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