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:

5. • A 4x4 image











































jj
jj
jj
jj
11
1111
11
1111
3366
3245
2889
8631
11
1111
11
1111
~
44 XFFX
• Compute its 2D-DFT:













3366
3245
2889
8631
X































jj
jj
jjjj
jjjj
11
1111
11
1111
5542134
6379
5542134
16192121

















jjjj
jj
jjjj
jj
811744594
1361113613
457481194
5235277
MATLAB function: fft2
lowest frequency
component
highest frequency
component

6. 
















jjjj
jj
jjjj
jj
811744594
1361113613
457481194
5235277
~
X
Real part:
















11454
611613
54114
23277
~
realX

















8749
130130
4789
5050
~
imagX













60.1306.840.685.9
32.141132.1413
4.606.860.1385.9
39.5339.577
~
magnitudeX

















628.005.137.115.1
138.10138.10
37.105.1628.015.1
19.1019.10
~
phaseX
Imaginary part:
Magnitude: Phase:

7. 














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






























jj
jj
jjjj
jj
jjjj
jj
jj
jj
11
1111
11
1111
811744594
1361113613
457481194
5235277
11
1111
11
1111
4
1~
244
**
FXF
• Compute the inverse 2D-DFT:
X













3366
3245
2889
8631































jjjj
jjjj
jj
jj
5542134
6379
5542134
16192121
11
1111
11
1111
4
1
MATLAB function: ifft2

9. +
Original High Pass Filtered

10. Original High Frequency Emphasis
Original
High Frequency
Emphasis

11. Original High pass Filter
High Frequency
Emphasis
High Frequency Emphasis
+
Histogram Equalization

12. 2D Image 2D Image - Rotated
Fourier Spectrum Fourier Spectrum