1. Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Total Variation and related Methods: Error Estimation

2. Martin Burger Total Variation 2 Cetraro, September 2008 Error Estimation Start with the quadratic case D generalizes gradient Optimality

3. Martin Burger Total Variation 3 Cetraro, September 2008 Error Estimation Estimate 1: Two Solutions of Variational Problems Difference Scalar product with

4. Martin Burger Total Variation 4 Cetraro, September 2008 Error Estimation Use Young‘s inequality

5. Martin Burger Total Variation 5 Cetraro, September 2008 Error Estimation Estimate 2: Asymptotic for exact data Need regularity for : Source condition

6. Martin Burger Total Variation 6 Cetraro, September 2008 Error Estimation Source Condition Equivalent to existence of saddle point for

7. Martin Burger Total Variation 7 Cetraro, September 2008 Error Estimation

8. Martin Burger Total Variation 8 Cetraro, September 2008 Error Estimation Estimate 3: Asymptotic for noisy data

9. Martin Burger Total Variation 9 Cetraro, September 2008 Error Estimation Similar estimation as above yields

10. Martin Burger Total Variation 10 Cetraro, September 2008 Error Estimation Nonlinear Variational Method Optimality condition

11. Martin Burger Total Variation 11 Cetraro, September 2008 Error Estimation Stability Estimate between two solutions ´ Same procedure as before: take difference and use duality product with

12. Martin Burger Total Variation 12 Cetraro, September 2008 Error Estimation Error measure: symmetric Bregman distance ´ Note: symmetric Bregman distance is sum of non-symmetric ones

13. Martin Burger Total Variation 13 Cetraro, September 2008 Bregman distance R smooth and strictly convex in some H-space Same for symmetric Bregman distance

14. Martin Burger Total Variation 14 Cetraro, September 2008 Error Estimation R nonsmooth: Bregman distance multivalued and depends on the choice of the subgradient Note: error estimate possible for any appropriate subgradient

15. Martin Burger Total Variation 15 Cetraro, September 2008 Error Estimation R not strictly convex: Bregman distance is not a strict distance, possibly

16. Martin Burger Total Variation 16 Cetraro, September 2008 Error Estimation Bregman distance example

17. Martin Burger Total Variation 17 Cetraro, September 2008 Error Estimation Sparsity measure

18. Martin Burger Total Variation 18 Cetraro, September 2008 Error Estimation Total Variation Contrast change

19. Martin Burger Total Variation 19 Cetraro, September 2008 Error Estimation Contrast Change

20. Martin Burger Total Variation 20 Cetraro, September 2008 Error Estimation Estimate 2: Asymptotic for exact data

21. Martin Burger Total Variation 21 Cetraro, September 2008 Error Estimation Asymptotic

22. Martin Burger Total Variation 22 Cetraro, September 2008 Error Estimation Source condition

23. Martin Burger Total Variation 23 Cetraro, September 2008 Error Estimation Error estimate in Bregman distance Analogous in the noisy case

24. Martin Burger Total Variation 24 Cetraro, September 2008 Error Estimation Multivalued estimate Note: error estimate holds for any Open interpretation for total variation and

25. Martin Burger Total Variation 25 Cetraro, September 2008 Error Estimation TV Subgradients and edges

26. Martin Burger Total Variation 26 Cetraro, September 2008 Error Estimation TV subgradients

27. Martin Burger Total Variation 27 Cetraro, September 2008 Error Estimation

28. Martin Burger Total Variation 28 Cetraro, September 2008 Error Estimation

29. Martin Burger Total Variation 29 Cetraro, September 2008 Error Estimation

30. Martin Burger Total Variation 30 Cetraro, September 2008 Error Estimation Mean Curvature Source condition means smoothness of edge sets !!

31. Martin Burger Total Variation 31 Cetraro, September 2008 Error Estimation Bregman distance

32. Martin Burger Total Variation 32 Cetraro, September 2008 Error Estimation Second term