BAE 790I / BMME 231 Fundamentals of Image Processing Class 5

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BAE 790I / BMME 231Fundamentals of Image
ProcessingClass 5

• Fourier space display
• Matrix representations for linear systems
• Sampling
• Point sampling
• Non-point apertures

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Matrix Representation
A linear system
Matrix with coefficients mapping each (x,y)
to each (x,y)
Pixel values in some order (lexical)
Pixel values in some order (lexical)

Vector g
Matrix H
Vector f
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Matrix Representation
A linear system
In matrix form
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Matrix Representation
What does H look like?
What if the system is LSI?
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Matrix Representation
The Fourier Transform as a matrix
Vector with each element Representing a basis
coefficient
Matrix with each row representing a discrete
Fourier basis function
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Sampling

• Conversion of a continuous image to a discrete
  image
• Point sampling One pixel has the intensity of a
  single point in the continuous image
• Non-point sampling One pixel has the intensity
  integrated over some region (the aperture) of the
  continuous image

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Non-point Apertures
f(x,y) Original continuous image
fap(x,y) Virtual continuous image
Convolve with aperture function
Point Sample
f(m,n) Discrete image
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Square Aperture
Integrate the continuous input image over this
region
L
L
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Fourier Basis Images
Aliasing

fx 11, fy 42
fx 245, fy 214
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Jaszczak phantom sampled with square aperture at
256, 128, 64, and 32 scale
1x1
2x2

The larger aperture decreases visible
detail, limiting resolution
4x4
8x8
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Gaussian blob sampled with square aperture at
256, 128, 64, and 32 scale
1x1
2x2

As the aperture size approaches the size of
the object to be viewed, detail is lost.
4x4
8x8
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Fourier transform magnitudes for Gaussian at 256,
128, 64, and 32 scale

Note the increasing persistence of
sinc function with larger apertures.