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http://www.stat.sc.edu/~dryden/course/ild-ch-04.pdf Statistical Shape Analysis Ian Dryden (University of Nottingham) Session I Dryden and Mardia (1998, chapters 1,2,3,4) ✁ Introduction Ian.Dryden@Nottingham.ac.uk � ild ✁ Motivation and applications http://www.maths.nott.ac.uk/ ✁ Size and shape coordinates ✁ Shape space ✁ Shape distances. 3e cycle romand de statistique et probabilit´ es appliqu´ es Les Diablerets, Switzerland, March 7-10, 2004. 1 2 An object’s shape is invariant under the similarity trans- formations of translation, scaling and rotation. In a wide variety of applications we wish to study the geometrical properties of objects. We wish to measure, describe and compare the size and shapes of objects Shape: location, rotation and scale information (simi- larity transformations) can be removed. [Kendall, 1984] � � Size-and-shape: location, rotation (rigid body trans- formations) can be removed. Two mouse second thoracic vertebra (T2 bone) out- lines with the same shape. 3 4

✁ Landmark: point of correspondence on each object that matches between and within populations. Different types: anatomical (biological), mathematical, pseudo, quasi From Galileo (1638) illustrating the differences in shapes of the bones of small and large animals. 5 6 ✁ Bookstein (1991) Type I landmarks (joins of tissues/bones) Type II landmarks (local properties such as maximal curvatures) Type III landmarks (extremal points or constructed land- marks) ✁ Labelled or un-labelled configurations T2 mouse vertebra with six mathematical landmarks (line junctions) and 54 pseudo-landmarks. 7 8

1 3 A 2 B 3 2 1 3 1 C Traditional methods 2 D 3 1 - ratios of distances between landmarks or angles sub- 2 2 mitted to multivariate analysis 3 3 E - the full geometry usually if often lost F 1 1 2 - collinear points? Six labelled triangles: A, B have the same size and - interpretation of shape differences in multivariate space? shape; C has the same shape as A, B (but larger size); D has a different shape but its labels can be permuted to give the same shape as A, B, C; triangle E can be reflected to have the same shape as D; triangle F has a different shape from A,B,C,D,E. 9 10 Geometrical shape analysis Rather than working with quantities derived from or- ganisms one works with the complete geometrical ob- ✁ Pioneers: Fred Bookstein and David Kendall ject itself (up to similarity transformations). In the spirit of D’Arcy Thompson (1917) who consid- Summaries of the field are given by Bookstein (1991, ered the geometric transformations of one species to Cambridge), Small (1996, Springer), Dryden and Mar- another dia (1998, Wiley), Kendall et al (1999, Wiley), Lele and Richstmeier (2001, Chapman and Hall). We conside a shape space obtained directly from the landmark coordinates, which retains the geometry of a point configuration at all stages. 11 12

The map of 52 megalithic sites (+) that form the ‘Old MR brain scan Stones of Land’s End’ in Cornwall (from Stoyan et al., 1995). 13 14 0.4 b 12 13 11 0.2 n 10 Braincase 9 l 8 0.0 7 6 na 5 Face -0.2 pr 4 1 3 2 st -0.4 ba o -0.4 -0.2 0.0 0.2 0.4 Ape cranium Handwritten digit 3 15 16

✂ 250 200 150 • • S S 100 • S S S • • S • • S S S • • 50 (a)� (b)� 0 0 50 100 150 200 250 Electrophoretic gel matching Face recognition 17 18 Proton density weighted MR image Cortical surface extracted from MR scan 19 20

✫ ☛ ✭ ✆ ✫ ✟ ✪ ☛ ✯ ★ ✮ ✩ ✪ ✆ ★ ✚ ✥ ✟ ✆ ✛ ☛ ✛ ☛ ✧ ★ ✩ ✪ ☞ ✧ ✩ ✫ ✪ ✚ ✆ ★ ✫ ✬ ✭ ✆ ✰ ✬ ✟ ✄ ✰ ✮ ✄ ✝ ✛ ✱ ✆ ✝ ✆ ✚ ✤ ✟ ✛ ✵ ✪ ✮ ✬ ✲ ✱ ✆ ✄ ✱ ☛ ✱ ✲ ☛ ✤ ✤ ✆ ✟ ✦ ✕ ✳ ✴ ✍ ✓ ✚ ✦ ✤ ✆ ✔ ✏ ✒ ✓ ✑ ✍ ✓ ✓ ✑ ✕ ✖ ✠ ✡ ✆ ✝ ✍ ✏ ✡ ☎ ✶ ✄ ☎ ✆ ✄ ✝ ✞ ✎ ✟ ✠ ✡ ☛ ☞ ✌ ✍ ✠ ☞ ✗ ✤ ✛ ✝ ✣ ✢ ✚ ✆ ✪ ✣ ✘ ✚ ✆ ✛ ✟ ✤ ✥ ✆ ✟ ✚ ✜ ✘ ✙ ✚ ☎ ✛ ✢ ✚ ✚ ✢ ✆ OUR FOCUS: landmarks in real dimensions is a matrix ( ) Invariance with respect to Euclidean similarity group (translation, scale and rotation) = Size.... Any positive real valued function ✛ such that ✣ . ✛ for a positive scalar 203 Pseudo-landmarks on the cortical surface of the brain 21 22 ✁ Centroid size: An alternative size measure is the baseline size , i.e. the length between landmarks 1 and 2: ✫ and where This was used as early as 1907 by Galton for normal- izing faces. Other size measures: square root of area, cube root ✛ - Euclidean norm, of volume ✄ identity matrix, ☛ - ☛ - ✱ vector of ones. 23 24

❙ ✸ ✚ ❅ ✟ ❆ ✸ ✶ ✶ ✶ ✄ ❄ ✛ ● ❍ ● ▼ ▲ ❑ ■ ✸ ✶ ❏ ✾ ❉ ❈ ❖ ◆ ✼ ✽ ✫ ✟ ✿ ❃ ❀ ✾ ❁ ✾ ❂ ❀ ✾ ❁ ✬ ❍ ❀ ❊ ❯ ✾ ❋ ❊ ✗ ✮ ❚ ✪ ❀ ❉ ❀ ❈ ✩ ❉ ❈ ❇ ❇ ❈ ❉ ● ❀ ❉ ❋ ❏ ❋ ■ ❍ ❏ ❀ ● ❏ ✗ ✾ ■ ❍ ✾ ❀ ❉ ✾ ❋ ■ ❍ ✩ ❋ ❖ ● ✷ ☛ ✹ ✺ ✁ ❍ ✾ ❀ ✾ ✶ ❋ ✩ ● ❍ ● ▼ ❚ ☛ ✸ ✶ ◗ ❋ ❘ ● ■ ❍ ❏ ❀ ● ❏ ✗ ✶ ● ■ ✷ ✪ ✸ ✷ ✮ ✸ ◆ ◗ ✾ ❏ ❏ ❀ ● ❏ ❋ ❀ ■ ❍ ❀ ■ ❉ ❏ ❋ ■ ❍ ✾ ❀ ● ❍ ❋ ◗ ● ✻ ◆ P ✩ ✫ ❯ ◆ ● ❍ ▼ ✾ ▲ ❑ ■ ❍ ✾ ✻ ❀ ❉ ◆ Landmarks: Shape coordinates: Bookstein shape coordinates (1984,1986) (For two Fixed coordinate system dimensional data) vs • 100 200 100 200 • • • • Local Coordinate system • • • • • Im(z) Im(z) • 0 0 • • -200 -100 -200 -100 • Are angles appropriate.....?? -200 -100 0 100 200 -200 -100 0 100 200 Re(z) Re(z) 3 1.0 100 200 2 • 0.5 • • • Im(z) Im(z) • • • • • 0 1 -200 -100 • 0.0 • • • 1 2 • -0.5 -200 -100 0 100 200 -0.5 0.0 0.5 3 Re(z) Re(z) 1 2 3 Shape: 25 26 3 20 • 20 • 3 10 2 10 • • • 1 • 0 0 2 • • 1 -20 -10 -20 -10 -20 -10 0 10 20 -20 -10 0 10 20 (a) (b) 1.0 20 3 10 0.5 • 3 • • • • 0 1 2 0.0 -20 -10 • • • 1 2 -0.5 -20 -10 0 10 20 -0.5 0.0 0.5 (c) (d) In real co-ordinates: The outline of a microfossil with three landmarks (from where , and Bookstein, 1986). . 27 28

❲ ❜ ❱ ✐ ✸ ✽ ❤ ✚ ❛ ❝ ✽ ❬ ❭ ❪ ❞ ❨ ❲ ❜ ❢ ❵ ❣ ❡ ❲ ❩ ❬ ❬ ❪ ❫ ❳ ❭ ❴ 0.8 0.32 0.36 0.40 0.44 100 • 9.2 • • • • 88 • 87 • • • 9.0 88 • 88 • • • • • • 84 • • • 0.7 • • • • • • 8.8 100 • slog • • • • • 8.6 • 92 • 65 • 75 • 84 • • • • • • • 76 • 8.4 • • 71 • • • • • • • 74 • 84 • 8.2 0.6 • • V 0.44 71 • • • • • • • 72 • • • • • • • • • • • 67 • • • • • • • • • 0.40 • •• • • • • • 64 • • U • 0.5 0.36 • • 65 • • • • • 0.32 60 • • • • • 0.75 • • •• • 0.4 • • • • • • 0.65 • • • • • • • • 0.2 0.3 0.4 0.5 0.6 • • • • • • • • U V • • 0.55 • • • • • • • • 0.45 • • 8.2 8.4 8.6 8.8 9.0 9.2 0.45 0.55 0.65 0.75 A scatter plot of (U+1/2) for the Bookstein shape vari- ables for some microfossil data. (Bookstein, 1986) 29 30 2 4 0.5 1 E 5 3 VB A O B 0 0.0 v 1 2 F -1 6 -0.5 -2 -2 -1 0 1 2 B U -0.5 0.0 0.5 u The shape space of triangles, using Bookstein’s co- A scatter plot of the Bookstein shape variables for the ordinates ✛ . All triangles could be relabelled T2 mouse data. and reflected to lie in the shaded region. 31 32