1. Low Pass Filter High Pass Filter Band pass Filter Blurring Sharpening Image Processing Image Filtering in the Frequency Domain

2. Frequency Bands Percentage of image power enclosed in circles (small to large) : 90, 95, 98, 99, 99.5, 99.9 ImageFourier Spectrum

3. Blurring - Ideal Low pass Filter 90%95% 98% 99% 99.5% 99.9%

6. 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

7. Image Filtering Low pass High pass Band pass Local pass Usages

8. Recall: The Convolution Theorem

9. Low pass Filter f(x,y) F(u,v) g(x,y) G(u,v) G(u,v) = F(u,v) H(u,v) spatial domainfrequency domain filter f(x,y)F(u,v) H(u,v)g(x,y)

10. H(u,v) - Ideal Low Pass Filter u v H(u,v) 0 D0D0 1 D(u,v) H(u,v) 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

11. Blurring - Ideal Low pass Filter 98.65% 99.37% 99.7%

12. Blurring - Ideal Low pass Filter 98.0% 99.4% 99.6% 99.7% 99.0% 96.6%

13. The Ringing Problem G(u,v) = F(u,v) H(u,v) g(x,y) = f(x,y) * h(x,y) Convolution Theorem sinc(x) h(x,y)H(u,v)  D0 D0  Ringing radius +  blur IFFT

14. The Ringing Problem Freq. domain Spatial domain

15. H(u,v) - Gaussian Filter D(u,v) 0D0D0 1 H(u,v) u v D(u,v) =  u 2 + v 2 H(u,v) = e -D 2 (u,v)/(2D 2 0 ) Softer Blurring + no Ringing

16. Blurring - Gaussain Lowpass Filter 96.44% 98.74% 99.11%

17. The Gaussian Lowpass Filter Freq. domain Spatial domain

18. Blurring in the Spatial Domain: Averaging = convolution with 1 = point multiplication of the transform with sinc: 050100 0 0.05 0.1 0.15 -50050 0 0.2 0.4 0.6 0.8 1 Gaussian Averaging = convolution with 1 2 1 2 4 2 1 2 1 = point multiplication of the transform with a gaussian. Image DomainFrequency Domain

19. 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 0 D0D0 1 D(u,v) H(u,v) u v

20. D(u,v) 0D0D0 1 D(u,v) =  u 2 + v 2 High Pass Gaussian Filter u v H(u,v) H(u,v) = 1 - e -D 2 (u,v)/(2D 2 0 )

21. High Pass Filtering OriginalHigh Pass Filtered

22. High Frequency Emphasis OriginalHigh Pass Filtered +

23. High Frequency Emphasis Emphasize High Frequency. Maintain Low frequencies and Mean. (Typically K 0 =1) H'(u,v) = K 0 + H(u,v) 0 D0D0 1 D(u,v) H'(u,v)

24. High Frequency Emphasis - Example OriginalHigh Frequency Emphasis OriginalHigh Frequency Emphasis

25. High Pass Filtering - Examples OriginalHigh pass Emphasis High Frequency Emphasis + Histogram Equalization

26. Band Pass Filtering H(u,v) = 1 D 0 -  D(u,v)  D 0 + 0 D(u,v) > D 0 + D(u,v) =  u 2 + v 2 D 0 = cut off frequency u v H(u,v) 0 1 D(u,v) H(u,v) D0-D0- w 2 D0+D0+ w 2 D0D0 0 D(u,v)  D 0 - w 2 w 2 w 2 w 2 w = band width

27. H(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 0 = local frequency radius Local Frequency Filtering u v H(u,v) 0 D0D0 1 D(u,v) H(u,v) -u 0,-v 0 u 0,v 0 D 2 (u,v) =  (u+u 0 ) 2 + (v+v 0 ) 2 u 0,v 0 = local frequency coordinates

28. H(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 0 = local frequency radius Band Rejection Filtering 0 D0D0 1 D(u,v) H(u,v) -u 0,-v 0 u 0,v 0 D 2 (u,v) =  (u+u 0 ) 2 + (v+v 0 ) 2 u 0,v 0 = local frequency coordinates u v H(u,v)

29. Demo