1. Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Total Variation and Related Methods II

2. Martin Burger Total Variation 2 Cetraro, September 2008 Variational Methods and their Analysis We investigate the analysis of variational methods in imaging Most general form:

3. Martin Burger Total Variation 3 Cetraro, September 2008 Variational Methods and their Analysis Questions: - Existence - Uniqueness - Optimality conditions for solutions (-> numerical methods) - Structural properties of solutions - Asymptotic behaviour with respect to

4. Martin Burger Total Variation 4 Cetraro, September 2008 Variational Methods and their Analysis Two simplifying assumptions: -Noise is Gaussian (variance can be incorporated into ) - A is linear ´ Y Hilbert space

5. Martin Burger Total Variation 5 Cetraro, September 2008 TV Regularization Under the above assumptions we have

6. Martin Burger Total Variation 6 Cetraro, September 2008 Mean Value Technical simplification by eliminating mean value

7. Martin Burger Total Variation 7 Cetraro, September 2008 Mean Value Eliminate mean value Hence, minimum is attained among those functions with mean value c

8. Martin Burger Total Variation 8 Cetraro, September 2008 Mean Value We can minimize a-priori over the mean value and restrict the image to mean value zero W.r.o.g.

9. Martin Burger Total Variation 9 Cetraro, September 2008 Structure of BV 0 Equivalent norm

10. Martin Burger Total Variation 10 Cetraro, September 2008 Poincare-Inequality Proof. Assume does not hold. Then for each natural number n there is such that

11. Martin Burger Total Variation 11 Cetraro, September 2008 Poincare-Inequality Proof (ctd).

12. Martin Burger Total Variation 12 Cetraro, September 2008 Poincare-Inequality Proof (ctd).

13. Martin Burger Total Variation 13 Cetraro, September 2008 Dual Space Property Define

14. Martin Burger Total Variation 14 Cetraro, September 2008 Dual Space Property

15. Martin Burger Total Variation 15 Cetraro, September 2008 Dual Space Property

16. Martin Burger Total Variation 16 Cetraro, September 2008 Dual Space Property

17. Martin Burger Total Variation 17 Cetraro, September 2008 Dual Space Property

18. Martin Burger Total Variation 18 Cetraro, September 2008 Existence Basic ingredients of an existence proof are -Sequential lower semicontinuity - Compactness

19. Martin Burger Total Variation 19 Cetraro, September 2008 Existence What is the correct topology ?

20. Martin Burger Total Variation 20 Cetraro, September 2008 Lower Semicontinuity Compactness follows in the weak* topology. Lower semicontinuity ?

21. Martin Burger Total Variation 21 Cetraro, September 2008 Lower Semicontinuity

22. Martin Burger Total Variation 22 Cetraro, September 2008 Lower Semicontinuity

23. Martin Burger Total Variation 23 Cetraro, September 2008 Lower Semicontinuity First term: analogous proof implies

24. Martin Burger Total Variation 24 Cetraro, September 2008 Existence Theorem: Let J be sequentially lower semicontinuous and be compact. Then there exists a minimum of J Proof.

25. Martin Burger Total Variation 25 Cetraro, September 2008 Existence Proof (ctd). Due to compactness, there exists a subsequence, again denoted by such that By lower semicontinuity Hence, u is a minimizer

26. Martin Burger Total Variation 26 Cetraro, September 2008 Uniqueness Since the total variation is not strictly convex and definitely will not enforce uniqueness, the data term should do Proof: