General Functions

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General Functions

• A non-periodic function can be represented as a
  sum of sins and coss of (possibly) all
  frequencies
• F(?) is the spectrum of the function f(x)

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Fourier Transform

• F(?) is computed from f(x) by the Fourier
  Transform

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Example Box Function
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Box Function and Its Transform
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Cosine and Its Transform
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-1
If f(x) is even, so is F(?)
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Sine and Its Transform
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If f(x) is odd, so is F(?)
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Delta Function and Its Transform
Fourier transform and inverse Fourier transform
are qualitatively the same, so knowing one
direction gives you the other
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Shah Function and Its Transform
Moving the spikes closer together in the spatial
domain moves them farther apart in the frequency
domain!
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Gaussian and Its Transform
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Qualitative Properties

• The spectrum of a functions tells us the relative
  amounts of high and low frequencies
• Sharp edges give high frequencies
• Smooth variations give low frequencies
• A function is bandlimited if its spectrum has no
  frequencies above a maximum limit
• sin, cos are bandlimited
• Box, Gaussian, etc are not

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Functions to Images

• Images are 2D, discrete functions
• 2D Fourier transform uses product of sins and
  coss (things carry over naturally)
• Fourier transform of a discrete, quantized
  function will only contain discrete frequencies
  in quantized amounts
• Numerical algorithm Fast Fourier Transform (FFT)
  computes discrete Fourier transforms

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2D Discrete Fourier Transform
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Filters

• A filter is something that attenuates or enhances
  particular frequencies
• Easiest to visualize in the frequency domain,
  where filtering is defined as multiplication
• Here, F is the spectrum of the function, G is the
  spectrum of the filter, and H is the filtered
  function. Multiplication is point-wise

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Qualitative Filters
F
G
H
Low-pass
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High-pass
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Band-pass
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Low-Pass Filtered Image
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High-Pass Filtered Image
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Filtering in the Spatial Domain

• Filtering the spatial domain is achieved by
  convolution
• Qualitatively Slide the filter to each position,
  x, then sum up the function multiplied by the
  filter at that position

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Convolution Example
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Convolution Theorem

• Convolution in the spatial domain is the same as
  multiplication in the frequency domain
• Take a function, f, and compute its Fourier
  transform, F
• Take a filter, g, and compute its Fourier
  transform, G
• Compute HF?G
• Take the inverse Fourier transform of H, to get h
• Then hf?g
• Multiplication in the spatial domain is the same
  as convolution in the frequency domain

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Sampling in Spatial Domain

• Sampling in the spatial domain is like
  multiplying by a spike function

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Sampling in Frequency Domain

• Sampling in the frequency domain is like
  convolving with a spike function

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Reconstruction in Frequency Domain

• To reconstruct, we must restore the original
  spectrum
• That can be done by multiplying by a square pulse

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Reconstruction in Spatial Domain

• Multiplying by a square pulse in the frequency
  domain is the same as convolving with a sinc
  function in the spatial domain

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Aliasing Due to Under-sampling

• If the sampling rate is too low, high frequencies
  get reconstructed as lower frequencies
• High frequencies from one copy get added to low
  frequencies from another

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Aliasing Implications

• There is a minimum frequency with which functions
  must be sampled the Nyquist frequency
• Twice the maximum frequency present in the signal
• Signals that are not bandlimited cannot be
  accurately sampled and reconstructed
• Not all sampling schemes allow reconstruction
• eg Sampling with a box

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More Aliasing

• Poor reconstruction also results in aliasing
• Consider a signal reconstructed with a box filter
  in the spatial domain (which means using a sinc
  in the frequency domain)

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