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Multi-Scale Gaussian Filtering in Image Analysis: Insights and Open Questions

This presentation explores the application of Gaussian filtering in image analysis, emphasizing the importance of scale in observing filtered data. It discusses the mathematical underpinnings related to regularization, convolving discrete data with Gaussian functions, and obtaining derivatives to reveal continuous functions. Key concepts include the multi-scale structure of images, directional derivatives, and the relationship to nonlinear partial differential equations. Open questions related to cellular structures and the behavior of filaments within lamellipodia are also examined, highlighting challenges in convergence and uniqueness.

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Multi-Scale Gaussian Filtering in Image Analysis: Insights and Open Questions

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Presentation Transcript

  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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