Frequency Domain Filtering

Frequency Domain Methods Spatial Domain Frequency 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 domaintime domain Example:

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

Band-pass filters 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. (real)

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

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)

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)

(Case 2) h(x,y) specified in spatial domain If h(x,y) is given in the spatial domain, 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) Recall these properties:

Example: h(x,y) is specified in the spatial domain 600 x 600 Sobel time frequency Warning: need to preserve odd symmetry when padding with zeroes  H(u,v) should be imaginary and odd (read details on pages 241 and 268 ) Example: 6 x 6 g(x,y)= -g(6-x,6-y) 602 x 602

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

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

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

Lowpass (LP) filtering (cont’d) In 2D, the cutoff frequencies are specified by a circle. Ideal

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) r=43 (95%)r=78 (99%) r=152 (99.5%) original r: radius

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

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

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

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

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

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

Gaussian LP filter (GLPF)

Gaussian: Frequency – Spatial Domains frequency domain spatial domain

Example: smoothing by GLPF (1)

6/13/ Examples of smoothing by GLPF (2) D 0 =100 D 0 =80

High Pass (LP) Filters Ideal high-pass filter (IHPF) Butterworth high-pass filter (BHPF) Gaussian high-pass filter (GHPF)

High-pass filtering Preserves high frequencies, attenuates low frequencies. H(u)

High-Pass filtering (cont’d) A high-pass filter can be obtained from a low-pass filter as follows: = 1 - D0D0

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 IHPFBHPFGHPF

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

Gaussian HP filter GHPF BHPF

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

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

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? 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.

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

Homomorphic Filtering: Example