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Summary and outlook

Summary and outlook. 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

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Summary and outlook

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  1. Summary and outlook

  2. 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. 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. 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. 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. More open questions • Where are the cells and how large are they?

  7. And more… • Trace the spot! • See http://cellix.imba.oeaw.ac.at/video_tour_19.html

  8. 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. 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. 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!

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