Lecture 10
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Frequency domain filtering involves modifying the Fourier transform of an image by multiplying it with a filter function. Periodic noise appears as unusually high magnitudes in the spectral coefficients corresponding to the noise frequency. This noise can be reduced or removed in the frequency domain by correcting these coefficients using techniques like thresholding or median filtering in local neighborhoods. Filtering in blocks instead of the whole image can better handle non-uniform quasi-periodic noise.
![Gaussian Median Notch Filter
The m x n vicinity of the peak is multiplied by the
following surface:
( ) ( ) ( )[ ]
1,...,0;1,...,0
1,
2
2
12
2
1
−=−=
−=
−− −+−−
mynx
AeyxG
mn yxB
⊗ =
Peak Gaussian-like Surface Filtered Peak](https://image.slidesharecdn.com/ppf7cxxqsaux4poxkjhh-signature-a5150aa5597b981705fbdc77d9e1399ff6de7cbf1122e68a1070e0378d302ee9-poli-160109133328/85/Lecture-10-21-320.jpg)
Lecture 10
- 1. FREQUENCY DOMAIN PROCESSING. PERIODIC NOISE FILTERING CS-467 Digital Image Processing 1
- 2. Frequency Domain Filtering • Filtering in the frequency domain consists of modifying the Fourier transform of an image and then computing the inverse transform to obtain the processed result where H(u,v) is a filter function 2 ( )( ) ( ) 1 1 ( , ) ( , ) ( , )ˆ( , ) ( , ) ( , ) F f x y f x y F F f x y F FH u v H u v u v− − =
- 3. Periodic Noise • Periodic Noise is a sinusoidal wave with the frequency r (period 1/r) added to a signal • In the spatial domain, this noise corrupts the entire signal • In the frequency domain, it corrupts only a few spectral coefficients corresponding to the frequency of the noisy wave. This kind of noise leads to unusually high magnitude of the corresponding spectral coefficients • Thus, in the frequency domain, this noise can be reduced or even completely removed if the corresponding spectral coefficients have been corrected 3
- 4. Image Its power spectrum Corrupted magnitudes of the spectral coefficients corresponding to the noise frequencies
- 5. Image Its power spectrum Corrupted magnitudes of the spectral coefficients corresponding to the noise frequencies
- 6. Moire Effect 6 © 1992-2008 R .C. Gonzalez & R.E. Woods
- 7. Moire Effect 7 © 1992-2008 R .C. Gonzalez & R.E. Woods
- 8. Moire Effect 8 © 1992-2008 R .C. Gonzalez & R.E. Woods
- 9. Types of Frequency Domain Filters 9 © 1992-2008 R .C. Gonzalez & R.E. Woods
- 10. “Ideal” Mask Generation ( ) ( ) ( ) ≥ < = Tk,lC Tk,lC lkM if,0 if,1 , MCC ⊗= ~ To filter, we do element-wise multiplication of C and M: Moreover if C(k,l) is larger than T for some (k,l), we not just put one 0 in the mask, we put a circle of zeroes with center at (k,l) to improve the result.
- 11. Image with periodic stripes Its power spectrum Filtered with the mask on the right Mask after thresholding with T=1.5
- 12. Laplacian Edge Detection Applied To The Filtering Results
- 13. Mean Filter in Frequency Domain ( ) ( ) ( ) ( ) ( ) ≤ = otherwise,/, , , if,, ,* dlkC T lkS lkC lkC lkC S(k,l) is the local mean in the moving window around the coefficient C(k,l). T is the threshold C*(k,l) is the filtered value of C(k,l) d - is the parameter specifying the strength of peak reduction
- 14. Mean Spectral Filter With T=5, d=50, 3x3 window Enhanced Power Spectrum
- 15. Enhanced Power Spectrum Mean Spectral Filter With T=5, d=50, 3x3 window
- 16. Enhanced Power Spectrum Mean Spectral Filter With T=5, d=50, 3x3 window
- 17. ( ) ( ){ } T jiCMED jiC ≥ , , Median Based Peak Detector C(i,j) is a peak if:
- 18. Images Enhanced Magnitude Detected Peaks (11x11 window, T=6) (11x11 window, T=8)
- 19. Comparison With Thresholding Spectra Thresholding Median Detector
- 20. Spectral Median Filter ( ) ( ){ } ( ) ( ){ } ( ) ≥ = otherwise,, , , if,, ,* lkC T lkCMED lkC lkCMED lkC The peak is substituted by median:
- 21. Gaussian Median Notch Filter The m x n vicinity of the peak is multiplied by the following surface: ( ) ( ) ( )[ ] 1,...,0;1,...,0 1, 2 2 12 2 1 −=−= −= −− −+−− mynx AeyxG mn yxB ⊗ = Peak Gaussian-like Surface Filtered Peak
- 22. Median filter, 11x11 window, T=4 Gaussian filter, 11x11 window, T=6 Thresholding, T=1,5 Image with periodic stripes
- 23. Median Gaussian Thresholding Laplacian Edge Detection
- 24. Clown Gaussian Median Filter, 11x11 window, T=8
- 25. Gaussian Median Filter, 11x11 window, T=11
- 26. Filtering With Overlapping Windows • If the image sizes are not the power of 2 (the limitation of FFT) it is better to filter it by blocks instead of extending to the closest power of 2. • When the noise is non-uniform (quasi- periodic) there’s a chance that in smaller blocks it should be uniform.
- 27. Block Extraction
- 29. Processed by blocks Processed as a whole