lecture 13 filtering in the frequency domain dr. … digital image processing lecture 13 filtering...
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EC-433 Digital Image Processing
Lecture 13
Filtering in the Frequency Domain
Dr. Arslan Shaukat
Acknowledgement: Lecture slides material from
Dr. Rehan Hafiz, Gonzalez and Woods
Highpass Filtering
),(1),( vuHvuH LPHP
Highpass Filters
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Highpass Filters - Spatial Domain
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IHPF
BHPF
GHPF
Do = 30, 60, 160
26/05/2011 EME (NUST) EC-433 Digital Image Processing
HPF and Thresholding
Application: Finger Print Enhancement
HP Filtered image lost the gray-level Zero DC Term
Dark tones pre-dominate in HP Filtered Images (-ve & +ve values)
Binary Thresholding
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Scaling considerations for:
– Can have different scaling
– Need to normalize
Solution:
– Pre-Scale f(x,y)
– Re-Scale Laplacian Image after DFT application
The Laplacian in the Frequency Domain
),(4),( 22 vuDvuH
26/05/2011 EME (NUST) EC-433 Digital Image Processing
)],(),([),( 12 vFvHyxf
)],([),(),( 2 yxfcyxfyxg
The Laplacian in the Frequency Domain
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Implementation
gmask(x,y) = f(x,y) - fLP(x,y)
g(x,y) = f(x,y)+ k*gmask(x,y)
Unsharp Masking k =1
Highboost Filtering k>1
Unsharp Masking and Highboost Filtering
),()],(1[*1),( 1 vuFvuHkyxg LP
),(),(*1),( 1 vuFvuHkyxg HP
26/05/2011 EME (NUST) EC-433 Digital Image Processing
)],(),([),( 1 vFvHyxf LPLP
DC Term is not forced to ZERO !!
High-Frequency Emphasis Filtering
High-pass filtering emphasizes edges but fine details in
the image (i.e., low frequencies) are lost.
Add a constant to H(u,v) to preserve low frequencies.
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Combining Spatial and Frequency Domain Techniques
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Homomorphic Filtering
Homomorphic Filtering
Consider the following model of image formation:
The illumination component is characterized by slow
spatial variations
The reflectance component tends to vary abruptly,
particularly at the junctions of dissimilar objects
Associate the low frequencies of the Fourier transform of
the logarithm of an image with illumination and the high
frequencies with reflectance
So probable solution is to specify H(u,v):
– Enhance high frequencies
– Attenuate low frequencies but preserve fine detail26/05/2011 EME (NUST) EC-433 Digital Image Processing
Separating Low from High Frequencies
Here, low and high frequencies from i(x,y) and r(x,y)
have been mixed together
Difficult to operate on low/high frequencies separately
Solution?
– Attempt to separate signals combined in a nonlinear way by
making the problem become linear (Homomorphic techniques)
26/05/2011 EME (NUST) EC-433 Digital Image Processing
)],([)],([)],([ yxryxiyxf
Homomorphic Filtering
Take the log( )Idea
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Homomorphic Filtering - STEPS
(1) Take Log
(2) Apply FT:
or
(3) Apply H(u,v)
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Homomorphic Filtering (cont’d)
(4) Take Inverse FT:
or
(5) Take exp( )
or
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Homomorphic Filtering (cont’d)
How to choose H(u,v)?
– If l<1 and H>1, the filter tends to decrease the contribution
made by the low frequencies (illumination) and amplify the
contribution made by high frequencies (reflectance)
]/),([ 22
1),( DovuDc
LH evuH
26/05/2011 EME (NUST) EC-433 Digital Image Processing
Application: PET Scan
Blurry Image
Low intensity features obscured by high intensity
of “hot spots”
High Detail
Reduction of effects of dominant illumination allows dynamic
range of lower intensity to be displayed properly
Reflectance components are sharpened26/05/2011 EME (NUST) EC-433 Digital Image Processing
Selective Filtering
Bandreject and Bandpass Filters
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Rejects/Passes a predefined neighborhood of frequencies
Zero phase shift filters must be symmetric about origin.
H(u,v)=H(-u,-v)
Can be defined as product of high pass filters centered at
notch location
Let Q be the no. of notches
Notch Filter
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),(),(),(1
vuHvuHvuH kk
Q
kNR
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Notch Filters - Applications
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Separability of the 2-D DFT
2-D DFT: Separability
The 2-D DFT can be computed using 1-D transforms
– Forward DFT:
– Inverse DFT:
26/05/2011 EME (NUST) EC-433 Digital Image Processing
DFT Properties: Separability (cont’d)
Rewrite F(u,v) as follows:
Let’s set:
Then:
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DFT Properties: (cont’d)
26/05/2011 EME (NUST) EC-433 Digital Image Processing