Announcements

1
Announcements

• Mid-term given out the week after next.
• Send powerpoint to me after presentation.

2
Thompson Comparison of Related Forms
3
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 -gt
  species are related. Many compelling examples.
• Morph between species, predict intermediate
  species.
• Can predict missing parts of skeleton.

4
Math is helpful for morphology

• Seems pretty obvious.
• This was a radical view in biology.

5
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.

6
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.

7
Cannon-bone of ox, sheep, giraffe
Simplest, subset of affine
8
Piecewise affine
9
Logarithmically varying eg., tapirs toes
10
Smooth amphipods (a kind of crustacean).
11
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).

12
No growth in one direction (as in a leaf on a
stem), growth increases in directions away from
this so r sin(q/2).
13
Asymmetric amounts of growth on two sides.
14
Related Species
15

• Lack of transformation -gt no straight line of
  descent.

16
Invention of Morphing?

• Given transformation between species, linearly
  interpolate intermediate transformations.
• Intermediate morphs predict intermediate species.

17
Pages 1070-71
18
Figure 537
19
Pages 1078-79
20
Conclusions

• Stress on homologies.
• Shape comparison through non-trivial
  transformations.
• Simplicity of transformation -gt similarity of
  shape.
• What is the simplest transformation? How do we
  find it?
• Transformation may leave some deviations, how are
  these handled?