1. Image deblurring Seminar inverse problems April 18th 2007 Willem Dijkstra

2. References Curtis R. Vogel: Computational methods for inverse problems, Ch. 5 James G. Nagy et al: Iterative methods for image deblurring: a Matlab object-oriented approach Per Christian Hansen et al : Deblurring images: matrices, spectra, and filtering

3. Blurring point source optical device image screen Not so sharp point

4. Mathematical model f is exact image of gray values on unit square: g is blurred image on unit square: Relation between f and g: k(x,y) is the Point Spread Function (PSF)

5. PSF as image PSF often represented as image f Kf

6. Astronomy Moffat PSF

7. Example blurring Blurring is multiplication by PSF: K= g f*

8. Frequency domain F(k) may have zeros If F(k) has small nonzero values  unstable k often not known analytically g may contain noise: F(f) the 2D Fourier transform of f

9. Discrete equations K is a large matrix: n 2 xn 2 f is a stacked vector:

10. K is a BTTB matrix: Block Toeplitz Toeplitz Block Toeplitz: constant along diagonals BTTB: same blocks along diagonals, each block Toeplitz Efficient matrix-vector multiplication: K need not to be stored BTTB matrices

11. Why BTTB? pixel in g = weighted sum of same pixel in f + some neighbour pixels continuous: discrete:

12. Why BTTB?

13. Boundary conditions Zero ReflexivePeriodic

14. Iterative methods Solve iteratively Use g as initial guess

15. Least squares No exact solution f of since g contains noise Introduce penalty function: Minimizer of T α satisfies:(1) L is penalty matrix: L=I or L=discrete Laplacian:

16. Fourier techniques can be used to solve (1) if K and L are Block Circulant Circulant Block (BCCB) matrices Circulant: each column is shift of 1 st column BCCB: each column with blocks is shift from 1 st column of blocks L can be chosen BCCB K can be made BCCB BCCB matrices

17. Algorithm For K and L BCCB: use fft2(.) and ifft2(.)

18. Example: CASA blurred blurred+noisyrestored

19. Example: grain exact blurred blurred+ noisy restored

20. Example: baby

21. Color image

22. Summary Efficient deblurring tool Exploiting BTTB and BCCB structure of PSF matrix Easy to extend to color images