1. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at Summary and outlook

2. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 2/10 Gaussians - Scale matters What we observe is filtered Gaussian filter is plausible it is uncommitted Mathematically it relates to regularization Convolve the discrete data with a Gaussian Holds for obtaining derivatives You obtain a continuous function Consider all scales! Scales cannot be taken too large or too small Equivalent to solving the heat equation –“everything blurs away” The multi-scale structure contains information

3. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 3/10 Geometry Build up with isophotes Directional (gauge) derivates are the relevant derivatives They are invariant combinations of Cartesian derivatives Higher order derivates express image structure –Gradient magnitude –Ridges –Edges

4. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 4/10 Nonlinear PDEs Steer the evolution Add image information Define some energy functional Take the variational derivative Use gauge coordinates! Need to choose a parameter –Set a stopping time – scale –Get the noise variance Existence / Uniqueness / convergence not always clear

5. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 5/10 Open questions What are the primary directions of the hairs (called filaments) at every point within the lamellipodium. How many filaments" show into each direction.

6. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 6/10 More open questions Where are the cells and how large are they?

7. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 7/10 And more… Trace the spot!Trace the spot See http://cellix.imba.oeaw.ac.at/video_tour_19.html

8. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 8/10 Even more! Implementation of the evolution by combinations of Lww and Lvv and tracing the critical points. Degenerated scale space saddles and iso manifolds through them

9. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 9/10 outlook Implementation with level sets: –Course 323.022 in the next semester Opportunity to see more of what’s going on in mathematical, industrial, and medical image analysis & processing: >12 invited talks in Linz 28 February – 2 March Details: http://www.flll.jku.at/IEEEembs/Workshop2007.htm

10. Johann Radon Institute for Computational and Applied Mathematics: www.ricam.oeaw.ac.at 10/10 exam Make an appointment We talk 30-45 minutes about –The literature mentioned on the web site –The presentation You can steer the discussion by talking about the problems mentioned in the “list of questions” The more problems solved, the more you steer! Good luck and thanks for participating!