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

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Frequency Domain Methods Spatial Domain Frequency 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 frequency domaintime domain Example:

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

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

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

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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.

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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 (cont’d) G(u,v)= F(u,v)H(u,v) = H(u,v) R(u,v) + jH(u,v)I(u,v) (case 1)

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

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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: 1.Form h p (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)

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Example: h(x,y) is specified in the spatial domain 600 x 600 Sobel Important: need to preserve odd symmetry (i.e., H(u,v) should be imaginary) (read details on page 268)

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Results of Filtering in the Spatial and Frequency Domains spatial domain filtering frequency domain filtering

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Low-pass (LP) filtering Preserves low frequencies, attenuates high frequencies. ideal in practice D 0 : cut-off frequency

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Lowpass (LP) filtering (cont’d) 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 (cont’d) Typically, most frequencies are concentrated around the center of the spectrum. r=8 (90% power)r=18 (93% power) r=43 (95%)r=78 (99%)r=152 (99.5%) original r: radius

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How does D 0 control smoothing? Reminder: multiplication in the frequency domain implies convolution in the time domain * = freq. domain time domain

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How does D 0 control smoothing? (cont’d) D 0 controls the amount of blurring r=8 (90%) r=78 (99%)

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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). h=f*g

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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 n=1 n=4n=16

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Spatial Representation of BLPFs n=1 n=2 n=5 n=20

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Comparison: Ideal LP and BLPF ILPF BLPF D 0 =10, 30, 60, 160, 460 n=2 D 0 =10, 30, 60, 160, 460

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Gaussian LP filter (GLPF)

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Gaussian: Frequency – Spatial Domains frequency domain spatial domain

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Example: smoothing by GLPF (1)

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5/4/ Examples of smoothing by GLPF (2) D 0 =100 D 0 =80

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High-Pass filtering A high-pass filter can be obtained from a low-pass filter using: = 1 - D0D0

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High-pass filtering (cont’d) 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 IHPFBHPFGHPF

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Comparison: IHPF and BHPF IHPF BHPF D 0 =30,60,160 n=2 D 0 =30,60,160

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Gaussian HP filter GHPF BHPF

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Comparison: BHPF and GHPF GHPF BHPF D 0 =30,60,160 n=2 D 0 =30,60,160

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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 (cont’d) 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) Frequency domain: previous definition:

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Revisit: Unsharp Masking and Highboost Filtering Highboost Filter

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Highboost and High-Frequency-Emphasis Filters 1 1+k k1k1 k 1 +k 2 HighboostHigh-emphasis

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High-Frequency Emphasis filtering Using Gaussian filter k 1 =0.5, k 2 =0.75 D 0 =40 Example GHPF High-emphasis and hist. equal.

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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 (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)

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How are frequencies mixed together? When applying filtering, it is difficult to handle low/high frequencies separately. Low and high frequencies from i(x,y) and r(x,y) are mixed together.

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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 (cont’d) (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

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Homomorphic Filtering: Example

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