Deconvolution, Deblurring and Restoration

1. T-61.182, Biomedical Image Analysis Seminar Presentation 14.4.2005 Seppo Mattila & Mika Pollari Deconvolution, Deblurring andRestoration

2. Overview (1/2) • Linear space-invariant (LSI) restoration filters - Inverse filtering - Power spectrum equalization - Wiener filter - Constrained least-squares restoration - Metz filter • Blind Deblurring

3. Overview (2/2) • Homomorphic Deconvolution • Space-variant restoration • Sectioned image restoration • Adaptive-neighbourhood deblurring • The Kalman filter • Applications - Medical - Astronomical

4. g(x,y) measured image f(x,y) true (ideal) image h(x,y) point spread function (PSF) (impuse response function) n(x,y) additive random noise Introduction • Find the best possible estimate of the original unknown image from the degraded image. • One typical degradation process has a form:

5. Image Restoration General • One has to have some a priori knowledge about the degragation process. • Usually one needs 1) model for degragation, some information from 2) original image and 3) noise. • Note! Eventhough one doesn’t know the original image some information such as power spectral density (PSD) and autocorreletion function (ACF) are easy to model.

6. Linear-Space Invariant (LSI) Restoration Filters • Assume: linear and shift-invariant degrading process • Random noise statistically indep. of image-generating process • Possible to design LSI filters to restore the image

7. (see Sect. 3.5.3 for details) (if no noise) Inverse Filtering • Consider degrading process in matrix form: • Given g and h, estimate f by minimising the squared error between observed image (g) and : where and are approximations of f and g • Set derivative of є2 to zero: (if noise present)

8. f' G Inverse Filtering Examples • Works fine if no noise but... • H(u,v) usually low-pass function. • N(u,v) uniform over whole spectrum. • High-freq. Noise amplified!! 0.4x 0.2x

9. PSD( (u,v)) = PSD(f(u,v)) Power Spectrum Equalization (PSE) i.e. • Want to find linear transform L such that: • Power spectral density (PSD) = FT(Autocorrelation function) . . .

10. The Wiener Filter (1/2) • Degradation model: • Assumtions: Image and noise are second-order-stationary random processes and they are statistically independent • Optimal mean-square error (MSE) criterion Find Wiener filter (L) which minimize MSE

11. The Wiener Filter (2/2) • Minimizing the criterion we end up to optimal Wiener filter. • The Wiener filter depends on the autocorrelation function (ACF) of the image and noise (This is no problem). • In general ACFs are easy to estimate.

12. Comparison of Inverse Filter, PSE, and Wiener Filter

13. Constrained Least-squares Restoration • Minimise: with constraint: where L is a linear filter operator • Similar to Wiener filter but does not require the PSDs of the image and noise to be known • The mean and variance of the noise needed to set optimally. If = 0 inverse filter . . .

14. The Metz Filter • Modification to inverse filter. • Supress the high frequency noise instead of amplyfying it. • Select factor so that mean-square error (MSE) between ideal and filtered image is minimized.

15. Motion Deblurring – Simple Model • Assume simple in plane movement during the exposure • Either PSF or MTF is needed for restoration

16. Blind Deblurring • Definition of deblurring. • Blind deblurring: models of PSF and noise are not known – cannot be estimated separately. • Degragated image (in spectral domain) consist some information of PSF and noise but in combined form.

17. Method 1 – Extension to PSE • Broke image to M x M size segment where M is larger than dimensions of PSF then • Average of PSD of these segments tend toward the true signal and noise PSD • This is combined information of blur function and noise which is needed in PSE • Finaly, only PSD of image is needed

18. Extension to PSE Cont...

19. Method 2 – Iterative Blind Deblurring • Assumptation: MTF of PSF has zero phase. • Idea: blur function affects in PSD but phase information preserves original information from edges.

20. Iterative Blind Deblurring Cont... • Fourier transform of restored image is • Note that smoothing operator S[] has small effect to smooth functions (PSF). This leads to iterative update rule

21. Examples of Iterative Blind Deblurring

22. Homomorphic deconvolution • Start from: • Convert convolution operation to addition: • Complex cepstrum: Complex cepstra related: • Practical application, however, not simple...

23. Steps involved in deconvolution using complex cepstrum

24. Space-variant Image Restoration • So far we have assumed that images are spatially (and temporaly) stationary • This is (generally) not true – at the best images are locally stationary • Techniques to overcome this problem: • Sectioned image restoration • Adaptive neighbourhood deblurring • The Kalman filter (the most elegant approach)

25. Sectioned Image Restoration • Divide image into small [P x P] rectangular, presumably stationary segments. • Centre each segment in a region, and pad the surrounding with the mean value. • For each segment apply separately image restoration (e.g. PSE or wiener).

26. Adaptive-neighborhood deblurring (AND) • Grow adaptive neighborhood regions: • Apply 2D Hamming window to each region: • Estimate the noise spectrum: Pixel locations within the region Centered on (m,n) A is a freq. domain scale factor that depends on the spectral characterisics of the region grown etc.

27. AND segmentation

28. Adaptive-neighborhood deblurring (AND) Cont… • Frequency-domain estimate of the uncorrupted adaptive-neighborhood region: • Obtain estimate for deblurred adaptive neigborhood region m,n(p,q) by FT-1 • Run for every pixel in the input image g(x,y) Deblurred image

29. Comparison of Sectioned and AND-technique

30. Kalman Filter • Kalman filter is a set of mathematical equations. • Filter provides recursive way to estimate the state of the process (in non-stationary environment), so that mean of squared errors is minimized (MMSE). • Kalman filter enables prediction, filtering, and smoothing.

31. Kalman Filter State-Space • Process Eq. • Observation Eq. • Innovation process:

32. Kalman Filter in a Nutshell (1/2) • Data observations are available • System parameters are known • a(n+1,n), h(n), and the ACF of driving and observation noise • Initial conditions • Recursion

33. Kalman Filter in a Nutshell (2/2) • Compute the Kalman gain K(n) • Obtain the innovation process • Update • Compute the ACF of filtered state error • Compute the ACF of predicted state error

34. Wiener Filter Restoration of Digital Radiography

35. Astronomical applications • Images blurred by atmospheric turbulence • Observing above the atmosphere very expensive (HST) • Improve the ground-based resolution by • Suitable sites for the observatory (@ 4 km height) • Real time Adaptive optics correction • Deconvolution

36. Point Spread Function (PSF) in Astronomy Iobserved = Ireal⊗PSF Easy to measure and model from several stars usually present in astro-images Determines the spatial resolution of an image Commonly used for image matching and deconvolution Ideal PSF if no atmosphere FWHM ~ 1.22x/D Atmospheric turbulence broadens the PSF Gaussian PSF with FWHM ~ 1" < 0.1" (8m telescope)

37. Richardson-Lucy deconvolution • Used in both fields: astronomy & medical imaging • Start from Bayes's theorem, end up with: • Takes into account statistical fluctuations in the signal, therefore can reconstruct noisy images! • In astronomy the PSF is known accurately • From an initial guess f0(x) iterate until converge

38. Astro-examples Observed PSF Inverse filter Richardson-Lucy