Image Processing Image Filtering in the Frequency Domain
- Slides: 29
Image Processing Image Filtering in the Frequency Domain • Low Pass Filter • High Pass Filter • Band pass Filter • Blurring • Sharpening
Frequency Bands Image Fourier Spectrum Percentage of image power enclosed in circles (small to large) : 90, 95, 98, 99. 5, 99. 9
Blurring - Ideal Low pass Filter 90% 95% 98% 99. 5% 99. 9%
The Power Law of Natural Images • The power in a disk of radii r=sqrt(u 2+v 2 ) follows: P(r)=Ar- where 2 Images from: Millane, Alzaidi & Hsiao - 2003
Image Filtering • • • Low pass High pass Band pass Local pass Usages
Recall: The Convolution Theorem
Low pass Filter spatial domain f(x, y) frequency domain F(u, v( filter G(u, v) = F(u, v) • H(u, v) g(x, y) f(x, y( G(u, v( F(u, v( • g(x, y( H(u, v(
H(u, v) - Ideal Low Pass Filter H(u, v = ( 1 D(u, v) D 0 0 D(u, v) > D 0 D(u, v) = u 2 + v 2 D 0 = cut off frequency H(u, v) v u 1 0 D(u, v)
Blurring - Ideal Low pass Filter 99. 7% 99. 37% 98. 65%
Blurring - Ideal Low pass Filter 96. 6% 98. 0% 99. 4% 99. 6% 99. 7%
The Ringing Problem G(u, v) = F(u, v) • H(u, v) Convolution Theorem g(x, y) = f(x, y) * h(x, y( IFFT H(u, v) D 0 sinc(x) h(x, y) Ringing radius + blur
The Ringing Problem Freq. domain Spatial domain
H(u, v) - Gaussian Filter H(u, v) 1 v u H(u, v) = 0 D 0 -D 2(u, v)/(2 D 20) e D(u, v) = u 2 + v 2 Softer Blurring + no Ringing D(u, v)
Blurring - Gaussain Lowpass Filter 99. 11% 98. 74% 96. 44%
The Gaussian Lowpass Filter Freq. domain Spatial domain
Blurring in the Spatial Domain: 1 1 Averaging = convolution with = point multiplication of the transform with sinc: Gaussian Averaging = convolution with 1 2 4 2 1 = point multiplication of the transform with a gaussian. 0. 15 1 0. 8 0. 6 0. 05 0. 4 0. 2 0 0 50 Image Domain 100 0 -50 0 Frequency Domain 50
Image Sharpening - High Pass Filter H(u, v) - Ideal Filter H(u, v = ( 0 D(u, v) D 0 1 D(u, v) > D 0 D(u, v) = u 2 + v 2 D 0 = cut off frequency H(u, v) v u 1 0 D(u, v)
High Pass Gaussian Filter H(u, v) 1 v u 0 H(u, v) = 1 - e D 0 -D 2(u, v)/(2 D 20) D(u, v) = u 2 + v 2 D(u, v)
High Pass Filtering Original High Pass Filtered
High Frequency Emphasis Original High Pass Filtered +
High Frequency Emphasis Emphasize High Frequency. Maintain Low frequencies and Mean. H'(u, v) = K 0 + H(u, v) (Typically K 0 =1) H'(u, v) 1 0 D(u, v)
High Frequency Emphasis - Example Original High Frequency Emphasis
High Pass Filtering - Examples Original High pass Emphasis High Frequency Emphasis + Histogram Equalization
Band Pass Filtering H(u, v = ( 0 D(u, v) D 0 - 1 D 0 - 0 D(u, v) > D 0 + w 2 D(u, v) D 0 + w 2 D(u, v) = u 2 + v 2 D 0 = cut off frequency w = band width H(u, v) v u 1 D(u, v) 0 D 0 - w 2 D 0+w 2
Local Frequency Filtering H(u, v) v u H(u, v) 1 0 -u 0, -v 0 H(u, v = ( D 0 u 0, v 0 D(u, v) 1 D 1(u, v) D 0 or D 2(u, v) D 0 0 otherwise D 1(u, v) = (u-u 0)2 + (v-v 0)2 D 2(u, v) = (u+u 0)2 + (v+v 0)2 D 0 = local frequency radius u 0, v 0 = local frequency coordinates
Band Rejection Filtering H(u, v) v u H(u, v) 1 0 -u 0, -v 0 H(u, v = ( D 0 u 0, v 0 D(u, v) 0 D 1(u, v) D 0 or D 2(u, v) D 0 1 otherwise D 1(u, v) = (u-u 0)2 + (v-v 0)2 D 2(u, v) = (u+u 0)2 + (v+v 0)2 D 0 = local frequency radius u 0, v 0 = local frequency coordinates
Demo
- Nnnnnf
- Ingress filtering vs egress filtering
- In digital image processing
- Intensity transformations and spatial filtering
- Linear filtering citra
- Z domain to frequency domain
- Frequency domain to time domain
- What is the z - transform of anu[n] and -anu[-n-1]
- The z transform of np
- Frequency filtering
- Frequency filtering
- Frequency filtering
- High pass filter radiology
- Frequency domain image
- Neighborhood processing in image processing
- Define point processing
- Histogram processing in digital image processing
- Neighborhood processing in digital image processing
- Point processing
- Morphological
- Image filtering
- Translate
- Optimum notch filter in image processing
- Image compression models in digital image processing
- Key stage in digital image processing
- Error free compression in digital image processing
- Image sharpening in digital image processing
- Image geometry in digital image processing
- The range of values spanned by the gray scale is called
- Digital image processing