1. Digital Image Processing Lecture 11: Image Restoration March 30, 2005 Prof. Charlene Tsai

2. Digital Image ProcessingLecture 11 2 Review  In last lecture, we discussed techniques that restore images in spatial domain.  Mean filtering  Order-statistics filering  Adaptive filering  Gaussian smoothing  We’ll discuss techniques that work in the frequency domain.  In last lecture, we discussed techniques that restore images in spatial domain.  Mean filtering  Order-statistics filering  Adaptive filering  Gaussian smoothing  We’ll discuss techniques that work in the frequency domain.

3. Digital Image ProcessingLecture 11 3 Periodic Noise Reduction  We have discussed low-pass and high-pass frequency domain filters for image enhancement.  We’ll discuss 2 more filters for periodic noise reduction  Bandreject  Notch filter  We have discussed low-pass and high-pass frequency domain filters for image enhancement.  We’ll discuss 2 more filters for periodic noise reduction  Bandreject  Notch filter

4. Digital Image ProcessingLecture 11 4 Bandreject Filters  Removing a band of frequencies about the origin of the Fourier transform.  Ideal filter where D(u,v) is the distance from the center, W is the width of the band, and D 0 is the radial center.  Removing a band of frequencies about the origin of the Fourier transform.  Ideal filter where D(u,v) is the distance from the center, W is the width of the band, and D 0 is the radial center.

5. Digital Image ProcessingLecture 11 5 Bandreject Filters (con’d)  Butterworth filter of order n  Gaussian filter  Butterworth filter of order n  Gaussian filter

6. Digital Image ProcessingLecture 11 6 Bandreject Filters: Demo Original corrupted by sinusoidal noise Fourier transform Butterworth filter Result of filtering

7. Digital Image ProcessingLecture 11 7 Notch Filters  Reject in predefined neighborhoods about the center frequency.  Due to the symmetry of the Fourier transform, notch filters must appear in symmetric pairs about the origin.  Given 2 centers (u 0, v 0 ) and (-u 0, -v 0 ), we define D 1 (u,v) and D 2 (u,v) as  Reject in predefined neighborhoods about the center frequency.  Due to the symmetry of the Fourier transform, notch filters must appear in symmetric pairs about the origin.  Given 2 centers (u 0, v 0 ) and (-u 0, -v 0 ), we define D 1 (u,v) and D 2 (u,v) as

8. Digital Image ProcessingLecture 11 8 Notch Filters: plots ideal Butterworth Gaussian

9. Digital Image ProcessingLecture 11 9 Notch Filters (con’d)  Ideal filter  Butterworth filter  Gaussian filter  Ideal filter  Butterworth filter  Gaussian filter

10. Digital Image ProcessingLecture 11 10 How to deal with motion blur? OriginalBlurred by motion

11. Digital Image ProcessingLecture 11 11 Convolution Theory: Review  Knowing the degradation function H(u,v), we can, in theory, obtain the original image F(u,v).  In practice, H(u,v) is often unknow.  We’ll discuss briefly one method of obtaining the degradation functions. For interested readers, please consult Conzalez, section 5.6 for other methods.  Knowing the degradation function H(u,v), we can, in theory, obtain the original image F(u,v).  In practice, H(u,v) is often unknow.  We’ll discuss briefly one method of obtaining the degradation functions. For interested readers, please consult Conzalez, section 5.6 for other methods. Filter (degradation function) Original image Degraded image

12. Digital Image ProcessingLecture 11 12 Estimation of H(u,v) by Experimentation  If equipment similar to the one used to acquire the degraded image is available, it is possible, in principle, to obtain the accurate estimate of H(u,v).  Step1: reproduce the degraded image by varying the system settings.  Step2: obtain the impulse response of the degradation by imaging an impulse (small dot of light) using the same system settings.  Step3: recalling that FT of an impulse is a constant (A)  If equipment similar to the one used to acquire the degraded image is available, it is possible, in principle, to obtain the accurate estimate of H(u,v).  Step1: reproduce the degraded image by varying the system settings.  Step2: obtain the impulse response of the degradation by imaging an impulse (small dot of light) using the same system settings.  Step3: recalling that FT of an impulse is a constant (A) What we want Degraded impulse image Strength of the impulse

13. Digital Image ProcessingLecture 11 13 Estimation of H(u,v) by Exp (con’d) An impulse of light (magnified). The FT is a constant A G(u,v), the imaged (degraded) impulse

14. Digital Image ProcessingLecture 11 14 Undoing the Degradation  Knowing G & H, how to obtain F?  Two methods:  Inverse filtering  Wiener filtering  Knowing G & H, how to obtain F?  Two methods:  Inverse filtering  Wiener filtering Filter (degradation function) Original image (what we’re after) Degraded image

15. Digital Image ProcessingLecture 11 15 Inverse Filtering  In the simplest form:  See any problems?  Division by small values can produce very large values that dominate the output.  In the simplest form:  See any problems?  Division by small values can produce very large values that dominate the output. Original Inverse filtering using Butterworth filter

16. Digital Image ProcessingLecture 11 16 Inverse Filtering (con’d)  Solutions?  There are two similar approaches:  Low-pass filtering with filter L(u,v):  Thresholding (using only filter frequencies near the origin)  Solutions?  There are two similar approaches:  Low-pass filtering with filter L(u,v):  Thresholding (using only filter frequencies near the origin)

17. Digital Image ProcessingLecture 11 17 Inverse Filtering: Demo Full filterd=40 d=70d=80

18. Digital Image ProcessingLecture 11 18 Inverse Filtering: Weaknesses  Inverse filtering is not robust enough.  It is even worse if the image has been corrupted by noise.  The noise can completely dominate the output.  Inverse filtering is not robust enough.  It is even worse if the image has been corrupted by noise.  The noise can completely dominate the output.

19. Digital Image ProcessingLecture 11 19 Wiener Filtering  What measure can we use to say whether our restoration has done a good job?  Given the original image f and the restored version r, we would like r to be as close to f as possible.  One possible measure is the sum-squared- differences  Wiener filtering: minimum mean square error:  What measure can we use to say whether our restoration has done a good job?  Given the original image f and the restored version r, we would like r to be as close to f as possible.  One possible measure is the sum-squared- differences  Wiener filtering: minimum mean square error: Specified constant

20. Digital Image ProcessingLecture 11 20 Comparison of Inverse and Wiener Filtering  Column 1: blurred image with additive Gaussian noise of variances 650, 65 and 0.0065.  Column 2: Inverse filtering  Column 3: Wiener filtering  Column 1: blurred image with additive Gaussian noise of variances 650, 65 and 0.0065.  Column 2: Inverse filtering  Column 3: Wiener filtering

21. Digital Image ProcessingLecture 11 21 Summary  Removal of periodic noise:  Bandreject  Notch filter  Deblurring the image:  Inverse filtering  Wiener filtering  Removal of periodic noise:  Bandreject  Notch filter  Deblurring the image:  Inverse filtering  Wiener filtering