Post on 05-Jan-2016
transcript
Discrete Fourier Transform in 2D – Chapter 14
Discrete Fourier Transform – 1D• Forward
• Inverse
MmugM
mG
M
mui
M
muug
MmG
M
uM
mui
M
u
e
0)(1
)(
2sin2cos)(1
)(
1
0
2
1
0
MumGM
ug
M
mui
M
mumG
Mug
M
mM
mui
M
m
e
0)(1
)(
2sin2cos)(1
)(
1
0
2
1
0
M is the length (number of discrete samples)
Discrete Fourier Transform – 2D• After a bit of algebraic manipulation we find that the 2D
Fourier Transform is nothing more than two 1D transforms
• Do a 1D DFT over the rows of the image• Then do a 1D DFT over the columns of the row-wise
DFT• This is for an MxN (columns by rows)
1
0
21
0
2),(
11),(
N
vN
nviM
uM
mui
eevugMN
nmG
1D DFT over row g(*,v)
What’s it all mean?
• Whereas for the 1D DFT we were adding together 1D sinusoidal waves…– For the 2D DFT we are adding together 2D
sinusoidal surfaces
• Whereas for the 1D DFT we considered parameters of amplitude, frequency, and phase– For the 2D DFT we consider parameters of
amplitude, frequency, phase, and orientation (angle)
Visualization
• A pixel in DFT space represents an orientation and frequency of the sinusoidal surface
• The corners each represent low frequency components which is inconvenient
Quadrant swapping
• Quadrant swapping brings all low frequency data to the center
A D
B C
C B
D A
• A pixel in DFT space represents an orientation and frequency of the sinusoidal surface
Visualization
Visualization
• The image is really a depiction of the frequency power spectrum and as such should be thought of as a surface
• Low frequencies are at the center, high frequencies are at the boundaries
Visualization
• Image coordinates represent the effective frequency…
• …and the orientation
Nn
Mm
nmf
22
1),(
is the sampling interval
nMmNnm ,),( tan1
Something interesting
• If the DFT space is square then rotation in the spatial domain is rotation in the frequency domain
Artifacts
• Since spatial signal is assumed to be periodic, drastic differences (large gradients) at the opposing edges cause a strong vertical line in the DFT
No border differences