Cut Loci in the Blink of an Eye

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Cut Loci in the Blink of an Eye

• Robert Sinclair
• University of the Ryukyus
• sinclair_at_math.u-ryukyu.ac.jp

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Where are the trees?
Ryoanji garden, Kyoto G.J. van Tonder, M.J.
Lyons Y. Ejima. Visual Structure of a Japanese
Zen Garden, Nature, 419359-360 (2002).
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Building plan (from 1681, white) Traditional
viewing point (red circle)
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What the Monkey saw
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Homogeneousinterior b
Boundary a
Boundary c
Medial axis responseexcited by a,b,c
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?!?
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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.

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The problem with skeletons
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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.

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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.

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(Blums) 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
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A Grass Fire
Attali, Boissonnat Edelsbrunner
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Piccasos Rite of Spring...
Lee, Mumford, Romero Lamme
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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?

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Heat flow on a circle
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Stochastic Analysis on Manifolds, E.P. Hsu AMS
Graduate Studies in Mathematics, Volume 38, 2002
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red
green
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The ratio as a function of time
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A skew torus
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The ratio on a skew torus
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The short-time limit of the ratio is
integer-valued
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The medial axis of an ellipsoid
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A problem - conjugate points
t 0.001
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The 2-sphere
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