Fourier Transform and Image
Enhancement in the
Frequency Domain
Gilad Lerman
Math 5467
(stealing slides from
Gonzalez&Woods, and Efros)
DFT of images
Camera man image
Magnitude
Phase of
of Centered
Centered DFT
DFT
How to interpret?
DFT of a simple signal
Discrete box signal
Magnitude of centered DFT
Phase of centered DFT
Recall the Continuous Case…
DFT of Simple images
More DFT of simple images
Phases of DFTs of Simple Images
Another DFT of a Simple Image
White-gray image
Magnitude of Centered DFT
Back to Camera Man
Camera man image
Magnitude of Centered DFT
|DFT| of Another Image
DFT of Natural Images
• The magnitude spectra of all natural
images quite similar
(Heavy on low-frequencies, falling off in high
frequences, sometimes lines of discontinuity)
• Most information in the image is carried in
the phase, not the amplitude
• We demonstrate this idea next…
This is the
magnitude
transform
of the
cheetah pic
This is the
phase
transform
of the
cheetah pic
This is the
magnitude
transform
of the zebra
pic
This is the
phase
transform
of the zebra
pic
Reconstruction
with zebra
phase, cheetah
magnitude
Reconstruction
with cheetah
phase, zebra
magnitude
Similar example from Textbook
Image Enhancement in the
Frequency Domain
Input image: x(k1 , k2 ), k1=0,…,M-1, k2=0,…,N-1
1. Compute DFT xˆ (n1, n2 ), n1=0,…,M-1, n2=0,…,N-1
2. Generate a real symmetric filter function hˆ(n1, n2 ),
centered at the origin and form the product
yˆ (n1, n2 ) = xˆ(n1, n2 ) ×hˆ(n1, n2 )
3. Compute IDFT of yˆ (n1 , n2 ) (take real part due to
possible numerical errors)
Image Enhancement 2
For visualization purposes you may have
k1 + k2
×x( k1 , k2 ).
1. For input: x (k1 , k2 ) := (- 1)
2. Center the real symmetric filter function hˆ(n1, n2 ),
at (M/2,N/2) and then form the product:
yˆ (n1, n2 ) = xˆ(n1, n2 ) ×hˆ(n1, n2 ).
k1 + k2
×y ( k1 , k2 ).
3. For output: y ( k1 , k2 ) := (- 1)
Filters ĥ (or H)
Ideal Lowpass Filter
Butterworth Lowpass Filter
Gaussian Lowpass Filter in
Spatial Domain
Ideal Lowpass Filter in
Spatial Domain
Butterworth Lowpass Filter
in Spatial Domain
Example: ILPF, BLPF & GLPF
Highpass Filters
hˆHP (n1, n2 ) = 1- hˆLP (n1, n2 )
Differentiation in Spatial and
Frequency Domains
Exercise: compute the filter in frequency domain
Laplacian in Frequency
Domains
Continuous Setting
In Frequency Domain
¶2
¶2
Ñ =
+
2
¶ x1
¶ x2 2
2
- 4p 2 (x12 + x22 )
For discrete Laplacian: exercise
Unsharp and Highboost
• Recall: gunsharp = f - f
• That is, g unsharp = f - f lowpass = f highpass
• Highboost: g highboost = f + k ×g unsharp
Bandpass & Bandreject Filters
Band-pass Example
Image Enhancement 3
• We correct the convolution by padding with zeros
Sampling and Aliasing
(clarifying figures in the textbook)