# Frequency Domain Filtering (Chapter 4)

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Frequency Domain Filtering (Chapter 4)
CS474/674 - Prof. Bebis

Frequency Domain Methods
Spatial Domain

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

Example Original signal Low-pass filtered High-pass filtered
Band-pass filtered Band-stop filtered

Low-pass filters (i.e., smoothing filters)
Preserve low frequencies - useful for noise suppression frequency domain time domain Example:

High-pass filters (i.e., sharpening filters)
Preserves high frequencies - useful for edge detection frequency domain time domain Example:

Band-pass filters Example: Preserves frequencies within a certain band
frequency domain time domain Example:

Band-stop filters How do they look like? Band-pass Band-stop

Frequency Domain Methods
Case 1: H(u,v) is specified in the frequency domain. Case 2: h(x,y) is specified in the spatial domain.

Frequency domain filtering: steps
F(u,v) = R(u,v) + jI(u,v)

Frequency domain filtering: steps (cont’d)
(case 1) G(u,v)= F(u,v)H(u,v) = H(u,v) R(u,v) + jH(u,v)I(u,v)

Example f(x,y) fp(x,y) fp(x,y)(-1)x+y F(u,v) H(u,v) - centered
G(u,v)=F(u,v)H(u,v) g(x,y) gp(x,y)

h(x,y) specified in spatial domain: how to generate H(u,v) from h(x,y)?
If h(x,y) is given in the spatial domain (case 2), we can generate H(u,v) as follows: Form hp(x,y) by padding with zeroes. 2. Multiply by (-1)x+y to center its spectrum. 3. Compute its DFT to obtain H(u,v) 13

Example: h(x,y) is specified in the spatial domain
Important: need to preserve odd symmetry (i.e., H(u,v) should be imaginary) (read details on page 268) In this example, we start with a spatial mask and show how to generate its corresponding filter in the frequency domain. Then, we compare the filtering results obtained using frequency domain and spatial techniques. We use the 3x3 Sobel vertical edge detector. The left one is a 600x600 pixel image, and its spectrum is shown on the right. Sobel 14

Results of Filtering in the Spatial and Frequency Domains
spatial domain filtering frequency domain filtering In this example, we start with a spatial mask and show how to generate its corresponding filter in the frequency domain. Then, we compare the filtering results obtained using frequency domain and spatial techniques. We use the 3x3 Sobel vertical edge detector. The left one is a 600x600 pixel image, and its spectrum is shown on the right. 15

Low-pass (LP) filtering
Preserves low frequencies, attenuates high frequencies. ideal in practice D0: cut-off frequency

Lowpass (LP) filtering (cont’d)
In 2D, the cutoff frequencies lie on a circle.

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.

Specifying a 2D low-pass filter (cont’d)
Typically, most frequencies are concentrated around the center of the spectrum. r=8 (90% power) r=18 (93% power) original r: radius r=43 (95%) r=78 (99%) r=152 (99.5%)

How does D0 control smoothing?
Reminder: multiplication in the frequency domain implies convolution in the time domain time domain freq. domain * =

How does D0 control smoothing? (cont’d)
D0 controls the amount of blurring r=78 (99%) r=8 (90%)

Ringing Effect Sharp cutoff frequencies produce an overshoot of image features whose frequency is close to the cutoff frequencies (ringing effect). h=f*g

Low Pass (LP) Filters Ideal low-pass filter (ILPF)
Butterworth low-pass filter (BLPF) Gaussian low-pass filter (GLPF)

Butterworth LP filter (BLPF)
In practice, we use filters that attenuate high frequencies smoothly (e.g., Butterworth LP filter)  less ringing effect n=1 n=4 n=16

Spatial Representation of BLPFs
n= n= n= n=20

Comparison: Ideal LP and BLPF
ILPF BLPF D0=10, 30, 60, 160, 460 D0=10, 30, 60, 160, 460 n=2

Gaussian LP filter (GLPF)
27

Gaussian: Frequency – Spatial Domains

Example: smoothing by GLPF (1)

Examples of smoothing by GLPF (2)
D0=100 D0=80 4/14/2017

High-Pass filtering A high-pass filter can be obtained from a low-pass filter using: = 1 - D0

High-pass filtering (cont’d)
Preserves high frequencies, attenuates low frequencies.

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

Butterworth high pass filter (BHPF)
In practice, we use filters that attenuate low frequencies smoothly (e.g., Butterworth HP filter)  less ringing effect

Spatial Representation of High-pass Filters
IHPF BHPF GHPF

Comparison: IHPF and BHPF

Gaussian HP filter GHPF BHPF

Comparison: BHPF and GHPF

Example: High-pass Filtering and Thresholding for Fingerprint Image Enhancement
BHPF (order 4 with a cutoff frequency 50)

Difference of Gaussians: Frequency – Spatial Domains
This is a high-pass filter!

Difference of Gaussians: Frequency – Spatial Domains (cont’d)
High-pass filter!

Frequency Domain Analysis of Unsharp Masking and Highboost Filtering
(alternative definition) previous definition: Frequency domain:

Revisit: Unsharp Masking and Highboost Filtering

Highboost and High-Frequency-Emphasis Filters
1 1+k k1 k1+k2 Highboost High-emphasis

Example D0=40 High-Frequency Emphasis filtering Using Gaussian filter
GHPF D0=40 High-emphasis High-emphasis and hist. equal. High-Frequency Emphasis filtering Using Gaussian filter k1=0.5, k2=0.75

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.

Homomorphic Filtering (cont’d)
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)

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.

Can we separate them? Idea: Take the ln( ) of

Steps of Homomorphic Filtering
(1) Take (2) Apply FT: or (3) Apply H(u,v)

Steps of Homomorphic Filtering (cont’d)
(4) Take Inverse FT: or (5) Take exp( ) or

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

Homomorphic Filtering: Example

Homomorphic Filtering: Example