1. Tirgul 4: Image Restoration in the Frequency Domain Example for Image Reconstruction (Tomography)‏

2. Degradation/Restoration Model
Goal: Reconstruct a degraded image using a priori knowledge of the degradation phenomenon.
Degradation
Function H +
Restoration
filters

3. Degradation/Restoration Model
Assumptions on H:
Linearity (additivity & homogeneity)‏
Space invariance: 
Linear, position-invariant degradation can be modelled as the convolution:

4. Degradation / Restoration Model
Linear, position-invariant degradation can be modelled as the convolution:
Equivalent frequency domain representation :

5. Estimating the Degradation function
We need to have some information on the degradation function
Observation
Experimentation
Mathematical modelling

6. Estimation by image observation
Look for information in the image itself : search for small section of the image, s, containing simple structure (edge, point)‏
High signal to noise ratio
Guess the image in that area:

7. Estimation by Experiment
If we have the equipment used to acquire the degraded image we can obtain accurate estimation of the degradation
Obtain an impulse response of the degradation using the same system setting

8. Estimation by modelling
Mathematical model of degradation can be for example atmospheric turbulence
Hufnagel & Stanley: k=
k=0.001
k=0.0025
Original

9. Estimation by modeling: Uniform motion blur

10. Estimation by modelling: Uniform motion blur
Assuming the image was blurred by a uniform linear motion, between the image and the sensor, during image acquisition.
The degradation : f(x)*recta(x)‏
In the Fourier domain: F(u)·sinc(u)‏

11. Noise Models Principal source of noise – acquisition / transmission
Assumptions:
White noise: Fourier spectrum of the noise is constant.
Noise is independent in spatial coordinates.
Uncorrelated with the image.
Zero mean

12. Estimating White Noise Spectrum
Assuming additive white noise, and given the linearity of Fourier transform, the transform of G will be sum of the transforms
H∙F + N = G

13. Inverse Filtering (Deconvolution Filters)‏
Problem: H may have very small values.
Conclusion: We need to find a way to deal with the noise N.

14. Minimum Mean Square Error (Wiener) Filtering
Goal: have an inverse filter that deals with the noise better than inverse filtering.
Try to minimize:
– power spectrum

15. Minimum Mean Square Error (Wiener) FilteringK
K can be chosen interactively to yield best visible results

16. (K is chosen interactively)‏
Examples
Modelling atmosphere turbulence:
Radially limited
Inverse filtering
Wiener filter
(K is chosen interactively)‏
Inverse filtering

17. Examples Images corrupted by motion blur and additive noise. Image
Inverse filtering
Weiner Filtering
Images corrupted by motion blur and additive noise.
Noise Variance

18. Reconstruction Example: Computerized Tomography
Goal : reconstruct 2D slices of 3D data given projection
Computerized Tomography
X-Ray : propagating beam
Photons are lost from the beam as they are absorbed in the body
Try to reconstruct linear attenuation coefficient representing the material

19. Parallel projection

20. Sum on f(x,y) along each ray AB:
Radon Transform
Sum on f(x,y) along each ray AB:

21. Fourier Slice Theorem
Assume w.l.o.g. that Ө = 0; the projection of f(x,y) on the x axis is
Thus,
This is true for each angle, so

22. Fourier Slice Theorem

23. The Inverse Radon f(x,y) reconstruction algorithm:
1. Compute all of the projections pθ(r).
2. Apply Fourier transform on p, and Reconstruct F(u,v).
3.Apply inverse Fourier transform and get the reconstructed function f(x,y).
Problem: Algorithm is very sensitive to noise.

24. Iterative Back Projection
Start with initial guess of the image.
For each ray – perform summation on the guess and on the real image.
Spread the difference between guess and observation uniformly for all pixels along this ray.
This process converges in many cases to the actual image .