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Cut Loci in the Blink of an Eye

Discover the mathematical significance of visual structures in nature, focusing on cut loci and skeletal objects computed by our eyes. This exploration delves into the intersection of biology, physics, and mathematics through the study of skeletons and their computational algorithms. Can we uncover new mathematical objects by identifying the skeletal patterns perceived by our eyes? Delve into the world of skeletal computations and their implications for mathematical study.

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Cut Loci in the Blink of an Eye

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  1. Cut Loci in the Blink of an Eye • Robert Sinclair • University of the Ryukyus • sinclair@math.u-ryukyu.ac.jp

  2. Where are the trees? Ryoanji garden, Kyoto G.J. van Tonder, M.J. Lyons & Y. Ejima. Visual Structure of a Japanese Zen Garden, Nature, 419:359-360 (2002).

  3. Building plan (from 1681, white) Traditional viewing point (red circle)

  4. What the Monkey saw

  5. Homogeneousinterior b Boundary a Boundary c Medial axis responseexcited by a,b,c

  6. ?!?

  7. What is the point of all this? • It seems that our eyes can compute some skeleton-like object. • Since we know that we can see well, this object must be well-behaved. • In computer vision, the fact that skeletons are not well-behaved is a well-known problem. • If we can identify the biological object and/or algorithm, we will solve this problem.

  8. The problem with skeletons

  9. Why this could be interesting for a pure mathematician • Traditionally, the needs of physics have driven much development in mathematics.( derivatives, distributions ) • What about biology? • If we can identify the skeleton-like object apparently “computed” by our eyes, then we may have a new object for mathematical study.

  10. Some skeletons • The cut locus from a point p in a Riemannian manifold is the closure of the set of points with more than one minimizer to p.

  11. (Blum’s) Medial Axis The medial axis of an object is the locus of the centres of maximal discs included in the shape [ D. Attali & A. Montanvert ]

  12. A Grass Fire [ Attali, Boissonnat & Edelsbrunner ]

  13. Piccaso’s Rite of Spring... [ Lee, Mumford, Romero & Lamme ]

  14. Algorithms • There are many ways to compute these various skeletons, but it is not clear if our “standard” methods include the biological algorithm. • This is a motivation to develop as complete a set of algorithms as possible. • The “standard” algorithms are based on deterministic geometry (differential, polyhedral), but what about a stochastic approach?

  15. Heat flow on a circle

  16. “Stochastic Analysis on Manifolds”, E.P. Hsu AMS Graduate Studies in Mathematics, Volume 38, 2002

  17. red: green:

  18. The ratio as a function of time

  19. A skew torus

  20. The ratio on a skew torus

  21. The short-time limit of the ratio is integer-valued

  22. The medial axis of an ellipsoid

  23. A problem - conjugate points t = 0.001

  24. The 2-sphere

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