Shape spaces. Shape usually defined explicitly to be residual: independent of size.
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1 Shape spaces Before define a shape space, must define shape. Numerous definitions of shape in relation to size. Many definitions of size (which we will explicitly define later). Shape usually defined explicitly to be residual: independent of size. All sets of size/shape definitions are models: neither right nor wrong, just more or less useful. Kendall s (1977) definition of shape is relevant to geometric morphometric methods: Shape is all the geometrical information that remains when location, scale, and rotational effects are filtered out from an object. Based directly on Procrustes superimposition. Defines scale as isometric size variation.
2 To understand shape spaces as used in geometric morphometrics, must consider several basic kinds of geometric spaces: Landmark space (=figure space): Space in which landmarks are digitized and plotted. Euclidean. Configuration space (=form space): Space in which landmark configurations are represented by single points. Euclidean. Shape space: Space in which configurations are plotted after scaling, translation, and rotation. Non-Euclidean. Tangent space: Euclidean approximation of a shape space. Non-Euclidean distance
3 Landmark space and configuration space Given a configuration of k landmarks in m dimensions: Landmark space is: 2D Geometric space of m dimensions. Each dimension corresponds to one spatial dimension of the point configuration. A landmark is a single point within the space. The centroid of each individual is the mean landmark position. The centroid is a single point within the space. Configuration space is: Geometric space of km (=kk m) dimensions. i Set of all possible km matrices. A configuration (i.e., an individual specimen) is a single point within the space. 32D
4 So, each landmark configuration (entire set of landmarks) is a point within the configuration space. Configurations may differ in location, orientation, or scale (or combinations of these). Ex: each column of triangles has the same shape. But plots in a different position of 6D configuration space.
5 Consider scale: Centroid size: k m 2 ij j for landmarks and dimensions CS x x k m i 1 j 1 Scale configuration to have a centroid size = 1. k m 2 ij j 1 CS x x i 1 j 1 k m i 1 j 1 x ij 2 x 1 because 1 1 j
6 Call dimensions of coordinates X and Y: 2 2 Then X Y 1 This is the equation of a unit circle in 2 dimensions. Radius = 1. Centered on the origin (0,0). Thus, if centroid size scaled to value of 1, the centroid is a single point lying on a unit circle. Position of point on circle represents shape of configuration. = Configuration fg space. 1 unit
7 For 3D, call dimensions of coordinates X, Y, and Z: Then X Y Z 1 This is the equation of a unit sphere in 3 dimensions. Radius = 1. Centered on the origin (0,0). Thus, if centroid size is scaled to value of 1, then the centroid is a single point lying on a unit sphere. = Configuration space.
8 Shape spaces Procrustes superimposition: iti Alters two or more configurations so as to remove differences in scale, position, and orientation. Configurations then occupy almost identical positions within the configuration space. = Shape space. Has fewer dimensions than original configuration space: 1 dimension lost in setting common scale. } Pre-shape space m dimensions i lost in translating ti to common centers. m(m-1)/2 dimensions lost in rotation to common orientation. Thus shape space has for k landmarks has: 2k-4 dimensions for 2D landmarks. 3k-7 dimensions for 3D landmarks.
9 Problem with shape space: Procrustes distance between two forms (landmark configurations) is distance along the surface of the shape space. But Procrustes distance isn t the shortest distance between two forms. Shorter distance is length of chord, = partial Procrustes distance (D P ). Shortest distance found by relaxing size constraint on one form, reducing to smaller centroid size (to position B). = Full Procrustes distance (D F ), = angle. Reference form 1 Procrustes distance Target form 2
10 Kendall s shape space, for triangles: Configurations a scaled to centroid size = 1. Distances between forms along surface shortened to full Procrustes distance rather than regular Procrustes distance. Thus, in Kendall s shape space: Individual triangles are single points on the 2k-4 = 2 dimensional surface of a 3D sphere. Sphere has a diameter (not radius) = 1.
11 Kendall s shape space for triangles: North pole represents an equilateral triangle. South pole represents reflection of the triangle. Thus, only need consider the northern hemisphere.
12 Ex: gorilla scapulas (Slice 2005): Triangles superimposed using least-squares squares Procrustes method (LSTRA). Similar forms tend to cluster together on surface of shape space.
13 Kendall s shape space is defined only for triangles. Principles of shape space extend to >3 landmarks. Extensive body of theoretical results (Small, 1996). Details messy. But: near reference configurations, properties of general shape space are similar il to those of Kendall s shape space.
14 Tangent space Problem: shape space is curved (non-euclidean). All conventional multivariate methods assume linear (Euclidean) relationships among forms. Solution: map locations in Kendall s shape space to corresponding locations in a linear Euclidean space. Analogous to 2D maps of the 3D earth s surface. Some distortion: Minimal for sets of similar forms. Highly dissimilar forms must be analyzed directly in Kendall s shape space.
15 Estimation of tangent space: Plane (or hyperplane) that is tangent to Kendall s hyperspherical shape space at a single point. Points on hypersphere h are then projected onto tangent plane. Major issues: At which point on hypersphere is tangent plane determined? How should points be projected? E.g., many possible, and very different, projects of earth s surface onto a 2D map. earth s surface onto a 2D map. Several different approaches.
16 Most common approach in morphometrics: Project to tangent space from reference form, or consensus of a set of superimposed forms. Project from center of Procrustes shape space, = south pole of Kendall s shape space. Projection represents equivalent positions within both space spaces simultaneously. l Distortion in projected positions in tangent space increases away from reference point. Kendall s shape space Procrustes shape space (Slice, 2001)
17 Pairwise distances among forms Pairwise distances between landmark configurations: Procrustes distance calculated as a result of superimposition: i i Measures the sum of squared distances between corresponding landmarks in two forms. Not an appropriate distance for multivariate analyses. Nonlinearly overestimates (i.e., distorts) distances between forms. Appropriate pairwise distances for multivariate analysis is full Procrustes distance. = Distance along curved surface in Kendall s shape space. Estimated by distances in tangent space. Used for cluster analysis, PCA, DFA, etc.
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