Frequency Domain Filtering (Chapter 4)

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Frequency Domain Filtering (Chapter 4)

• CS474/674 - Prof. Bebis

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Frequency Domain Methods
Frequency Domain
Spatial Domain
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Major filter categories

• Typically, filters are classified by examining
  their properties in the frequency domain
• (1) Low-pass
• (2) High-pass
• (3) Band-pass
• (4) Band-stop

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Example
Original signal
Low-pass filtered
High-pass filtered
Band-pass filtered
Band-stop filtered
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Low-pass filters(i.e., smoothing filters)

• Preserve low frequencies - useful for noise
  suppression

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High-pass filters(i.e., sharpening filters)

• Preserves high frequencies - useful for edge
  detection

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Band-pass filters

• Preserves frequencies within a certain band

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Band-stop filters

• How do they look like?

Band-stop
Band-pass
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Frequency Domain Methods
Case 1 h(x,y) is given in the spatial
domain. Case 2 H(u,v) is given in the frequency
domain.
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Frequency domain filtering steps
F(u,v) R(u,v) jI(u,v)
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Frequency domain filtering steps (contd)
G(u,v) F(u,v)H(u,v) H(u,v) R(u,v)
jH(u,v)I(u,v)
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Example
fp(x,y)
f(x,y)
fp(x,y)(-1)xy
F(u,v)
G(u,v)F(u,v)H(u,v)
H(u,v) - centered
g(x,y)
gp(x,y)
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h(x,y) specified in spatial domainHow to
generate H(u,v) from h(x,y)?

• If h(x,y) is given in the spatial domain, we can
• generate H(u,v) as follows
• Form hp(x,y) by padding with zeroes.
• 2. Multiply by (-1)xy to center its spectrum.
• 3. Compute its DFT to obtain H(u,v)

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Example h(x,y) is specified in the spatial domain

600 x 600
Important need to preserve odd symmetry (i.e.,
H(u,v) should be imaginary) (read details on
page 268)
Sobel
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Results of Filtering in the Spatial and
Frequency Domains

spatial domain filtering
frequency domain filtering
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Example

• Consider a filter H(u,v) that is 0 at the center
  of the transform and 1 elsewhere
• What can you say for the output image?

zero average intensity
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Low-pass (LP) filtering

• Preserves low frequencies, attenuates high
  frequencies.

ideal
in practice
D0 cut-off frequency
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Lowpass (LP) filtering (contd)

• In 2D, the cutoff frequencies lie on a circle.

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Specifying a 2D low-pass filter

• Specify cutoff frequencies by specifying the
  radius of a circle centered at point (N/2, N/2)
  in the frequency domain.
• The radius is chosen by specifying the percentage
  of total power enclosed by the circle.

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Specifying a 2D low-pass filter (contd)

• Typically, most frequencies are concentrated
  around the center of the spectrum.

r8 (90 power)
r18 (93 power)
original
r radius
r43 (95)
r78 (99)
r152 (99.5)
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How does D0 control smoothing?

• Reminder multiplication in the frequency domain
  implies convolution in the time domain

time domain
freq. domain


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How does D0 control smoothing? (contd)

• D0 controls the amount of blurring

r78 (99)
r8 (90)
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Ringing Effect

• Sharp cutoff frequencies produce an overshoot of
  image features whose frequency is close to the
  cutoff frequencies (ringing effect).

hfg
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Low Pass (LP) Filters

• Ideal low-pass filter (ILPF)
• Butterworth low-pass filter (BLPF)
• Gaussian low-pass filter (GLPF)

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Butterworth LP filter (BLPF)

• In practice, we use filters that attenuate high
  frequencies smoothly (e.g., Butterworth LP
  filter) ? less ringing effect

n1
n4
n16
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Spatial Representation of BLPFs

n1 n2 n5
n20
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Comparison Ideal LP and BLPF

BLPF
ILPF
D010, 30, 60, 160, 460
D010, 30, 60, 160, 460
n2
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Gaussian LP filter (GLPF)

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Gaussian Frequency Spatial Domains

spatial domain
frequency domain
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Example smoothing by GLPF (1)
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Examples of smoothing by GLPF (2)
D0100
D080
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High-Pass filtering

• A high-pass filter can be obtained from a
  low-pass filter using

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High-pass filtering (contd)

• Preserves high frequencies, attenuates low
  frequencies.

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High Pass (LP) Filters

• Ideal high-pass filter (IHPF)
• Butterworth high-pass filter (BHPF)
• Gaussian high-pass filter (GHPF)
• Difference of Gaussians
• Unsharp Masking and High Boost filtering

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Butterworth high pass filter (BHPF)

• In practice, we use filters that attenuate low
  frequencies smoothly (e.g., Butterworth HP
  filter) ? less ringing effect

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Spatial Representation of High-pass Filters

IHPF
BHPF
GHPF
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Comparison IHPF and BHPF

IHPF
D030,60,160
D030,60,160
BHPF
n2
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Gaussian HP filter
GHPF
BHPF
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Comparison BHPF and GHPF

D030,60,160
BHPF
n2
D030,60,160
GHPF
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Example High-pass Filtering and Thresholding
for Fingerprint Image Enhancement

BHPF (order 4 with a cutoff frequency 50)
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Difference of Gaussians Frequency Spatial
Domains

This is a high-pass filter!
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Difference of Gaussians Frequency Spatial
Domains (contd)

spatial domain
frequency domain
High-pass filter!
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Frequency Domain Analysis of Unsharp Masking and
Highboost Filtering

Unsharp Masking
Highboost filtering (alternative definition)
previous definition
Frequency domain
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Revisit Unsharp Masking and Highboost Filtering

Highboost Filter
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Highboost and High-Frequency-Emphasis Filters

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Example

GHPF
D040
High-emphasis
High-emphasis and hist. equal.
High-Frequency Emphasis filtering Using Gaussian
filter k10.5, k20.75
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Homomorphic filtering

• Many times, we want to remove shading effects
  from an image (i.e., due to uneven illumination)
• Enhance high frequencies
• Attenuate low frequencies but preserve fine
  detail.

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Homomorphic Filtering (contd)

• Consider the following model of image formation
• In general, the illumination component i(x,y)
  varies slowly and affects low frequencies mostly.
• In general, the reflection component r(x,y)
  varies faster and affects high frequencies mostly.

i(x,y) illumination r(x,y) reflection
IDEA separate low frequencies due to i(x,y)
from high frequencies due to r(x,y)
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How are frequencies mixed together?

• Low and high frequencies from i(x,y) and r(x,y)
• are mixed together.

• When applying filtering, it is difficult to
  handle
• low/high frequencies separately.

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Can we separate them?

• Idea

Take the ln( ) of
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Steps of Homomorphic Filtering

• (1) Take
• (2) Apply FT
• or
• (3) Apply H(u,v)

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Steps of Homomorphic Filtering (contd)

• (4) Take Inverse FT
• or
• (5) Take exp( ) or

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Example using high-frequency emphasis

Attenuate the contribution made by illumination
and amplify the contribution made by reflectance
Attenuate the contribution made by illumination
and amplify the contribution made by reflectance
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Homomorphic Filtering Example

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Homomorphic Filtering Example