Announcements Mid-term given out the week after next. Send powerpoint to me after presentation.
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Announcements
• Mid-term given out the week after next.
• Send powerpoint to me after presentation.
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Thompson: Comparison of Related Forms
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Key Points
• Math is helpful for morphology.• Homologous structures necessary: correspondence.• Given these, compute transformations of plane.• Uses:
– Nature of transformation gives clues to forces of growth.– Shapes related by simple transformation -> species are
related. Many compelling examples.– Morph between species, predict intermediate species.– Can predict missing parts of skeleton.
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Math is helpful for morphology
• Seems pretty obvious.
• This was a radical view in biology.
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Homologies
• Had a long tradition– Aristotle: Save only for a difference in the way of
excess or defect, the parts are identical in the case of such animals as are of one and the same genus.
– In biology, study of homologous structures in species preceded and provided background for Darwin.
• Homologous structures explained by God creating different species according to a common plan.
• Ontogeny provided clues to homology.
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Transformations
• Given matching points in two images, we find a transformation of plane.
• Homeomorphism (continuous, one-to-one)• This is underconstrained problem
– Implicitly, seeks simple transformation.– Not well defined here, will be subject of much
future research.– Intuitively pretty clear in examples considered.
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Simplest, subset of affine
Cannon-bone of ox, sheep, giraffe
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Piecewise affine
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Logarithmically varying: eg., tapir’s toes
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Smooth: amphipods (a kind of crustacean).
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Descriptions of shape: Clues to Growth
• Somewhat different topic, shape descriptions relevant even without comparison.– Introduces fourier descriptors.
• Equal growth in all directions leads to circle (or sphere).
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No growth in one direction (as in a leaf on a stem), growth increases in directions away from this so r = sin(.
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Asymmetric amounts of growth on two sides.
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Related Species
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• Lack of transformation -> no straight line of descent.
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Invention of Morphing?
• Given transformation between species, linearly interpolate intermediate transformations.
• Intermediate morphs predict intermediate species.
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Pages 1070-71
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Figure 537
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Pages 1078-79
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Conclusions
• Stress on homologies.• Shape comparison through non-trivial
transformations.• Simplicity of transformation -> similarity of
shape.• What is the simplest transformation? How do
we find it?• Transformation may leave some deviations,
how are these handled?