Image Processing Basics

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Image Processing Basics
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What are images?

• An image is a 2-d rectilinear array of pixels

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Pixels as samples

• A pixel is a sample of a continuous function

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Images are Ubiquitous

• Input
• Optical photoreceptors
• Digital camera CCD array
• Rays in virtual camera (why?)
• Output
• TVs
• Computer monitors
• Printers

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Properties of Images

• Spatial resolution
• Width pixels/width cm and height pixels/ height
  cm
• Intensity resolution
• Intensity bits/intensity range (per channel)
• Number of channels
• RGB is 3 channels, grayscale is one channel

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Image errors

• Spatial aliasing
• Not enough spatial resolution
• Intensity quantization
• Not enough intensity resolution

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Two issues

• Sampling and reconstruction
• Creating and displaying images while reducing
  spatial aliasing errors
• Halftoning techniques
• Dealing with intensity quantization

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Sampling and reconstruction
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Aliasing

• Artifacts caused by too low sampling frequency
  (undersampling) or improper reconstruction
• Undersampling rate determined by Nyquist limit
  (Shannons sampling theorem)

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Aliasing in computer graphics

• In graphics, two major types
• Spatial aliasing
• Problems in individual images
• Temporal aliasing
• Problems in image sequences (motion)

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Spatial Aliasing

• Jaggies

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Spatial aliasing
Ref SIGGRAPH aliasing tutorial
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Spatial aliasing

• Texture disintegration

Ref SIGGRAPH aliasing tutorial
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Temporal aliasing

• Strobing
• Stagecoach wheels in movies
• Flickering
• Monitor refresh too slow
• Frame update rate too slow
• CRTs seen on other video screens

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Antialiasing

• Sample at a higher rate
• What if the signal isnt bandlimited?
• What if we cant do this, say because the
  sampling device has a fixed resolution?
• Pre-filter to form bandlimited signal
• Low pass filter
• Trades aliasing for blurring
• Non-uniform sampling
• Not always possible, done by your visual system,
  suitable for ray tracing
• Trades aliasing for noise

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Sampling Theory

• Two issues
• What sampling rate suffices to allow a given
  continuous signal to be reconstructed from a
  discrete sample without loss of information?
• What signals can be reconstructed without loss
  for a given sampling rate?

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Spectral Analysis

• Spatial (time) domain

• Frequency domain

Any (spatial, time) domain signal (function) can
be written as a sum of periodic functions
(Fourier)
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Fourier Transform
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Fourier Transform

• Fourier transform
• Inverse Fourier transform

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Sampling theorem

• A signal can be reconstructed from its samples if
  the signal contains no frequencies above ½ the
  sampling frequency.
• -Claude Shannon
• The minimum sampling rate for a bandlimited
  signal is called the Nyquist rate
• A signal is bandlimited if all frequencies above
  a given finite bound have 0 coefficients, i.e. it
  contains no frequencies above this bound.

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Filtering and convolution

• Convolution of two functions ( filtering)
• Convolution theorem
• Convolution in the frequency domain is the same
  as multiplication in the spatial (time) domain,
  and
• Convolution in the spatial (time) domain is the
  same as multiplication in the frequency domain

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Filtering, sampling and image processing

• Many image processing operations basically
  involve filtering and resampling.
• Blurring
• Edge detection
• Scaling
• Rotation
• Warping

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Resampling

• Consider reducing the image resolution

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Resampling

• The problem is to resample the image in such a
  way as to produce a new image, with a lower
  resolution, without introducing aliasing.
• Strategy-
• Low pass filter transformed image by convolution
  to form bandlimited signal
• This will blur the image, but avoid aliasing

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Ideal low pass filter

• Frequency domain
• Spatial (time) domain

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Image processing in practice

• Use finite, discrete filters instead of infinite
  continous filters
• Convolution is a summation of a finite number of
  terms rather than in integral over an infinite
  domain
• A filter can now be represented as an array of
  discrete terms (the kernel)

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Discrete Convolution
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Finite low pass filters

• Triangle filter

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Finite low pass filters

• Gaussian filter

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Edge Detection

• Convolve image with a filter that finds
  differences between neighboring pixels

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Scaling

• Resample with a gaussian or triangle filter

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Image processing

• Some other filters

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Summary

• Images are discrete objects
• Pixels are samples
• Images have limited resolution
• Sampling and reconstruction
• Reduce visual artifacts caused by aliasing
• Filter to avoid undersampling
• Blurring (and noise) are preferable to aliasing