3. The Frequency Domain
• Any wave shape can be approximated by a
sum of periodic (such as sine and cosine)
functions.
• a--amplitude of waveform
• f-- frequency (number of times the wave
repeats itself in a given length)
• p--phase (position that the wave starts)
• Usually phase is ignored in image processing
240-373: Chapter
3
6. The Hartley Transform
• Discrete Hartley Transform (DHT)
– The M x N image is converted into a second
image (also M x N)
– M and N should be power of 2 (e.g. .., 128, 256,
512, etc.)
– The basic transform depends on calculating the
following for each pixel in the new M x N array
1
H (u , v) =
MN
M −1 N −1
∑∑
x =0 y =0
ux vy
ux vy
f ( x, y ) ⋅ cos(2π ) − + sin( 2π ) −
M N
M N
240-373: Chapter
6
7. The Hartley Transform
where f(x,y) is the intensity of the pixel at position
(x,y)
H(u,v) is the value of element in frequency domain
– The results are periodic
– The cosine+sine (CAS) term is call “the kernel of
the transformation” (or ”basis function”)
240-373: Chapter
7
8. The Hartley Transform
• Fast Hartley Transform (FHT)
– M and N must be power of 2
– Much faster than DHT
– Equation:
H (u, v) = { T (u , v) + T ( M − u, v) + T (u , N − v) − T ( M − u , N − v)} / 2
240-373: Chapter
8
9. The Fourier Transform
• The Fourier transform
– Each element has real and imaginary values
– Formula:
F (u , v) = ∫∫ f ( x, y )e −2iπ ( ux +vy ) dxdy
– f(x,y) is point (x,y) in the original image and F(u,v)
is the point (u,v) in the frequency image
240-373: Chapter
9
10. The Fourier Transform
• Discrete Fourier Transform (DFT)
ux vy
− 2 iπ +
1 M −1 N −1
F (u, v) =
f ( x, y )e M N
∑∑
MN x =0 y =0
– Imaginary part
1 M −1 N −1
ux vy
Fi (u, v) = −
f ( x, y ) sin 2π +
∑∑
MN x =0 y =0
M N
– Real part
1
Fr (u , v) =
MN
M −1 N −1
∑∑
x =0 y =0
ux vy
f ( x, y ) cos 2π +
M N
– The actual complex result is Fi(u,v) + Fr(u,v)
240-373: Chapter
10
11. Fourier Power Spectrum and Inverse
Fourier Transform
• Fourier power spectrum
F (u , v) = Fr (u , v) 2 + Fi (u , v) 2
• Inverse Fourier Transform
1
f ( x, y ) =
MN
240-373: Chapter
M −1 N −1
∑∑ F (u, v)e
ux vy
2 iπ +
M N
x =0 y =0
11
12. Fourier Power Spectrum and Inverse
Fourier Transform
• Fast Fourier Transform (FFT)
– Much faster than DFT
– M and N must be power of 2
– Computation is reduced from M2N2 to
MN log2 M . log2 N (~1/1000 times)
240-373: Chapter
12
13. Fourier Power Spectrum and Inverse
Fourier Transform
• Optical transformation
– A common approach to view image in frequency
domain
A
D
B
C
Original image
240-373: Chapter
C
B
D
A
Transformed image
13
14. Power and Autocorrelation Functions
• Power function:
[
1
P (u , v) = F (u , v) = H (u , v) 2 + H (−u ,−v) 2
2
• Autocorrelation function
F (u , v)
2
– Inverse Fourier transform of
or
1
[ H (u, v) 2 + H (−u,−v) 2 ]
– Hartley transform of 2
240-373: Chapter
14
]
17. Applications of Frequency Domain
Processing
• Convolution in the frequency domain
240-373: Chapter
17
18. Applications of Frequency Domain
Processing
– useful when the image is larger than 1024x1024
and the template size is greater than 16x16
– Template and image must be the same size
1
1 1 1
⇒
1 1 0
0
240-373: Chapter
1
1
0
0
0
0
0
0
0
0
0
0
18
19. – Use FHT or FFT instead of DHT or DFT
– Number of points should be kept small
– The transform is periodic
• zeros must be padded to the image and the template
• minimum image size must be (N+n-1) x (M+m-1)
240-373: Chapter
19
22. Convolution using the Fourier transform
Technique 1: Convolution using the Fourier
transform
USE: To perform a convolution
OPERATION:
– zero-padding both the image (MxN) and the
template (m x n) to the size (N+n-1) x (M+m-1)
– Applying FFT to the modified image and template
– Multiplying element by element of the transformed
image against the transformed template
240-373: Chapter
22
23. Convolution using the Fourier transform
OPERATION: (cont’d)
– Multiplication is done as follows:
F(image)
(r1,i1)
F(template)
F(result)
(r2, i2)
(r1r2 - i1i2, r1i2+r2i1)
i.e. 4 real multiplications and 2 additions
– Performing Inverse Fourier transform
240-373: Chapter
23
24. Hartley convolution
Technique 2: Hartley convolution
USE: To perform a convolution
OPERATION:
– zero-padding both the image (MxN) and the
template (m x n) to the size (N+n-1) x (M+m-1)
image
0 1
2
template
0
1 2 0 0
4 5 6 0
8 9 10 0
0 0 0 0
7 3 0 0
0 0 0 0
0 0 0 0
240-373: Chapter
24
26. Hartley convolution
– Multiplying them by evaluating:
I (u , v)T (u, v) + I (u , v)T ( N − u , M − v)
NM
R (u , v) =
+ I ( N − u , M − v)T (u , v)
2
− I ( N − u , M − v)T ( N − u, M − v)
240-373: Chapter
26
29. Deconvolution
• Convolution
R = I * T
• Deconvolution I = R *-1 T
• Deconvolution of R by T = convolution of R
• by some ‘inverse’ of the template T (T’)
240-373: Chapter
29
