Fourier Transform - Cornell University

CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
1
Frequency Domain Operations
Spatial frequencies, Frequency domain filters
Fourier Transform, Fast Fourier Transform (FTT)
Fourier Transform
A
f(x)
0
x
f(x) = A0 + A1 sin (2ax)
d
where a = /d
A0+A1
A0
A1
A
A0-A1
A0 + A1
d/2
0
d
k
x
0 1/d
A0 + A1
A0
7/d
7/d
9/d
5/d
7/d
7/d
9/d
7/d
7/d
9/d
A1
A1
3
0
d/2
d
x
k
0 1/d
3/d
f(x) = A0 + A1 [sin (ax) + 1/3 sin (3ax) +1/5 sin (5ax)]
A0
A0
A0 - A1
5/d
f(x)=A0+A1[sin(ax)+1/3sin(3ax)]
A0
A0 - A1
3/d
0
d/2
d
x
A1
0 1/d
A1
3
3/d
A1
5
5/d
k
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
2
The next term in the Fourier Series is:
( )
[
(
or, more generally
)
(
(
)
)
(
∑
)
(
(
)]
)
n = 1, 3, 5, . . .
Bn = A1/n
ky
y
kx
1
d
x
3
5d
5
Image
7
/d
/d
/d
y
/d
kN = 0.5
cycles / sample
ky0
Fourier Transform
5d
1
/d
kx
2
/d
d
Image
x
d/4
Fourier
Transform
kN = 0.5 cycles/sample
high frequency
FOVy
kN
y
sy
kox
low
frequency
sx
FOVx
kN
kmax
Image
koy
x
Fourier Transform
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
3
kN
y
kNx
kmax < kNx ~ kNy
The sampling interval defines the Nyquist frequency -- the high frequency limit of the imaging
system:
kNx = 1/(2sx)
kNy = 1/(2sy)
The edge of the Fourier transform image represents the Nyquist frequency. The highest
frequency is along the diagonal:
kN
m ax


kN
x

2

 kN
y

2
The low frequency "resolution" limit is related to the image FOV:
kox = 1/(FOVx)
koy = 1/(FOVy)
collimated
Source image
light
(transparency)
source
FT
Lens
FT
Plane
(filter
position)
Inverse
FT
Lens
For a similar, intuitive description of the Fourier Transform, see:
http://sharp.bu.edu/~slehar/fourier/fourier.html
Final
(filtered)
image
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
4
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
Optical FT: random square/circular apertures
5
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
Optical FT: arrays of square/circular apertures
6
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
Digital FFT: wave spectra
Optical FT: airphoto subsets
FFT
7
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
8
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
Digital FFT: Ideal filter and ringing
FFT
9
CEE 6150: Digital Image Processing
W. Philpot, Cornell University
FFT
10

Download Report