Fourier transformation
Transcript of Fourier transformation
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