fourier transformation
TRANSCRIPT
BackgroundThe 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.
2D DFT and Inverse DFT (IDFT)
NjN 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:
Real Part, Imaginary Part, Magnitude, Phase, Spectrum
Real part:
Imaginary part:
Magnitude-phase
representation:Magnitude(spectrum
):Phase
(spectrum):
PowerSpectrum:
Computation of 2D-DFT• To compute the 1D-DFT of a 1D signal x (as a
vector):
NNXFFX ~
*2
~1NNN
FXFX *
xFx N~
xFx * ~1NN
To compute the inverse 1D-DFT:
• To compute the 2D-DFT of an image X (as a matrix):
To compute the inverse 2D-DFT:
Computation of 2D-DFT: Example
• 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
Computation of 2D-DFT: Example
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:
Computation of 2D-DFT: Example
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
High Frequency Emphasis
+
Original High Pass Filtered
High Frequency EmphasisOriginal High Frequency Emphasis
OriginalHigh Frequency Emphasis
Original High pass Filter
High Frequency Emphasis
High Frequency Emphasis +
Histogram Equalization
High Frequency Emphasis
2D Image 2D Image - Rotated
Fourier Spectrum Fourier Spectrum
Rotation