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Statistical Shape Analysis Ian Dryden (University of Nottingham) 3E Cycle

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Statistical Shape Analysis Ian Dryden (University of Nottingham) 3E Cycle Statistical Shape Analysis Ian Dryden (University of Nottingham) Session I Dryden and Mardia (1998, chapters 1,2,3,4) [email protected] ¡ Introduction ¡ http://www.maths.nott.ac.uk/ ild Motivation and applications ¡ Size and shape coordinates ¡ Shape space 3e cycle romand de statistique et probabilites´ ¡ Shape distances. appliques´ Les Diablerets, Switzerland, March 7-10, 2004. 1 2 An object’s shape is invariant under the similarity trans- In a wide variety of applications we wish to study the formations of translation, scaling and rotation. 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) ¡ T2 mouse vertebra with six mathematical landmarks Labelled or un-labelled configurations (line junctions) and 54 pseudo-landmarks. 7 8 1 3 A 2 B 3 2 1 3 1 C 2 D Traditional methods 3 1 2 2 - ratios of distances between landmarks or angles sub- 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- ject itself (up to similarity transformations). ¡ Pioneers: Fred Bookstein and David Kendall 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 MR brain scan The map of 52 megalithic sites (+) that form the ‘Old Stones of Land’s End’ in Cornwall (from Stoyan et al., 1995). 13 14 0.4 b 12 13 11 0.2 10 n 9 Braincase l 7 8 0.0 6 5 na Face 4 pr -0.2 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 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 ¥ 203 Pseudo-landmarks on the cortical surface of the for a positive scalar . brain 21 22 ¡ Centroid size: An alternative size measure is the baseline size, i.e. " " ¥ ¥ ¥ ¥ the length between landmarks 1 and 2: # & ' ( & ¨ ¨ ! ) # $ % & $ % 0 ¥ ¥ ¥ % ' % % ¥ ¥ ¨ 1 ( & # & # $ % * ¨ ) where and ) 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 % 3 3 3 4 5 2 1 1 1 2 Landmarks: 2 Shape coordinates: ) ¡ Bookstein shape coordinates (1984,1986) (For two Fixed coordinate system dimensional data) vs • 6 6 200 • • 200 • • 100 • 100 • • Local Coordinate system • • 0 0 Im(z) Im(z) • • -100 -100 -200 -200 Are angles appropriate.....?? -200 -100 0 100 200 -200 -100 0 100 200 3 Re(z) Re(z) 1.0 200 • • 2 100 • • 0.5 • • 0 • • Im(z) Im(z) 1 • • • 0.0 1 -100 2 • -200 -0.5 -200 -100 0 100 200 -0.5 0.0 0.5 3 Re(z) Re(z) 1 @ £ ' > 3 3 3 2 3 & 8 ¨ 9 : ; 9 < 1 ? ¨ A 1 1 1 Shape: 7 < 9 = ; 9 25 26 3 • 20 20 • 2 3 10 • 10 • 1 • 0 0 2 • 1 -10 -10 -20 -20 -20 -10 0 10 20 -20 -10 0 10 20 (a) (b) 1.0 20 3 10 • 3 0.5 • 0 • • 1 2 • • 0.0 -10 1 2 -20 -0.5 -20 -10 0 10 20 -0.5 0.0 0.5 (c) (d) In real co-ordinates: % $ B C D D D G E F F F J J F J J ; 9 G ; 9 H I 9 ; 9 H I G ; H I ; H I K L M H G N ) $ O C D D D G E F F J J F J J F 9 G ; 9 H I ; H I ; G ; H I 9 ; 9 H I K L M H G N $ P $ & The outline of a microfossil with three landmarks (from where , and G G G R S F F J J M 9 G ; 9 H I G ; H I N Q Q Q N G H Bookstein, 1986). B O C C D D . ; T U U T N 27 28 0.8 W 100• 0.32 0.36 0.40 0.44 • • • • 88• 9.2 87• • • X 88• 88• 9.0 84• • • • • ••• • • • • • 0.7 100• 8.8 slog • • • • • • 92• 65• 75• 84• 8.6 • • • • • • 76• • • 71• • • • • 8.4 74• 84• • • V • • 8.2 V 0.6 71• • • 0.44 • • • • 72• • • • • • • • • • 67• • • • • •• • • • •• • • • 0.40 64• • • • U • 0.5 • • 65• 0.36 • • • • 60• Y • • 0.32 • • • • • • 0.75 • • • • • 0.4 • • • • • • • • • • 0.2 0.3 0.4 0.5 0.6 • • • • • • • 0.65 U • • V • • • • • • 0.55 • • • • 0.45 W 8.2 8.4 8.6 8.8 9.0 9.2 W 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 a 4 0.5 1 d E ` 5 _ 3 g f b B 0 V A O B v \ ] 0.0 1 2 e F -1 ^ 6 -0.5 -2 [ \ ] -2 -1 0b 1 2 c [ [ B -0.5 0.0 0.5 U Z u The shape space of triangles, using Bookstein’s co- h A scatter plot of the Bookstein shape variables for the 3 i 8 ordinates 8 . All triangles could be relabelled T2 mouse data. and reflected to lie in the shaded region. 31 32 ¡ Kendall’s shape sphere (1983) (triangles only) Isosceles triangles 3 Equilateral (North pole) 1 2 θ=0 Unlabelled Right-angled Kendall’s shape coordinates φ=2π/3 φ=5π/3 θ=π/2 φ=0 φ=π/3 % 3 3 % k ¨ l j m ¨ j 1 1 1 j Remove location j Flat triangles - (Equator) ; θ=π 1 2 & % j @ £ 3 3 3 & o p q & 7 n n ¨ ¨ A 1 1 1 1 ; % Reflected equilateral (South pole) j A mapping from Kendall’s shape variables to the sphere Simple 1-1 linear correspondence with Booklstein S.V. is (equ. 2.11 of book) ' s q ' 3 P P 7 n n ) A A ! For triangles Kendall’s SV sends baseline to ! , 3 u 3 , r , r 2 ¨ ¨ j ¨ t s s s o o o ) ) ) , , , s o q P P ¨ 7 n n ) ) and ) , so that % u o o v 2 j ¨ ) ) ) . 33 34 w Kendall’s Bell 3 x Kendall’s spherical shape shape variables are then given by the usual polar coordinates w w w , x 3 u , x 3 , 3 2 ¨ ¨ j ¨ t t t w y y z > y x { where > is the angle of latitude and t z is the angle of longitude. 35 36 Bookstein coordinates - 3D ¥ 3 3 # % # # P # 2 2 2 The Schmidt net for 1/12 sphere Landmarks ¨ - ) w } 3 3 & % & & P & | | t t t 7 ¨ 7 7 7 3 x > y y 3 > y { z - ¨ ¨ ! 1 t ~ ) ¥ ¥ % o @ £ ¥ , ' 3 3 3 & ) ¨ ¨ A 1 1 ¥ ¥ t C ' % • 0.0 A B C C ) • • A BA B C C • C C • A B A B A • AB • B C C ¦ -0.5 A • BC C C AC • B A • • A A AB •AB •AB B where is a rotation matrix C C • • ¥ ¥ ¥ A B C CA B 3 3 P A • B C C • % A • BA • BA B C C A • B A • B (a function of ) and C C A • B C C A • B ) A • B A • B -1.0 A C• B A • CB ¥ ¥ t A C• B A •CB t ' 3 > 3 > 3 3 > 3 > 3 A C • B A •CB A •CB A • CB % r - , r - , A C • B A • CB ) A • C B A • C B -1.5 ¥ 3 3 > P % P P -0.5 0.0 0.5 P 7 ¨ 7 7 - ) @ ¥ > > P P P & & 7 ¨ 7 ¨ where 7 , and for ) £ 3 3 1 1 1 .
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