スライド表紙.indd

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1 MINAKA Nobuhiro

6 Ernst Haeckel

10 D'Arcy Wentworth ompson

12 On Growth and Form (1917) Proc. R. Soc. Edinburgh (1915)

16 (geometric morphometrics)

17 Fred Bookstein = landmark-based method, deformation analysis, linear statistics,... David Kendall = landmark-based method, Riemannian manifold, nonlinear statistics,...

19 (landmark)

21 Landmark-based morphometrics

22 Z 3 Z 4 Z 1 = 4 4 i= 1 Z i Z 1 Z 2

26 size-and-shape size shape

27 km 図形空間 (figure space) 重心への変位 km m 前形態空間 (pre-form space) km m m ( m 1) 2 形態空間 (form space) 回転による整列重心サイズによるスケーリング km m 1 前形状空間 (pre-shape space) 重心サイズによるスケーリング回転による整列 Kendall 形状空間 (Kendall's shape space) km m m ( m 1) 2 1

28 k zi (i=1, 2,..., k) z zi C 1 i=1, 2,..., k z z' (z z' C k )

29 z centroid = (1/k) [i] zi z z' = z z centroid [i] z ' i = 0

30 S 2 = z'z' * =(z z centroid )(z z centroid ) * * S z' z''= z'/s z''z'' * = 1

31 km 図形空間 (figure space) km m 前形態空間 (pre-form space) km m m ( m 1) 2 形態空間 (form space) 回転による整列重心への変位重心サイズによるスケーリング km m 1 前形状空間 (pre-shape space) 重心サイズによるスケーリング回転による整列 Kendall 形状空間 (Kendall's shape space) km m m ( m 1) 2 1

32 compact Lie group

34 Riemannian submersion

35 David Kendall (1984) Bull. London Math. Soc.

38 Kendall 形状空間 ( 超球 ) 中心接点接部分空間 ( 接平面 )

39 Homo sapiens x 2 x 3 x 2 x 3 x 1 x 1 x 2 x 3 x 1 x 2 x 3

40 Homo Australopithecus Neanderthal Chimpanzee

41 x 3 Homo R 3 x 2 Neanderthal Australopithecus Chimpanzee x 1 figure space

42 x 3 Homo x 2 Neanderthal Australopithecus Chimpanzee x 1

43 x 3 Homo x 2 Neanderthal Australopithecus Chimpanzee x 1

44 x 3 x 2 Homo Neanderthal Australopithecus Chimpanzee x 1 pre-form space

45 x 2 x 3 Homo pre-shape space Neanderthal Australopithecus 半径 1 Chimpanzee x 1 pre-form space

46 x 3 x 2 Homo pre-shape space Neanderthal x Australopithecus Chimpanzee pre-form space x 1

47 x 3 x 2 Homo pre-shape space Neanderthal x Australopithecus 半径 1/2 shape space Chimpanzee pre-form space x 1

48 x 3 x 2 tangent subspace Neanderthal Homo shape space x Australopithecus Chimpanzee x 1

49 pre-shape space tangent subspace

50 MINAKA Nobuhiro TEL / FAX mailto:minaka@affrc.go.jp

53 Fred Bookstein (1989) IEEE Trans. Pattern Anal. Machine Intell.

55 y 1 x y x -1

57 m

66 (a) N. americanum (b) N. muelleri

67 (a) アフィン変形 (b) 非アフィン変形 (c) 部分歪み 1 (d) 部分歪み 2

73 Kendall 形状空間 ( 超球 ) 中心接点接部分空間 ( 接平面 )

76 Directional statistics Spherical statistics Circular statistics Ian L. Dryden & K. V. Mardia 1998 K. V. Mardia 1972 Shape statistics

77 CN k (m ) f(z)={ k } -1 exp -(z - m) * -1 (z - m)}, z C k C k ={z=(z 1, z 2,...,z k ) T } Bingham CB k-1 (A) f(z)=c(a) -1 exp(z * Az), z CS k-1 CS k-1 ={z=(z 1, z 2,...,z k ) T : zz * =1}

78 J. T. Kent 1994 If w CN k (0 ), then w ww*=1 CB k-1 (- -1 ). Pre-form Pre-shape k=3 ww*=1 Bingham

82 All these people are asking questions about shape. It is not appropriate, however, to think of shapes as points in a Euclidean space. They are odd creatures, and live in peculiar and quite particular spaces most of which occur in no other context. Thus what is required is a revised version of multidimensional statistics that takes the nature of the shape space fully into account. Huiling Le & David G. Kendall (1993) The Annals of Statistics, 21(3), p