High-Pass and Band-Pass Filtering
High-Pass and Band-Pass Filtering
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
High-Pass and Band-Pass Filtering
Introduction
Sharpening
Blurring is low-pass filtering, so deblurring is high-pass filtering.
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Explicit high-pass filtering
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Unsharp Masking
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Deconvolution
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Edge Detection
Tradeoff:
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Reduces blur, but
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Increases noise
High-Pass and Band-Pass Filtering
Introduction
High-Pass Filtering
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1 if u > uc
0 otherwise
Ideal:
H(u) =
Gaussian:
H(u) = 1 − e− 2 u
Butterworth:
H(u) =
1
2 /u 2
c
1
n
1+(uc 2 /u 2 )
High-Pass and Band-Pass Filtering
Introduction
High-Pass Filters
Ideal
Butterworth
Gaussian
High-Pass and Band-Pass Filtering
Ideal Filters
Ideal Filters and Ringing
High-Pass and Band-Pass Filtering
Ideal Filters
Ideal Filters and Ringing (cont’d)
High-Pass and Band-Pass Filtering
Ideal Filters
Ideal Filters and Ringing
High-Pass and Band-Pass Filtering
Gaussian Filters
Gaussian High-Pass Filters
1
H(u) = 1 − e− 2 u
2 /u 2
c
1
0.8
0.6
0.4
0.2
20
40
60
80
100
120
Sound familiar?
This is what we did with unsharp masking to get the edges!
High-Pass and Band-Pass Filtering
Gaussian Filters
Gaussian High-Pass Filters
High-Pass and Band-Pass Filtering
Gaussian Filters
High-Boost Filtering
H(u) = 1 + α HP(u)
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HP(u) is a high-pass filter
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α controls how much to boost the higher frequencies
2
1.75
1.5
1.25
1
0.75
0.5
0.25
20
40
60
80
100
120
This is why unsharp masking in the spatial domain works!
High-Pass and Band-Pass Filtering
Butterworth Filters
Butterworth High-Pass Filter
Controllable sharpness of the frequency-domain cutoff uc :
H(u) =
1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
20
40
60
80
100
n
1 + (uc 2 /u 2 )
120
20
40
60
80
100
120
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The “cutoff” frequency uc controls where the cutoff occurs.
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The parameter n controls the sharpness of the cutoff.
High-Pass and Band-Pass Filtering
Butterworth Filters
Butterworth High-Pass Filters
High-Pass and Band-Pass Filtering
Band-Pass Filters
Band-Pass Filtering
Tradeoff: Blurring vs. Noise
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Low-pass reduces noise but accentuates blurring
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High-pass reduces blurring but accentuates noise
A compromise:
Band-boost filtering boosts certain midrange freqencies and
partially corrects for blurring, but does not boost the very high
(most noise corrupted) frequencies.
High-­‐Pass and Band-­‐Pass Filtering Band-­‐Pass Filtering Band-Pass: Frequency Domain
(http://paulbourke.net/miscellaneous/imagefilter/)
Band-­‐Pass Filtering High-­‐Pass and Band-­‐Pass Filtering Band-Pass: Space Domain
(http://paulbourke.net/miscellaneous/imagefilter/)
High-­‐Pass and Band-­‐Pass Filtering High-­‐Pass Filtering Example Ideal (Sharp) Cut-Off: Note Ringing
Smooth cut-off: ringing reduced
(http://paulbourke.net/miscellaneous/imagefilter/)
High-Pass and Band-Pass Filtering
Summary
Summary: Filtering
Consider what you want to do in the frequency domain
(the shape of the filter)
Convert signal to/from
frequency domain
Convert filter to equivalent
convolution operations
THEN
THEN
Implement in the
frequency domain
Implement in the
spatial domain