Sampling and Reconstruction The sampling and reconstruction process

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Sampling and Reconstruction

  • The sampling and reconstruction process

    • Real world: continuous
    • Digital world: discrete
  • Basic signal processing

    • Fourier transforms
    • The convolution theorem
    • The sampling theorem
  • Aliasing and antialiasing

    • Uniform supersampling
    • Nonuniform supersampling

Camera Simulation

    • Sensor response
    • Lens
    • Shutter
    • Scene radiance

Imagers = Signal Sampling

  • All imagers convert a continuous image to a discrete sampled image by integrating over the active “area” of a sensor.

  • Examples:

    • Retina: photoreceptors
    • CCD array
  • Virtual CG cameras do not integrate, they simply sample radiance along rays …

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:

    • DACs: sample and hold
    • Cathode ray tube: phosphor spot and grid

Sampling in Computer Graphics

  • Artifacts due to sampling - Aliasing

    • Jaggies
    • Moire
    • Flickering small objects
    • Sparkling highlights
    • Temporal strobing
  • Preventing these artifacts - Antialiasing


Fourier Transforms

  • Spectral representation treats the function as a weighted sum of sines and cosines

  • Each function has two representations

  • The Fourier transform converts between the spatial and frequency domain

Spatial and Frequency Domain


  • 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.

Sampling: Spatial Domain

Sampling: Frequency Domain

Reconstruction: Frequency Domain

Reconstruction: Spatial Domain

Sampling and Reconstruction

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 samples
    • without 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

Undersampling: Aliasing

Sampling a “Zone Plate”

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,

    • The sinc has infinite extent and we must use simpler filters with finite extents. Physical processes in particular do not reconstruct with sincs
    • The sinc may introduce ringing which are perceptually objectionable

Sampling a “Zone Plate”

Mitchell Cubic Filter

Antialiasing by Prefiltering


  • Antialiasing = Preventing aliasing

  • Analytically prefilter the signal

    • Solvable for points, lines and polygons
    • Not solvable in general
      • e.g. procedurally defined images
  • Uniform supersampling and resample

  • Nonuniform or stochastic sampling

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

Point vs. Supersampled

Analytic vs. Supersampled

Distribution of Extrafoveal Cones

Non-uniform Sampling

  • Intuition

  • Uniform sampling

    • The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes
    • Multiplying the signal by the sampling pattern corresponds to placing a copy of the spectrum at each spike (in freq. space)
    • Aliases are coherent, and very noticable
  • Non-uniform sampling

    • Samples at non-uniform locations have a different spectrum; a single spike plus noise
    • Sampling a signal in this way converts aliases into broadband noise
    • Noise is incoherent, and much less objectionable

Jittered Sampling

Jittered vs. Uniform Supersampling

Analysis of Jitter

Poisson Disk Sampling

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