8. Lecture. Image restoration: Fourier domain

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1 8. Lecture Image restoration: Fourier domain 1

2 Structured noise 2

3 Motion blur 3

4 Filtering in the Fourier domain ² Spatial ltering (average, Gaussian,..) can be done in the Fourier domain (convolution theorem) ² Speed gain, as convolution becomes a multiplication in the Fourier domain 4

5 Fourier transform of the filter kernel ² A convolution of two signals with length A and B results in a signal of length A + B 1 ² This has to be ensured for the ltering in the Fourier domain also ² The size of the Fourier domain image for an image of size A B and a lter kernel of C D therefore has to be: P A + C 1 Q B + D 1 ² This can be achieved by extending the spatial signals to the size P Q by padding 5

6 Gauss filtering in the Fourier domain Original image Gaussian filter kernel Original image (Fourier domain) Gaussian filter kernel (Fourier domain) 6

7 The effect of padding Infinite periodicity without padding Infinite periodicity with padding 7

8 The effect of padding Original image Averaging filter without padding (vertical borders don t show averaging effect) Averaging filter with proper padding (all borders show the same averaging effect) 8

9 Summary of filter process 9

10 Standard filter ² Standard lters in the Fourier domain are highpass and lowpass lters ² Lowpass: A lowpass lter lets low frequencies pass and cuts of high frequencies. This has an averaging e ect. It can be used instead of the Gaussian average lter. ² Highpass: A highpass lter lets high frequencies pass and cuts of low frequencies. The remaining image contains the edges only. 10

11 High pass filter Ideal high pass Butterworth high pass Gaussian highpass 11

12 Example: High pass filtering 12

13 Example: Low pass filtering Lowpass filter in Fourier domain 13

14 Image restoration ² Two methods to remove additive noise in the Fourier domain: g(x; y) = f(x; y) + (x; y) and G(u; v) = F (u; v) + N(u; v) ² Estimate N(u; v) and subtract it from G(u; v) (di±cult) ² Identify noise frequencies by inspection and design a frequency domain lter that suppresses the noise frequencies ^F (u; v) = H(u; v)g(u; v) 14

15 Image restoration Original corrupted noise Fourier transform Band reject filter Result of filtering 15

16 Image restoration filters ² Bandreject lter ² Bandpass lter ² Notch lter (notch reject lter) ² Notch pass lter 16

17 Bandreject filter Removes or attenuates a band of frequencies around the origin in the Fourier domain. Ideal bandreject filter D(u,v) is the distance from the origin. W is the width of the band and D 0 is the distance from the center of the band to the origin. 2 W D, if 1 2, 2 if 0 2, if 1, v D u W D v D u W D W D v D u v u H 17

18 More bandreject filters Butterworth filter of order n Gaussian filter n D v u D W v D u v u H ,, 1 1, ,, 2 1 1, W v u D D v u D e v u H 18

19 Example: Bandreject filter 19

20 Example: Bandreject filter 20

21 Band pass filter ² A bandpass lter is the opposite of a bandreject lter, it blocks all the frequencies outside of the band ² The lter function H(u; v) can be computed from the bandreject lter equation. H bp = 1 H br 21

22 Example: Bandpass filter ² Here bandpass ltering makes the image noise visible. 22

23 Notch reject filter ² A notch reject lter rejects frequencies in prede ned neighborhoods about a center frequency ² Notch lters must appear in symmetric pairs about the origin (except the one at origin) due to the symmetry of the Fourier transform ² Ideal notch reject lter: H(u; v) = 8 < : 0 if D 1 (u; v) D 0 or D 2 (u; v) D 0 1 otherwise D 1 (u; v) = [(u M=2 u 0 ) 2 + (v N=2 v 0 ) 2 ] 1 2 D 2 (u; v) = [(u M=2 + u 0 ) 2 + (v N=2 + v 0 ) 2 ]

24 Ideal notch reject filter 24

25 Notch reject filter ² Butterworth notch reject lter: H(u; v) = 1 + [ 1 D 2 0 D 1 (u;v)d 2 (u;v) ]n ² Gaussian notch reject lter: H(u; v) = 1 e 1 2 D [ 1(u;v)D 2(u;v) D 0 2 ] 25

26 Notch pass filter ² Notch reject lter can also be turned into notch pass lters with the relation: H np (u; v) = 1 H nr (u; v) ² If u 0 = v 0 = 0 then the lter has one cone in the origin: { A notch reject lter becomes a highpass lter { A notch pass lter becomes a lowpass lter 26

27 Example: Ideal notch reject filter 27

28 Degradation function ² So far we have concentrated on the noise term (x; y). Now we deal with ways to remove the degradation function H: ² This can be done by inverse ltering 28

29 Inverse filtering ² Use the simpli ed model without noise: F (u; v) = G(u; v) H(u; v) ² If the degradation H is known F (u; v) can be computed by elementwise division. ² Degradation H(u; v) can be computed as follows: { Observation in the image { Experimentation { Mathematical modelling 29

30 Estimation by image observation ² Here you select a subpart of the image from which you know how it should look like. ² Then you lter the subpart, so that you get a satisfying result. ² E.g. in the case of motion blur, look for an edge and apply an edge sharpening operation. ² Now H(u; v) can be computed for this small part by G(u; v)=f (u; v) ² Because of the shift invariance property of the DFT this degradation function can get applied to the complete image. 30

31 Estimation by experimentation ² Estimate the degradation from a reference image. ² The degradation can be computed from an image of an impulse (small bright dot) ² The Fourier transform of an impulse is constant and therefore: G(u; v) H(u; v) = A where A is a constant value (the Fourier transform of the undegraded impuls). 31

32 Estimation by experimentation Reference image (the impulse leads to Fourier spectrum of constant value) Degraded impulse 32

33 Estimation by mathematical modelling 33

34 Example: Motion blur 34

35 Wrong focus, atmospheric turbulences ² Wrong focus: H(u; v) = J 1(ar) ar r 2 = u 2 + v 2 where J 1 is a rst order Bessel function and a is a displacement parameter ² Atmospheric turbulences: H(u; v) = e k(u2 +v 2 ) 5=6 where k is a constant depending on the nature of the turbulence 35

36 Practical inverse filtering ² For inverse ltering we need to compute ^F (u; v) = G(u; v)=h(u; v) ² Problem: In practice H(u; v) might contain zeros or values close to zero. This might come from approximating H(u; v) but because of the division the in uence is very high. ² Solution: In practice this small values occur in high frequency regions only and therefore a lowpass lter applied to the image function G(u; v) can decrease the e ect 36

37 Example: Inverse filtering without lowpass cutoff radius d=40 cutoff radius d=70 cutoff radius d=80 37

38 Wiener filtering ² Is inverse ltering in the presence of noise ² Wiener ltering seeks an estimate ^f that minimizes the statistical error function: e 2 = Ef(f ^f) 2 g where E is the expected value operator. ² The solution to this expression in the Fourier domain is: ^F (u; v) = jh(u; v)j 2 H(u; v) jh(u; v)j 2 5 G(u; v) + S (u; v)=s f (u; v) 3 38

39 Wiener filtering ^F (u; v) = jh(u; v)j 2 H(u; v) jh(u; v)j 2 5 G(u; v) + S (u; v)=s f (u; v) 3 ² H(u; v) is the degradation function ² jh(u; v)j 2 = H (u; v)h(u; v) with H (u; v) the complex conjugate of H ² S (u; v) = jn(u; v)j 2 is the power spectrum of the noise ² S f (u; v) = jf (u; v)j 2 is the power spectrum of the undegraded image 39

40 Parametric Wiener filtering ² Main problem is that the power spectrum S f (u; v) is seldom known ² A solution for this is to replace the noise ratio with a constant K 2 1 jh(u; v)j ^F 2 3 (u; v) = 4 H(u; v) jh(u; v)j 2 5 G(u; v) + K ² K can be user selected and be varied interactively to nd a setting that produces a good result ² If K = 0 then Wiener ltering simpli es to direct inverse ltering 40

41 Example: Wiener filtering 41

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