Computer Graphics Sampling

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Computer Graphics- Sampling -

• Marcus Magnor

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Overview

• Today
• Signal Processing
• Sampling
• Next lecture
• Anti-Aliasing

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Motivation
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The Digital Dilemma

• Nature continuous signal (2D/3D/4D with time)
• Defined at all points
• Acquisition sampling
• Rays, pixel/texel, spectral values, frames, ...
• Representation discrete data
• Discrete points, discretized values
• Reconstruction interpolation
• Mimic continuous signal
• Impression natural
• Hopefully similar to the original signal, no
  artefacts

, not
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Sensors

• Sampling of signals
• Conversion of a continuous signal to discrete
  samples by integrating over the sensor field
• Required by physical processes
• Examples
• Photo receptors in the retina
• CCD cells
• Virtual cameras in computer graphics
• Inegration usually avoided
• too expensive
• Ray tracing mathematically ideal point samples
• Origin of alias !

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Aliasing

• Ray tracing
• Textured plane
• Checkerboard period becomes smaller than two
  pixels
• Nyquist limit
• One ray for each pixel (say, at pixel center)
• Hits textured plane at only one point, black or
  white by chance
• (correct integrate over pixel pre-image texture
  mapping lecture)

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Spatial Frequency

• Frequency period length of some structure in an
  image
• Unit 1/pixel
• Range -0.50.5 (-??)
• Lowest frequency
• Image average
• Highest frequency Nyquist
• In nature light wavelength
• In graphics image resolution

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Nyquist Frequency

• Highest (spatial) frequency that can be
  represented
• Determined by image resolution (pixel size)

Spatial frequency lt Nyquist
Spatial frequency Nyquist 2 samples / period
Spatial frequency gt Nyquist
Spatial frequency gtgt Nyquist
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Fourier Transformation

• Any continuous function f(x) can be expressed as
  an integral over sine and cosine waves
• f(x) ? a(?) sin(? x) b(?) cos(? x) d?
• ? c(?) exp(-i ? x) d?
• a(?) ? sin(? x) f(x) dx
• b(?) ? cos(? x) f(x) dx
• c(?) ? exp(i ? x) f(x) dx
• Sine cosine orthonormal basis functions
• a(?), b(?) weighting functions
• c(?) (complex-valued) Fourier transform

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Fourier Transformation

• Any periodic, continuous function can be
  expressed as the sum of an (infinite) number of
  sine or cosine waves
• f(x)?k ak sin(2?kx) bk cos(2?kx)
• k frequency band
• k0 mean value
• k1 function period, lowest possible frequency
• k1.5 ? not possible, periodic function f(x)
  f(x1)
• kmax ? band limit, no higher frequency present
  in signal
• ak,bk (real-valued) Fourier coefficients
• Even function f(x)f(-x)
• ak 0
• Uneven function f(x) -f(-x)
• bk 0

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Fourier Synthesis Example

• Periodic, uneven function square wave
• f(x) 0.5 ? 0 lt (x mod 2?) lt ?
  -0.5 ? ? lt (x mod 2?) lt 2?
• ak ? sin(kx) f(x) dx f(x)?k
  ak sin(kx)
• a0 0
• a1 1
• a2 0
• a3 1/3
• a4 0
• a5 1/5
• a6 0
• a7 1/7
• a8 0
• a9 1/9

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Spectral Analysis

• Fourier coefficients
• ak 1/2? ? sin(2?kx) f(x) dx , bk 1/2? ?
  cos(2?kx) f(x) dx
• Periodic function f(x)
• integral over function period length (normalized
  to 1)
• Even function ak 0
• Uneven function bk 0
• Decomposition of signal into different frequency
  bands
• Representation of a function as weighted sum of
  sine and cosine functions
• Equivalent representations of a periodic function
• Spatial/temporal domain f(x)
• Frequency domain F(k) ak , bk

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Discrete Fourier Transform

• Equally-spaced function samples
• Function values known only at discrete points
• Physical measurements
• Pixel positions in an image !
• Fourier Analysis
• ak 1/ N ?i sin(2? k i / N) fi , bk 1/ N
  ?i cos(2? k i / N) fi
• Sum over all measurement points N
• k0,1,2, , ? Highest possible frequency ?
• Nyquist frequency
• Sampling rate Ni
• 2 samples / period ? 0.5 cycles per pixel
• k ? N / 2
• Above Nyquist repetition, periodic fourier
  transform function

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Spatial vs. Frequency Domain

• Examples (pixel vs
• cycles per pixel)
• Sine wave with positive offset
• 
  
• Square wave
  
• Scanline of animage

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2D Fourier Transform

• 2 separate 1D Fourier transformations along x-
  and y-direction
• Discontinuities orthogonal direction in Fourier
  domain !

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2D Fourier Transforms
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Spatial vs. Frequency Domain

• Important basis
  functions
• Box ?? sinc
• Wide box ? small sinc
• Negative values
• Infinite support
• Triangle ?? sinc2
• Gauss ?? Gauss

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Spatial vs. Frequency Domain

• Transform behavior
• Example box function
• Fourier transform sinc
• Wide box narrow sinc
• Narrow box wide sinc

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Convolution

• Two functions f, g
• Shift one function against the other by x
• Multiply function values
• Integrate
• overlapping region
• Numerical convolutionExpensive operation
• For each x integrate over non-zero domain

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Convolution and Filtering

• Technical Realisation
• In image domain
• Pixel mask with weights
• OpenGL Convolution
• Problems (e.g. sinc)
• Large filter support
• Large mask
• A lot of computation
• Negative weights
• Negative light?

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Convolution Theorem

• convolution in image domain multiplication in
  Fourier domain
• convolution in Fourier domain multiplication in
  image domain
• Multiplication much cheaper than convolution !

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Filtering

• Low-pass filtering
• Convolution with sinc inspatial domain, or
• multiplication with box in frequency domain
• High-pass filtering
• Only high frequencies
• Band-pass filtering
• Only intermediate
  frequencies

Low-pass filtering in frequency domain
multiplication with box
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Low-Pass Filtering

• Blurring

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High-Pass Filtering

• Enhances discontinuities in image
• Useful for edge detection

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Sampling

• Constant ?-Function
• flash
• Comb/Shah
  function

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Sampling

• Constant ?-Function
• Duality
• And vice versa
• Comb function
• Duality The dual of a comb function is again a
  comb function
• Inverse wave length, amplitude scales with
  inverse wave length

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Sampling

• Continuous function
• Band-limited Fourier transform
• Sampled at discrete points
• Multiplication with Comb function in space domain
• Corresponds to convolution in Fourier domain
• Frequency bands overlap ?
• No good
• Yes bad, aliasing

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Reconstruction

• Only original frequency band desired
• Filtering
• In Fourier domain multiplication with windowing
  function around origin
• In Space domain convolution with Fourier
  transform of windowing function
• Optimal filtering function
• Hat function in Fourier domain
• Corresponds to sinc in space domain
• Unlimited region of support
• In space domain only filter approximations
  possible

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Wrap-Up

• Fourier transformation
• Equivalent representation of transformed signal
• Spectral analysis shows signals frequency
  components
• Convolution
• Filtering
• Sampling
• Multiplication with comb function
• Only at discrete points no integration over
  signal
• Frequency spectrum replicated
• Replication distance sampling rate