Post on 06-Aug-2020
Page 1
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling and Reconstruction
The sampling and reconstruction processn Real world: continuous
n Digital world: discrete
Basic signal processing
n Fourier transforms
n The convolution theorem
n The sampling theoremAliasing and antialiasing
n Uniform supersampling
n Nonuniform supersampling
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Imagers = Signal Sampling
All imagers convert a continuous image to a discrete sampled image by integrating over the active “area” of a sensor.
Examples:n Retina: photoreceptorsn CCD arrayn Vidicon: phosphors
Virtual “computer graphics” cameras do not integrate,instead they simply sample ...
∫ ∫∫ ∫∫=t A
dtddAtSxPtxLRω
ωθω cos)()(),,(
Page 2
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Displays = Signal Reconstruction
All physical displays recreate a continuous image from a discrete sampled image by using a finite sized source of light for each pixel.
Examples:
n DAC’s: sample and holdn Cathode ray tube: phosphor spot and grid
DAC CRT
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling in Computer Graphics
Artifacts due to sampling - Aliasingn Jaggies
n Moire
n Flickering small objects
n Sparkling highlights
n Temporal strobing
Preventing these artifacts - Antialiasing
Page 3
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Jaggies
Retort sequence by Don Mitchell
Staircase pattern or jaggies
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Spectral Analysis / Fourier Transforms
Spectral representation treats the function as a weighted sum of sines and cosines
Each function has two representations
n Spatial (time) domain - normal representationn Frequency domain - spectral representation
The Fourier transform converts between the spatial and frequency domain
SpatialDomain
FrequencyDomain
∫
∫∞
∞−
∞
∞−
−
=
=
ωωπ
ω
ω
ω
deFxf
dxexfF
xi
xi
)(21
)(
)()(
Page 4
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Spatial and Frequency Domain
Spatial Domain Frequency Domain
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Convolution
Definition
Convolution Theorem: Multiplication in the frequency domain is equivalent to convolution in the space domain.
Symmetric Theorem: Multiplication in the space domain is equivalent to convolution in the frequency domain.
GFgf ×↔⊗
GFgf ⊗↔×
∫ ′′−′=⊗= xdxxgxfgfxh )()()(
Page 5
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling: Spatial Domain
× =
∑∞=
−∞=
−=n
n
nTxx )()III( δ Tf
Tf sss
ππω
22
1===
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling: Frequency Domain
⊗
∑∞=
−∞=
−=n
nsn )()III( ωωδω
=
Page 6
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Reconstruction: Frequency Domain
× =
>≤
=2
1
21
01
)II(xx
x
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Reconstruction: Spatial Domain
=⊗
xx
xπ
πsinsinc =
Page 7
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling and Reconstruction
×⊗
= =
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling Theorem
This result if known as the Sampling Theorem and is due to Claude Shannon who first discovered it in 1949
A signal can be reconstructed from its sampleswithout loss of information, if the original signal
has no frequencies above 1/2 the sampling
frequency
For a given bandlimited function, the rate at which it must be sampled is called the Nyquist Frequency
Page 8
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Undersampling: Aliasing
⊗ ×
= =
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Sampling a “Zone Plate”
22sin yx +y
x
Zone plate:
Sampled at 128x128Reconstructed to 512x512
Left rings: part of signalRight rings: prealiasingMiddle rings: postaliasing
Page 9
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Ideal Reconstruction
Ideally, use a perfect low-pass filter - the sinc function -to bandlimit the sampled signal and thus remove all copies of the spectra introduced by sampling
Unfortunately,
n The sinc has infinite extent and we must use simpler filters with finite extents. Physical processes in particular do not reconstruct with sincs
n The sinc may introduce ringing which are perceptually objectionable
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Mitchell Cubic Filter
otherwise
xx
CBxCBxCBxCBBxCBxCB
xh 211
0
)248()4812()306()6()26()61218()6912(
61)( 23
23
<<<
++−−+++−−
−+++−+−−=
Properties:
1)( =∑∞=
−∞=
n
n
xh
From Mitchell and NetravaliLook at other figures in that paper
)3/1,3/1(:Good
)2/1,0(:Rom-Catmull)0,1(:spline-B
Page 10
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Aliasing and Antialiasing
If the scene contains frequencies greater than theNyquist Frequency, then we have a aliasing problem
Results of aliasing:n Jaggiesn Moiren Flickering small objectsn Sparkling highlightsn Temporal strobing
Preventing aliasing or antialiasing:1. Analytically prefilter the signal2. Uniform supersampling and resample3. Nonuniform or stochastic sampling
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Antialiasing by Prefiltering
⊗ ×
= =
Page 11
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Antialiasing by Prefiltering
Ideally, low-pass with a perfect filter (a sinc function) tobandlimit the function to the Nyquist sampling rate.
Unfortunately, the sinc has infinite extent and we must use simpler filters (like a box filter, or area average).
Practically:n Constant colored polygonal fragments doable
n Complex environments not doable
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Uniform Supersampling
Increasing the sampling rate moves each copy of the spectra further apart, potentially reducing the overlap and thus aliasing
Resulting samples must be resampled (filtered) to image sampling rate
Samples Pixel
∑ ⋅=s
ss SamplewPixel
Page 12
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Point vs. Supersampled
Point 4x4 Supersampled
Checkerboard sequence by Tom Duff
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Analytic vs. Supersampled
Exact Area 4x4 Supersampled
Page 13
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Distribution of Extrafoveal Cones
Monkey eye cone distribution Fourier transform
Yellot theory
n Aliases replaced by noise
n Visual system less sensitive to high frequency noise
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Non-uniform Sampling
Intuition
Uniform sampling
n The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes
n Multiplying the signal by the sampling pattern corresponds to placing a copy of the spectrum at each spike (in freq. space)
n Aliases are coherent, and very noticable
Non-uniform sampling
n Samples at non-uniform locations have a different spectrum; a single spike plus noise
n Sampling a signal in this way converts aliases into broadband noise
n Noise is incoherent, and much less objectionable
Page 14
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Jittered Sampling
Add uniform random jitter to each sample
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Jittered vs. Uniform Supersampling
Jittered Sampling 4x4 Uniform
Page 15
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Analysis of Jitter
∑∞=
−∞=−=
n
nnxxxs )()( δ
nn jnTx +=
ωω sinc)(
2/102/11
)(
)(~
=
>≤
=
J
xx
xj
xjjn
[ ][ ] )(sinc1
1
)2
()(2
)(11
)(
2
2
2
2
ωδω
πωδωπωω
+−=
−+−= ∑−∞=
−∞=
T
Tn
JT
JT
Sn
n
Non-uniform sampling Jittered sampling
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Poisson Disk Sampling
Dart throwing algorithm
Page 16
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Filtering Nonuniform Samples
1. Reconstruction
2. Filtering further attenuates high frequency noise
3. Problematical if samples are clumpy
∑∑ ⋅
=
sss
sss
ss
yxw
yxSyxwP
),(
),(),(
Note: For jittered samples may approximate with filter coefficients at centers of each cell
CS348B Lecture 7 Pat Hanrahan, Spring 2000
Tiling