The document discusses Fourier analysis and its applications. It introduces:
1) Fourier series and transforms, which express any function as a sum of sines and cosines, allowing the original function to be reconstructed without loss of information.
2) The discrete Fourier transform (DFT), which is used to process digital images and signals, representing them in the frequency domain.
3) An example calculating the 2D DFT of a sample image and recovering the original from the inverse DFT.
4) Common representations of DFT outputs and filters that emphasize different frequencies.
1. The Analytic Theory of Heat, 1822, Jean Baptiste
Joseph Fourier
Any function that periodically repeats itself can be
expressed as the sum of sines and/or cosines of
different frequencies, each multiplied by a different
coefficient (Fourier Series)
Even non periodic functions can be expressed as the
integral of sines and/or cosines multiplied by a
weighting function (Fourier Transform)
The important characteristic that a function, expressed
in either a Fourier series or transform, can be
reconstructed (recovered) completely via an inverse
process, with no loss of information.
2. Nj
N eW /2
M, N: image size
x, y: image pixel position
u, v: spatial frequency
f(x, y) F(u, v)
often used
short notation:
3. Real Part, Imaginary Part,
Magnitude, Phase, Spectrum
Real part:
Imaginary part:
Magnitude-phase
representation:
Magnitude
(spectrum):
Phase
(spectrum):
Power
Spectrum:
4. • To compute the 1D-DFT of a 1D signal x (as a vector):
NN XFFX
~
*
2
~1
NN
N
FXFX *
xFx N~
xFx * ~1
N
N
To compute the inverse 1D-DFT:
• To compute the 2D-DFT of an image X (as a matrix):
To compute the inverse 2D-DFT:
