CSE 681 Supersampling

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CSE 681Supersampling
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Aliasing Recap
Fourier Transform
samplinggrid
(multiplication)
(convolution)
Fourier Transform
sampledsignal
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Aliasing Recap

• Aliasing due to
• Sampling
• The sampling process introduces frequencies out
  to infinity
• We have lost the function f (x) and now only have
  discrete samples
• Overlap in the frequency domain the higher
  frequencies for the copy at 1/T intermix with the
  low frequencies centered at the origin
• Reconstruction
• Convolution in the spatial domain is the same as
  multiplication in the frequency domain

No aliasing
Aliasing
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Anti-aliasing

• Sampling Increase sampling frequency
• Improves results at a computational cost
• Doesnt solve the problem in most cases
• Reconstruction Better filter with larger
  neighborhoods
• Improves results at a computational cost
• Bad filters are most cost effective but just
  blur bad results

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Anti-aliasing Supersampling

• OK, we decide to pay the computational cost
• Just add more samples initiate more rays per
  pixel
• Reconstruction becomes more expensive also
  which filter?

Idea
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Supersample and Reconstruct

• Subdivide the pixel into a grid of subpixels and
  define new subpixel centers
• Weight each sample according to your
  reconstruction filter
• The box filter will weight each sample equally
  according to the size of the neighborhood
• What about a tent filter or truncated gaussian?

filter weight
sample color
pixel
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Anti-aliasing Supersampling
Jaggies
Anti-aliasing
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Supersampling Schemes

• There are many ways to subdivide a pixel

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Full Scene Anti-Aliasing(FSAA)Hardware Supported
AA

• Borrowed from Answers.com

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Quincunx Multisampling

• Wikipedia definition
• A quincunx is the arrangement of five units in
  the pattern corresponding to the five-spot on
  dice, playing cards, or dominoes. The quincunx is
  named after the Roman coin of the same name.
• Uses
• A quincunx was the standard tactical formation
  for elements of a Roman legion.
• A quincunx is a standard pattern for planting an
  orchard, especially in France.
• Quincunxes are used in modern computer graphics
  as a sampling pattern for anti-aliasing.
• The spots on 5th side of a dice form a quincunx.
• In astrology (and less commonly in astronomy), a
  quincunx is an angle of five-twelfths of a
  circle, or 150, between two objects (the Sun,
  Moon or planets).
• The points on each face of a unit cell of a
  face-centred cubic lattice form a quincunx.
• A quincuncial map is a conformal map projection
  that maps the poles of the sphere to the centre
  and four corners of a square, thus forming a
  quincunx.

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Quincunx Hardware Supported AA

• Define only two samples/pixel
• reconstruct with five samples

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Quincunx Hardware Supported AA

• Quality comparable to 4xFSAA (full scene) at half
  the cost!

4X FSAA
4X Quincunx
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Adaptive Supersampling

• Observation Aliasing is worse where the function
  has higher frequencies
• Idea Target our supersampling. Only do it when
  we need to!

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Adaptive Supersampling

• A Recursive Algorithm
• Sample Phase (Top-Down)
• Trace rays at the pixel corners (initial area)
• Trace ray at pixel center
• If center is different from corners
• Subdivide into 4 sub-pixels
• Recurse on the sub-pixels
• Reconstruction Phase (Bottom-Up)
• Pixel color is ¼ of each quadrant
• Each subpixel (quadrant) is average of higher
  level center point and opposite corner point
• Collect colors from the bottom up

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Adaptive Supersampling

• Only do extra work when we think we need it
• For example, additional rays are traced along
  edges

Needs more samples
Andrew Zaferakis - http//www.cs.unc.edu/andrewz/
comp238/hw2/index.html
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Stochastic Sampling A Monte-Carlo Approach

• Monte-Carlo method any method that uses random
  numbers
• Random number generators on the computer must use
  an algorithm and thus can only give us
  pseudo-random numbers
• Unix has generators such as rand, random,
  drand48
• Go look them up using man pages
• A seed value will determine a specific sequence
  of random values

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A Modelling ExampleFractal Mountains

• Fractals represent controlled randomness to
  produce natural looking behavior Random height
  field

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A Monte-Carlo Method to Approximate p

• Take a unit circle (centered at the origin) and a
  bounding box (square with side length 2)
• Acircle p
• Asquare 4
• Take a point (x, y) inside the square
• The probability ((x, y) -1 x 1 and -1 y
  1 is inside the circle) chance the point
  will be inside the circle

outside
inside
For example, what is the chance of a heads if you
flip a coin?
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A Monte-Carlo Method to Approximate p

• OK, lets flip a coin
• Consider the following procedure
• Initialize Ncircle 0 and Nsquare 0
• Choose a random point (x, y)
• If (x, y) is inside the circle then increment
  Ncircle
• Increment Nsquare
• Repeat

Run this N times
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A Monte-Carlo Method to Approximate p

• The probability a random point is inside the unit
  circle
• Choose N random points and let M Ncircle
• Approximate p

As N Þ a better approximation
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Estimate p Using Monte-Carlo

• Results
• N 10,000 p 3.1388
• N 100,000 p 3.1452
• N 1,000,000 p 3.14164
• N 10,000,000 p 3.1422784

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Choosing Random Numbers

• Does it matter how the N random numbers are
  chosen?
• Choose random numbers evenly across the space

p 4
p 4
p 0
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Estimate p Using Monte-Carlo

• double x, y, pi
• const int m_nMaxSamples 100000000
• int count 0
• for (int k 0 k lt m_nMaxSamples k)
• x 2.0 drand48() 1.0 // Map to the range
  -1,1
• y 2.0 drand48() 1.0
• if (xx yy lt 1.0) count
• pi 4.0 (double) count / (double)m_nMaxSamples

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Stochastic Sampling

• OK, back to our story supersampling
• Observation Throwing more and more samples at
  problem areas (very high frequencies) seems like
  a losing battle
• We are just blurring our results anyway when we
  reconstruct
• Idea Reduce aliasing with noise
• Our visual system likes to enhance regular
  patterns
• Our eyes are less sensitive to non-ordered
  sampling patterns
• If it is going to look noisy anyway, lets just
  make it better looking noise

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Stochastic Sampling

• We cannot get rid of the noisy look so convert
  it to noise with the correct average intensity

Basic checkerboard
Checkerboard with noise
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Stochastic Sampling

• Simple as choosing sample locations in a pixel
  randomly . . . right?
• What can go wrong?
• Clustering

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Spatial Jittering

• Jittering Apply a noisy offset to pre-determined
  sample locations

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Jittering A Regular Grid

• Use pixel subdivision followed by jittering

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Reconstruction

• How do we reconstruct the function with
  irregularly spaced samples?
• Box filter (easy)
• Average all the contributions with equal weight
  within the pixel region
• Non-uniform weighted filters (more work)
• Weight each sample according to its distance from
  the area center
• Sum the weighted samples and normalize by
  dividing by the number of filter values

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Effects of Spatial Jittering

• High frequencies are converted to noise instead
  of incorrect patterns
• High frequencies are attenuated
• The intensity of the resulting noise is
  proportional to the lost energy

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Stochastic Sampling Hardware Supported

• ATI - SMOOTHVISIONä
• Jittered sampling pattern
• Each pixel has (2x, 4x, 8x) pre-programmed
  jittered sampling locations

Possible pixel sample locations for
the SMOOTHVISION 4x setting
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Quick Comparison
2x SmoothVision
Quincunx
Borrowed from http//www.anandtech.com
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Poisson DistributionNon-Uniform Stochastic
Sampling

• A stochastic sampling such that there is a
  minimum distance between neighboring samples
• I.e., any two samples are separated by at least a
  distance e
• Avoids clustering
• Similar to the distribution of vision receptors
  in your eye
• This is the ideal stochastic sampling to use, but
  not easy to define