Image Processing Image Filtering in the Frequency Domain

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Image Processing Image Filtering in the Frequency Domain • Low Pass Filter • High

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

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%

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

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

Image Filtering • • • Low pass High pass Band pass Local pass Usages

Recall: The Convolution Theorem

Recall: The Convolution Theorem

Low pass Filter spatial domain f(x, y) frequency domain F(u, v( filter G(u, v)

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)

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 99. 7% 99. 37% 98. 65%

Blurring - Ideal Low pass Filter 96. 6% 98. 0% 99. 4% 99. 6%

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,

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

The Ringing Problem Freq. domain Spatial domain

H(u, v) - Gaussian Filter H(u, v) 1 v u H(u, v) = 0

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%

Blurring - Gaussain Lowpass Filter 99. 11% 98. 74% 96. 44%

The Gaussian Lowpass Filter Freq. domain Spatial domain

The Gaussian Lowpass Filter Freq. domain Spatial domain

Blurring in the Spatial Domain: 1 1 Averaging = convolution with = point multiplication

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 =

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

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 Pass Filtering Original High Pass Filtered

High Frequency Emphasis 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) =

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 Frequency Emphasis - Example Original High Frequency Emphasis

High Pass Filtering - Examples Original High pass Emphasis High Frequency Emphasis + Histogram

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

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

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

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

Demo