Resolution
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Pixels Display Hardware

• The value at a point sample is spread across a
  pixel
• A pixel is determined by the display device
• Shape and size?

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Aliasing in 2D Texture Mapping

• Texture sampling starts from a pixel in screen
  space
• Texture image resolution is usually greater than
  the screen pixel resolution
• Camera motion such as translation and zoom in/out

zoom out
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Aliasing in 2D Texture Mapping

• Texture sampling starts from a pixel in screen
  space
• Texture image resolution is usually greater than
  the screen pixel resolution
• Camera motion such as translation and zoom in/out
  (called temporal aliasing)

zoom in
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The Classic Checkerboard Example

• A checkerboard texture is a 2D function with lots
  of high frequencies (thus, difficult to sample
  properly but great to check your anti-aliasing
  method)

Aliasing increases when texels per pixel
increases
This texture has more checkers
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Texture Anti-aliasing

• Take a pixel center from screen space, map to
  texture space, and filter the texture with the
  neighboring texels
• This produces the renderings on the previous
  slide
• How can we perform texture anti-aliasing?
• Map the four corners of the pixel to texture space

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Texture Anti-aliasing

• Map the four corners of the pixel to texture
  space
• Pixel maps to a quadrilateral (called its
  pre-image)

image courtesy of D. Cohen-Or
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Magnification/Minification

• Magnification
• Pre-image covers the area of one texel or less
• Minification
• Pre-image covers the area of greater than one
  texel

from Angel
Magnification
Minification
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How Do We Anti-alias?

• In minification, we can combine (average!) many
  texel values to compute a pixel color

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Mipmaps

• Approximate the quadrilateral with just a
  rectangle
• I.e., just assume we have an axis-aligned
  rectangle
• Construct a hierarchy of textures
• Sample the texture at difference scales
  (resolutions)
• This is called a Level of Detail (LOD)
• Assume your texture has a power of two resolution
• Compute a new texture by averaging four
  neighboring texel value together
• Do step 2 again with the new texture
• Each new texture is a pre-filtered
  (reconstructed?) version of the original texture
  where a texel covers larger and larger areas

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Mipmaps

• Mipmap A hierarchy of pre-filtered textures
• Match your pre-image to a pre-computed average
• Compute the area of your pre-image
• Index to the level you need, i.e. which LOD
  texture
• Nearest or linear filtering in the LOD texture

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Mipmap Construction Storage

• Simple construction
• OpenGL will automatically generate this for you
• Easy storage
• Fits into a single 2D texture

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Anti-aliasing With Mipmaps
Aliased
Anti-aliased
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Common Approaches To Do Anti-Aliasing

• Quantization
• Use higher precision values
• Sampling
• Use a higher sampling rate
• Reconstruction
• Use higher order filters
• For example, mipmapping

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Higher Sampling Rate

• Raytracing
• One ray/pixel
• Pixel centers
• Computational Cost?
• 512 512 262,144 rays
• Well just primary rays

Poor ray sampling due to the view angle and
perspective projection
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Higher Sampling Rate

• Supersampling
• 16 rays/pixel
• Spread across a pixel
• Computational Cost?
• 512 512 16 4,194,304 rays

High frequencies are diminishing! Looks averaged
some what
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Higher Sampling Rate

• Supersampling
• 256 rays/pixel
• Spread across a pixel
• Computation time?
• 512 512 256 67,108,864 rays

High frequencies are diminishing! Looks more
averaged and the speckly artifacts are greatly
diminished.
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Signal Theory

• Signal Theory (an unpopular term) can help us
• Understand why an anti-aliasing method works or
  does not work
• Discover methods

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Sampling Theorem

• Questions you should ask yourself
• Why does supersampling work?
• How many samples are enough?
• Shannons Sampling Theorem
• Lets assume regularly (equally?) spaced samples
• Almost always done in practice
• 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. This maximum
  frequency is called the Nyquist limit.

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A Cosine Example

• Consider the function f (x) cos(2px) and a
  regular sampling

• Given the samples, how do we reconstruct the
  function?

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A Cosine Example

• Consider the function f (x)cos(12px) at a higher
  frequency, but same sampling rate

• OK, we get aliasing. Sampling rate is too low.

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A Cosine Example

• Take our first function and reduce the sampling
  frequency
• Nyquist limit ½ sampling frequency

• Can we reconstruct the function exactly?
• The sampling theorem says yes, since our
  sampling rate is sufficient compared with the
  Nyquist limit
• What reconstruction filter should we use?

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Sampling Grid

• We can generate the table values by multiplying
  the continuous image function by a sampling grid
  of Kronecker delta functions.

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Sampling Density

• How densely must we sample an image?
• If the scene is well understood then we have a
  chance
• If it has complex textures and such, then it is
  difficult
• If we under-sample the signal, we won't be able
  to accurately reconstruct it

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More Theory Fourier Analysis

• So, what are the frequency characteristics of a
  function?
• Fourier Analysis defines a feature space (called
  Fourier Space or the Frequency Domain) that can
  show us the frequency characteristics of a
  function
• Lets define this space

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Fourier Analysis Harmonics

• This formulation is defined around periodic
  signals
• All periodic signals can be represented as a
  summation of sinusoidal waves

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Fourier Transform

• The Fourier transform defines a conversion
  between our function (spatial domain, i.e. f (x,
  y)) into Fourier space (frequency domain, i.e.
  F(u, v)) and vice versa

spatial domain
frequency domain
Fourier Transform
Inverse Fourier Transform
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Frequency Space

• Every periodic signal in the spatial domain has a
  dual in the frequency domain
• The u-axis tells us that a particular frequency
  is present in the function and the v-axis tells
  us how much
• A bandlimited function
• A function which has no frequencies above some
  threshold

frequency domain
spatial domain
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The Convolution Operator

• The convolution operator g f Ä h
• Combines two functions to produce a third
  function by shifting and multiplying one function
  with another

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Duality of Convolution And Multiplication

• Convolution in the spatial domain is the same as
  multiplication in the frequency domain
• Convolution in the frequency domain is the same
  as multiplication in the spatial domain

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Sampling And Fourier Space
Fourier Transform
samplinggrid
(multiplication)
(convolution)
Fourier Transform
sampledsignal
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Aliasing Due To Insufficient Sampling

• Aliasing due to sampling is seen in the frequency
  domain as overlapping of the function copies
• Fourier Analysis tells us that as the sampling
  rate is increased there is less overlap between
  the copies

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Sampling Theorem Revisited

• In order to have any hope of accurately
  reconstructing a function from a periodically
  sampled version of it, two conditions must be
  satisfied
• The function must be bandlimited
• The sampling frequency, fs, must be at least
  twice the maximum frequency, fmax, of the
  function (Nyquist Limit)

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Reconstruction

• Given a sampled version of a function, how do we
  evaluate (reconstruct?) the function at
  non-sampled locations?
• Filtering Combine the sample(s) to compute a
  function value

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Reconstruction

• We have two possible scenarios
• Problem How do we best filter the samples to
  reconstruct?

No aliasing
Aliasing
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Reconstruction

• Solution Extract a copy of the original signal
  from the frequency domain of the sampled signal,
  then we can reconstruct the original signal!
• This just gets rid of the high frequencies wait
  we did this before?

No aliasing
Aliasing
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Reconstruction Filters

• Reconstruction filter A method to combine one of
  more samples to compute the function at any
  location
• Most filters usually use a neighborhood of
  samples
• A weight is applied to each neighborhood sample
  and then added together ... sounds like averaging!

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Reconstruction Filters

• Goal Come up with a filter we can use in the
  spatial domain that will only pass our original
  function in the frequency domain
• If we didnt sample high enough, we are out of
  luck
• but can at least get rid of as much of the
  higher frequencies as we can

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Nearest Neigbhor Interpolation (Box Filter)

• Box filter Weight each sample with equal weight
• Nearest neighbor uses a neighborhood of one sample

spatial
Our Goal!
frequency
High frequencies are still present L
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The Box Filter is a Bad Filter To Use Why?

• Reconstruct a function in a pixel with a single
  value a pixel is not a square

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The Box Filter is a Bad Filter To Use Why?

• Nasty artifacts

Original high-resolution image
ugly horizontal banding
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Bilinear Interpolation (Tent Filter)

• Tent filter Reduce the weight of each sample as
  a linear ramp from the neighborhood center

spatial
frequency
High frequencies still present L But diminished!
Smoothing
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Filter Problems

• Visible artifacts are caused by poor
  reconstruction filters
• Two forces at play
• Pass-band attenuation cut off too narrow
• Cut out too many needed higher frequencies
  results in blurry images
• High-frequency leakage filter extends too much
• Let in unwanted higher frequencies results in
  "ringing" and can accentuate the sampling grid

frequency
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Reconstruction Filter Terminology

• Low-pass filter Reduces high frequency leakage
• For example, averaging smoothing
• High-pass filter Increases high frequency
  leakage
• For example, nearest neighbor
• What is a good filter?
• Look at its frequency domain profile and see what
  you get

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Better Filters?

• Gaussian filter Reduce the weight of each sample
  as an exponential drop off from the neighborhood
  center

spatial
frequency
What do you notice?
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Gaussian Filter

• We can change the width of the Gaussian
• A control for pass-band attenuation and
  high-frequency leakage

spatial
frequency
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A Pixel Is Not A Square

• Not on your CRT display anyway each pixel is
  reconstructed with a Gaussian filter

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Bi-Cubic Interpolation (Cubic Filter)

• Cubic filter The weights fit a cubic function

spatial
frequency
What do you make of this one?
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The Ideal Filter (The Sinc Function)

• OK is there a filter that will give us exactly
  what we want???
• Answer Yes! ?
• Why didnt I just give you that?
• Its impractical to use L

spatial
frequency
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Reconstruction A Summary

• Order reconstruction filters according to ease of
  computation
• Box
• Tent
• Gaussian
• Cubic
• Sinc
• Infinite extent means need all samples in the
  neighborhood
• Expensive to almost impossible to compute!
• Reconstruction depends upon how you performed
  sampling
• You didnt sample enough means you have
  overlapping copies means you will have
  aliasing!
• I.e., no reconstruction filter is going to come
  to your rescue

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Temporal Aliasing

• Have you ever noticed when looking at a fast
  rotating object that it reverses direction?
• An electric fan or a car wheel on the highway
• We have an under-sampling problem in time