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