What is lost when lowering resolution?

Fourier's crazy idea:

Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies

Spatial versus frequency domain

• The image plane itself
• Direct manipulation of pixels
• Changes in pixel position correspond to changes in the scene

• Fourier transform of an image
• Directly related to rate of changes in the image –
• Changes in pixel position correspond to changes in the frequency

Frequency domain overview

• High frequencies correspond to rapidly changing intensities across pixel
• Low frequency components correspond to large-scale image structures
• Frequency domain image processing via the Fourier transform

Definition of the Fourier transform (1D)

• Forward Fourier transform

$F(u) = \int_{-\infty}^{\infty} f(x)e^{-i2\pi ux}dx$

• Inverse transform

$f(x) = \int_{-\infty}^{\infty} F(u)e^{i2\pi ux}du$

• Note:
  • $e^{iwx} = cos(\omega x) + i sin(\omega x)$
  • when $\omega = 2 \pi u$ is radial frequency
  • u is spatial frequency
  • $i = \sqrt{-1} $

Properties of the Fourier transform

Definition of the Fourier transform

• forward Fourier transform
  • $ F(u, v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} f(x,y) e^{-2 \pi (ux + vy)} dx dy $
• Inverse Frourier transform
  • f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} F(u, v) e ^{i2\pi (ux + vy) dudv $
• uses complex valued sinusoids

Discrete Fourier transform (DFT)

• Forward discrete Fourier transform
  • $ F(u,v) = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} f(x,y) e^{-i2\pi (\frac{ux}{M} + \frac{vy}{N})} $
• Inverse discrete Fourier transform
  • $ f(x,y) = \frac{1}{MN} \sum_{u=0}^{M-1} \sum_{v=0}^{N-1} F(u,v) e^ {i2\pi ( \frac{ux}{M} + \frac{vy} {N} )}$

Frequency domain filtering

• Each $𝐹(𝑢, 𝑣)$ depends on all $f(x,y) $
• •Frequencies in the Fourier domain correspond to patterns in the image •
• The value of $𝐹(0,0)$ is the average intensity over all pixels of the image •
• Low frequencies correspond to slowly varying intensities across pixels •
• High frequencies correspond to rapid intensity changes across pixels •
• Noise typically corresponds to fluctuations in the highest frequencies