4.1.1 Procrustes distances

We shall initially describe the partial and full Procrustes distances. Consider two configuration matrices from k points in m dimensions X₁ and X₂ with pre-shapes Z₁ and Z₂. First we minimize over rotations to find the closest Euclidean distance between Z₁ and Z₂.

Definition 4.1 The partial Procrustes distance d_P is obtained by matching the pre-shapes Z₁ and Z₂ of X₁ and X₂ as closely as possible over rotations. So,

where Z_j = HX_j/||HX_j||, j = 1, 2.

Here ‘inf’ denotes the infimum, and we will use ‘sup’ for the supremum.

Result 4.1 The partial Procrustes distance is given by:

(4.1)

where λ₁ ≥ λ₂ ≥ … ≥ λ_{m − 1} ≥ |λ_m| are the square roots of the eigenvalues of Z^T₁Z₂Z^T₂Z₁, and the smallest value λ_m is the negative square root if and only if det(Z^T₁Z₂) < 0.

A proof of Result 4.1 is given after two useful lemmas.

Lemma 4.1

(4.2)

Proof: We follow the proof of Kendall (1984). First, consider a singular value decomposition of Z^T₂Z₁ given by:

(4.3)

where U, V ∈ SO(m), Λ = diag(λ₁, …, λ_m) with λ₁ ≥ λ₂ ≥ … ≥ λ_{m − 1} ≥ |λ_m| and

(4.4)

In this case the sequence of eigenvalues is called ‘optimally signed’ (Kent and Mardia 2001). Hence

(4.5)

where (r₁₁, …, r_mm) are the diagonals of R ∈ SO(m). Now the set of diagonals of R in SO(m) is a compact convex set with extreme points

with an even number of minus signs (Horn 1954). Hence, it is clear in our case that the supremum is achieved when r_ii = 1, i = 1, …, m, and hence Equation (4.2) follows.   □

Lemma 4.2 The optimal rotation is

(4.6)

where U, V are given in Equation (4.3), and the optimal rotation is unique if the singular values satisfy

(4.7)

in which case Kent and Mardia (2001) denote the sequence as ‘non-degenerate’.

Proof: The lemma follows because

where is the Procrustes rotation of Z₁ onto Z₂. For further details see Le (1991b) and Kent and Mardia (2001).   □

Proof (of Result 4.1): Note that

(4.8)

and ||Z₁|| = 1 = ||Z₂||. The supremum of trace(Z^T₂Z₁Γ) over Γ ∈ SO(m) is given by Lemma 4.1, and the optimal rotation is given by Lemma 4.2. Hence

and the result follows.   □

Definition 4.2 The full Procrustes distance between X₁ and X₂ is

(4.9)

where Z_r = HX_r/||HX_r||, r = 1, 2.

Result 4.2 The full Procrustes distance is

(4.10)

Proof: Note that

and ||Z₁|| = 1 = ||Z₂||. The optimization over rotation is given by Lemma 4.1, which is exactly the same as for the partial Procrustes distance. Hence

(4.11)

By differentiation we have

which is clearly a minimum as ∂²(d²_F)/∂β² > 0. Substituting into Equation (4.11) leads to the expression for the full Procrustes distance, as required.   □

We shall often use the full Procrustes distance in the shape space for practical statistical inference. Note that 0 ≤ ∑^m_{i = 1}λ_i ≤ 1 and so

The Procrustes distances are types of extrinsic distances as they are actually distances in an embedding of shape space. Another type of distance is an intrinsic distance defined in the space itself (see Section 3.1), and we introduce this distance in Section 4.1.4.

4.1.2 Procrustes

The term ‘Procrustes’ is used because the above matching operations are identical to those of Procrustes analysis, a commonly used technique for comparing matrices (up to transformations) in multivariate analysis (see Mardia et al. 1979, p. 416). In Procrustes analysis the optimal transformation parameters are estimated by minimizing a least squares criterion. The expression ‘Procrustes analysis’ was first used by Hurley and Cattell (1962) in factor analysis. In Greek mythology Procrustes was the nickname of a robber Damastes, who lived by the road from Eleusis to Athens. He would offer travellers a room for the night and fit them to the bed by stretching them if they were too short or chopping off their limbs if they were too tall. The analogy is rather tenuous but we can regard one configuration as the bed and the other as the person being ‘translated’, ‘rotated’ and possibly ‘rescaled’ so as to fit as close as possible to the bed. Procrustes was eventually captured and killed by Theseus, who fitted Procrustes to his own bed.

4.1.3 Differential geometry

The rotated Z on the pre-shape sphere is called a fibre of the pre-shape space S^k_m. Fibres on the pre-shape sphere correspond one to one with shapes in the shape space, and so we can think of a fibre as representing the shape of a configuration. The pre-shape sphere is partitioned into fibres by the rotation group SO(m) and the fibre is the orbit of Z under the action of SO(m). The fibres do not overlap.

In Figure 4.1 we see a diagrammatic view of the pre-shape sphere. Since the pre-shape sphere is a hypersphere embedded in we can consider the great circle distance between two points on a sphere. The great circle distance is an intrinsic distance, defined as the shortest geodesic distance between any two points on the pre-shape sphere.

Since the shapes of configurations are represented by fibres on the pre-shape sphere, we can define the distance between two shapes as the closest great circle distance between the fibres on the pre-shape sphere. This shape distance is called the Riemannian distance in shape space and is denoted by ρ. The Riemannian distance ρ(X₁, X₂) is the distance inherited from the projection of the fibres on the pre-shape sphere to points in the shape space. The projection from pre-shape sphere to shape space is isometric, because distances are preserved, and is termed a Riemannian submersion. The Riemannian distance is an intrinsic distance in the shape space. The shape space is a type of quotient space, where the rotation has been quotiented out from the pre-shape sphere using optimization. We write Σ^k_m = S_m^k/SO(m) and more formally we say that Σ^k_m is the quotient space of S^k_m under the action of SO(m).

For m > 2 there are singularities in the shape space when the k points lie in an m − 2 dimensional subspace (see Kendall 1989). Here we assume that our shapes are away from any such singularities. If our data are modelled by a continuous probability distribution, then the Lebesgue measure of the singularity set is zero. For m = 2 there is no such problem with singularities.

Further discussion of geometrical properties of the shape space is given by Kendall (1984, 1989); Le and Kendall (1993); Small (1996); Kendall et al. (1999); and Kent and Mardia (2001).

4.1.4 Riemannian distance

In Figure 4.1 two minimum distances have been drawn diagrammatically on the pre-shape sphere between the fibres (shapes): ρ is the closest great circle distance; and d_P is the closest chordal distance. The Riemannian distance is measured in radians.

Definition 4.3 The Riemannian distance ρ(X₁, X₂) is the closest great circle distance between Z₁ and Z₂ on the pre-shape sphere, where Z_j = HX_j/||HX_j||, j = 1, 2. The minimization is carried out over rotations.

Result 4.3 The Riemannian distance ρ is

(4.12)

where the eigenvalues λ_i, i = 1, …, m are defined in Equation (4.3) and Equation (4.4).

Proof: From trigonometry on sections of the pre-shape sphere one can see that the Riemannian distance is related to the partial Procrustes distance as follows:

as

   □

The Riemannian distance is also known as an intrinsic distance, as it is the shortest geodesic distance within the shape space. The partial Procrustes and full Procrustes distances are both extrinsic distances, which are shortest distances in an embedding of the shape space. Hence the extrinsic distances are shortest paths which are typically outside the space for non-Euclidean spaces, whereas the intrinsic distances are shortest paths that remain wholly within the space.

4.1.5 Minimal geodesics in shape space

It of interest to study the shortest path in shape space between two shapes – the minimal geodesic path – the length of which is the Riemannian distance. In Euclidean space the geodesic path is just a straight line, but in a non-Euclidean space the geodesic path is curved. Working with geodesics directly on shape space is complicated (Le 1991a). However, a practical way to proceed is to work with the pre-shape sphere (which is just an ordinary sphere) and then find an isometric (i.e. distance preserving) representation on the pre-shape sphere of a geodesic on the shape space. This representation corresponds to horizontal lifts to the pre-shape sphere of minimal geodesics in the shape space.

A geodesic on the pre-shape sphere is a great circle. Any two orthogonal points p and w on the pre-shape sphere define a unit speed geodesic Γ_{(p, w)} parameterized by:

where and trace(pw^T) = 0. If in addition pw^T = wp^T, then the geodesic is known as a ‘horizontal’ geodesic, which is invariant to rotations of p and w. See Chapter 6 of Kendall et al. (1999) for a full description of geodesics in shape space. The quantity |s| is the great circular arc distance from Γ_{(p, w)}(0) = p to Γ_{(p, w)}(s), and for a horizontal geodesic the length of the great circular arc is equal to the Riemannian distance ρ([p], [w]).

4.1.6 Planar shape

In the planar shape case with m = 2 then the use of complex arithmetic helps to simplify the expressions for distances and geodesics.

Result 4.4 If p and w are two complex (k − 1)-vectors on the pre-shape sphere, then the Riemannian distance is given by:

(4.13)

where p* is the transpose of the complex conjugate of p, and is the modulus of z.

Proof: We have

The minimum is given when the last two terms are real, that is when θ = −Arg(p*w), and hence

which in turn leads to

as required.   □

The horizontal geodesic on the pre-shape sphere is:

(4.14)

with conditions p*p = w*w = 1 and p*w = 0, with the final condition ensuring that the geodesic is horizontal.

The exponential map is a bijection if ||v|| ∈ [0, π/2) and is given by:

where s = ||v||, with conditions p*p = 1 and p*v = 0. The inverse exponential map is:

with w*p > 0.

If we parallel transport a vector along the shortest geodesic γ from z₁ to z₂ the result in is:

where z*₂z₁ > 0. Note that here the parallel transport is a linear isometry (i.e. a linear transformation which preserves distance).

The Riemannian curvature tensor (or Ricci curvature tensor) (Su et al. 2012) for X, Y, Z ∈ T_z(M) is given by:

(4.15)

where and < Y, Z > =Z*Y.

4.1.7 Curvature

Le and Kendall (1993) studied the curvature of the shape space and gave explicit formulae for the sectional curvatures, scalar curvature and an average curvature. The curvatures depend on the singular values of the preshape λ₁, …, λ_m, assuming k > m. Although in general the formulae are complicated, an expression for an average section curvature when k = 4 and m = 3 has a concise form, and is given by:

It is clear that the average curvature is greater than 1. Also, if λ₂ → 0 and hence λ₃ → 0 then the average curvature blows up to infinity, which occurs at each singularity of the shape space.

4.2.1 Relationships

In Figure 4.2 we see a cross-section of the pre-shape sphere illustrating the relationships between d_F, d_P and ρ, where

Note the Riemannian distance ρ can be considered as the smallest angle (with respect to rotations of the pre-shapes) between the vectors corresponding to Z₁ and Z₂ on the pre-shape sphere.

Important point: For shapes which are close together there is very little difference between the shape distances, since

Consequently for many practical datasets with small variability there is very little difference in the analyses when using different Procrustes distances. However, the distinction between the distances is worth making and the terminology is summarized in Table 4.1. We discuss some practical implications of different choices of distance in estimation in Chapter 6.

4.2.2 Shape distances in R

The shape distances can be calculated in R using the command riemdist from the shapes library. We consider three triangles, which are two equilateral triangles (which are reflections of each other) and a collinear triangle.

> a1<-matrix( c(-0.5,0.5,0,0,0,sqrt(3)/2),3,2)
> a2<-matrix( c(-0.5,0.5,0,0,0,-sqrt(3)/2),3,2)
> a3<-matrix( c(-0.5,0.5,0,0,0,0),3,2)
> a1
    [,1] [,2]
[1,] -0.5 0.0000000
[2,] 0.5 0.0000000
[3,] 0.0 0.8660254
> a2
    [,1] [,2]
[1,] -0.5 0.0000000
[2,] 0.5 0.0000000
[3,] 0.0 -0.8660254
> a3
    [,1] [,2]
[1,] -0.5 0
[2,] 0.5 0
[3,] 0.0 0

The Riemannian distances between the three pairs are:

> riemdist(a1,a2)
[1] 1.570796
> riemdist(a1,a3)
[1] 0.7853982
> riemdist(a2,a3)
[1] 0.7853982

and so here the first two triangles are maximally far apart at ρ = π/2. The partial and full Procrustes distances between the last two triangles are:

# partial Procrustes
> 2*sin(riemdist(a2,a3)/2)
[1] 0.7653669
# full Procrustes
> sin(riemdist(a2,a3))
[1] 0.7071068

A useful method of displaying data on Riemannian manifolds is to use classical multidimensional scaling (MDS) with a suitable choice of distance (e.g. Mardia et al. 1979, p. 397). The technique, also known as principal coordinate analysis (Gower 1966), involves representing the data in a low dimensional Euclidean subspace, such that the distances between pairs of observations in the original space are well approximated by Euclidean distances in the lower dimensional subspace. We display the first two principal coordinates of the schizophrenia dataset of Section 1.4.5 using the Riemannian distance and the full Procrustes distance in Figure 4.3. Clearly the plots are extremely similar as the distances are so similar.

data(schizophrenia)
distmat1 <- matrix( 0, 28, 28)
distmat2 <- matrix( 0, 28, 28)
for (i in 1:28){
for (j in 1:28){
rho <- riemdist( schizophrenia$x[,,i],schizophrenia$x[,,j] )
distmat1[i,j] <- rho
distmat2[i,j] <- sin(rho)
}
}
par(mfrow=c(1,2))
plot(cmdscale(distmat1),pch=as.character(schizophrenia$group))
plot(cmdscale(distmat2),pch=as.character(schizophrenia$group))

Also, there are other functions in the shapes library which compute shape distances, for example the routine procGPA includes the Riemannian distances of each observation to an estimated mean shape.

If it is of interest to compare configurations with reflection invariance then the option reflect=TRUE is used. For example, comparing the two equilateral triangles with reflection invariance:

> riemdist(a1,a2,reflect=TRUE)
[1] 1.490116e-08

which is effectively zero, and so the two equilateral triangles have the same reflection shape.

4.2.3 Further discussion

The minimization over rotations (and reflections) in the Procrustes distance calculations could also be obtained using calculus (e.g. see Mardia et al. 1979, p. 416) and we consider this calculation for minimizing over rotations in Section 7.2.1.

We illustrate the geometrical steps involved in obtaining the full and partial Procrustes distance in Figure 4.4. The first row shows the original figure X₁ (left), the centred X₁ (middle) and the rescaled X₁, which we call Z₁ (right). The second row shows the original figure X₂ (left), the centred X₂ (middle) and the rescaled X₂, which we call Z₂ (right). In the third row (left) Z₂ is rotated to Z₁ to minimize the sum of squared distances between pairs of landmarks – the partial Procrustes distance is the Euclidean distance between these fitted configurations. In the third row (middle) Z₂ is rotated and rescaled to Z₁ to minimize the sum of squared distances between pairs of landmarks – the full Procrustes distance is the Euclidean distance between these fitted configurations.

4.3.1 Complex arithmetic

We now consider the case where the k ≥ 3 landmarks are in m = 2 dimensions. In this case complex arithmetic enables us to deal with shape analysis very effectively. Consider k ≥ 3 landmarks in , z^o = (z^o₁, …, z_k^o)^T which are not all coincident, where z^o_j = x_j^o + iy^o_j, j = 1, …, k, . Location is removed by pre-multiplying by the Helmert submatrix H giving the complex Helmertized landmarks

where H is defined in (2.10). The centroid size is:

(4.16)

where (z^o)* denotes the complex conjugate of the transpose of z^o and C is the centring matrix of (2.3). Hence the complex pre-shape z is obtained by dividing the Helmertized landmarks by the centroid size,

We see that the pre-shape space S^k₂ is the complex sphere in k − 1 complex dimensions:

(4.17)

which is the same as the real sphere of unit radius in 2k − 2 real dimensions, S^{2k − 3}. In order to remove rotation we identify all rotated versions of z with each other, that is the shape of z^o is:

The complex sphere which has points z identified with ze^iθ (0 ≤ θ < 2π) is the complex projective space , as we see later in Result 4.5.

Example 4.1 Consider one particular T2 mouse vertebra in Figure 4.5, which has k = 6 landmarks in m = 2 dimensions, and is viewed at an observed location, scale and rotation. The full dataset has already been described in Section 1.4.1. The original complex coordinates of this vertebra are:

The Helmertized landmarks are:

The centroid size is S(z^o) = 228.85 and so the pre-shape is given by z = z_H/228.85. The shape of z^o is the set {ze^iθ: 0 ≤ θ < 2π}. We can plot the figure by considering a suitable icon. A choice of icon is the centred pre-shape given by:

where C is the centring matrix of Equation (2.3). In Figure 4.5 we see a plot of the centred pre-shape icon, with its centroid at the origin and with unit size – the rotation is unchanged from the original configuration z^o.   □

4.3.2 Complex projective space

Kendall (1984) has shown that the shape space Σ^k₂ with Riemannian metric ρ can be regarded as , the complex projective space with maximal sectional curvature 4, and ρ(X₁, X₂) is the Fubini–Study metric on (4) (see Kobayashi and Nomizu, 1969, Chapter IX).

Result 4.5 In the 2D case the shape space is:

the complex projective space with maximal sectional curvature 4.

Proof: Let the original landmarks be z^o = (z^o₁, …, z_k^o)^T. Consider the Helmertized complex landmarks,

(4.18)

(not allowing complete coincidence of landmarks). Multiplying z_H by some non-zero complex number λ = βe^iθ, , θ ∈ [0, 2π), is the same as rescaling and rotating z_H, and so λz_H has the same shape as z_H. Thus, the shape of z^o is represented by the set:

(4.19)

a complex line through the origin (but not including it) in k − 1 dimensions. The union of all such sets is the complex projective space and the maximal sectional curvature is 4 to make sure the Riemannian distance ρ(X₁, X₂) is the same in both pre-shape and shape space. Full details are given by Kendall (1984) and Stoyan et al. (1995).   □

From Equation (4.13) the Riemannian distance (and hence the partial and full Procrustes distance) is particularly simple to calculate for m = 2 dimensional data when we can use complex notation.

Consider the centred, unit size configurations (centred pre-shapes) y = (y₁, …, y_k)^T and w = (w₁, …, w_k)^T, where ||y|| = 1 = ||w|| and y*1_k = 0 = w*1_k. It follows from Equation (4.13) that

(4.20)

where d_F is the full Procrustes distance between the two shapes. Hence, we can regard

as the modulus of the complex correlation between the y_j and w_j.

For planar data the pre-shape space is a complex sphere of unit radius in k − 1 complex dimensions, defined in Equation (4.17). The angle between the complex pre-shapes y and w is:

(4.21)

This quantity is unaffected by rotations of y and w. Hence, we can explicitly see that the Riemannian distance ρ is the angle between complex pre-shapes y and w. Also, since the radius of the pre-shape sphere is 1 we can consider ρ to be the great circle distance on the pre-shape sphere, using simple geometry.

The squared chordal distance between pre-shapes y and w is:

(4.22)

and d_P is also invariant under rotations of y and w. The partial Procrustes distance d_P can be regarded as the chordal distance between the complex pre-shapes y and w.

Important point: We have a clear analogy with directional data analysis (Mardia and Jupp, 2000). The angle between the directions l and ν on the real sphere is given by θ = arccos(l^Tν), and the chordal distance between l and ν is 2(1 − l^Tν) = 2(1 − cos θ). Hence, from Equation (4.21) and Equation (4.22) the rôle of the cosine of Riemannian distance cos ρ = |y*w| is similar to that of

in directional data analysis, where |cos θ| is useful when axial or bipolar data are available. Hence, one can see that the complex pre-shapes y and w are closely analogous to the axes ± l and ± ν on the real sphere. As we shall see, techniques in shape analysis using the pre-shape sphere often have similarities with techniques in directional data analysis.

4.3.3 Kent’s polar pre-shape coordinates

Kent (1994) proposed some non-standard polar coordinates on the pre-shape sphere for 2D data. Given a point (z₁, …, z_{k − 1})^T on we transform to (s₁, …, s_{k − 2}, θ₁, …, θ_{k − 1}), where

(4.23)

for j = 1, …, k − 1, s_j ≥ 0, 0 ≤ θ_j < 2π and s_{k − 1} = 1 − s₁ − … − s_{k − 2}. A description of this and other polar coordinate systems was given by Shelupsky (1962). The coordinates s₁, …, s_{k − 2} are on the k − 2 dimensional unit simplex, . By identifying the complex pre-shape sphere with we have the volume measure of as:

(4.24)

The total volume is

since the volume of the j-dimensional simplex is 1/j!, j = 1, 2, 3, ….

Important point: This set of coordinates has the advantage that the uniform density on the pre-shape sphere is uniform in these coordinates (see Section 10.1). Hence, the coordinates are particularly useful for distributional results.

Shape coordinates can be obtained by rotating z to a fixed axis. For example, considering the rotation information of the original figure to be in θ_{k − 1}, then the 2k − 4 shape coordinates are:

(4.25)

where ϕ_j = θ_j − θ_{k − 1},  j = 1, …, k − 2. So the volume measure on the shape space is:

(4.26)

and the total volume is:

4.3.4 Triangle case

Working with shape spaces for triangles in two dimensions is particularly simple.

Result (Kendall 1983) The shape space for triangles in a plane is:

the sphere in three dimensions with radius

The proof follows from the result in differential geometry that the complex projective space can be identified with (Kendall 1983).

The radius is required to have length in order to ensure that the Procrustes (smallest great circle) distance between two fibres on the pre-shape sphere is equal to the great circle distance between corresponding shape points on the shape sphere. This important result enables us to work with the shapes of triangles as points on . Hence, statistical shape analysis for triangles in a plane is equivalent to directional data analysis on a sphere in with radius . Mardia (1989b) explored this connection statistically after identification of the spherical shape space by Kendall (1983). Mardia (1989b) gave an explicit expression for the transformation from Bookstein or Kendall coordinates to the sphere [see Equation (2.15)]. For an alternative simple proof that the triangle shape space is see Green (1995) and Mardia (1996b).

Result 4.7 The shape space for triangles in m ≥ 3 dimensions is:

the hemisphere in three dimensions with radius

Proof: If k = 3 and m ≥ 3, then, since in this case the triangle shapes are invariant under reflection (see Section 3.5.2), there is a one to one correspondence between shapes in the upper and lower hemispheres of the shape space (they are reflections and hence 3D rotations of each other), and so , the hemisphere in three dimensions of radius .   □

We now illustrate the various distances for the triangle case. In Figure 4.6 we have a section of the shape space sphere (with radius half) for triangles in m = 2 dimensions. In this case, we see that the chordal distance is given by:

which is the full Procrustes distance and note that ρ is measured in radians. The partial Procrustes distance d_P between two shapes [z^o] and [w^o] on is the sum of the Euclidean chord lengths from each shape point to half way along the great circle route from [z^o] to [w^o], see Figure 4.6.

Important point: Note that ρ is the great circle distance, but because the radius of the sphere is the angle between [z^o] and [w^o] is 2ρ.

Further discussion of coordinates for the triangle case is given in Section 2.6.2.

A measure of shape distance, such as the full Procrustes distance, gives us a numerical measure of overall shape comparison. However, a global shape measure such as shape distance does not indicate locally where the objects differ and the manner of the difference. The deformation methods of Chapter 12 are particularly useful for describing such differences.

4.4.1 Tangent spaces

The tangent space to shape space is the linearized version of the shape space in the vicinity of a particular point of shape space (the pole of the tangent projection). The pole is usually chosen to be an average shape obtained from the dataset of interest, and hence this choice of coordinates depends on the dataset under study. It turns out that working with the tangent space to shape space directly is complicated (Le 1991a) so instead we consider a tangent space to the pre-shape sphere, and identify the ‘horizontal part’ which is invariant to rotation. The Euclidean distance in the tangent space is a good approximation to the Procrustes distances d_F, d_P and ρ in shape space in the vicinity of the pole. In fact we will show that the Euclidean distance in the tangent space to the pole of the projection is exactly the same as the full Procrustes distance to the pole. So, if most of the objects in a dataset are quite close in shape, then using the Euclidean distance in the tangent space will be a good approximation to the shape distances in the shape space. Hence, for practical shape analysis we shall see that the tangent space can be extremely important and useful.

For illustration purposes we first discuss the tangent space to a real sphere, and then concentrate on tangent spaces to the pre-shape sphere which has an added layer of complication.

Example 4.2 Consider a real sphere . Let p ∈ S^{q − 1} and v ∈ T_p(S^{q − 1}). Let δ = ||v|| ≥ 0. The geodesic γ with γ(0) = p and is given by the great circle:

and the exponential map is:

The inverse exponential map exp ^{− 1}_p is the mapping from the sphere to the tangent space, which satisfies exp ^{− 1}_p(exp _p(v)) = v. The Riemannian distance between p and exp _p(v) is the arc length of the minimal geodesic, and is given by δ.

If q = 3 and we use spherical coordinates (θ, ϕ) where θ is angle with the north pole and ϕ is the latitude then the Riemannian metric is:

and the metric tensor is:

There are other choices of tangent coordinates for the sphere, for example using an orthogonal projection. The tangent normal decomposition of x at μ is given by:

where ξ^Tμ = 0. It is clear that (1 − t²)^1/2 = x^Tμ and so the orthogonal projection tangent coordinates for x are:

(4.27)

The matrix I_q − μμ^T is the projection matrix onto the space orthogonal to μ. See Figure 4.7 for a geometrical explanation. Note that the inverse exponential map tangent coordinates v and the orthogonal projection tangent coordinates v_⊥ are related by:

A disadvantage of the orthogonal projection is that points at distance further than π/2 are closer in the projection than a point at π/2.   □

4.4.2 Procrustes tangent coordinates

The tangent space to the shape space itself is complicated (Le 1991a), but it is convenient to work with tangent coordinates to the pre-shape sphere. The tangent space to the pre-shape sphere is exactly as given in Equation (4.27) with q = (k − 1)m. In order to use the space for shape analysis the tangent space can be decomposed into two complementary subspaces: the horizontal tangent space of dimension (k − 1)m − m(m − 1)/2 − 1 which does not depend on rotation, and the vertical tangent space of dimension m(m − 1)/2 which contains the rotation information. For shape analysis we work with coordinates in the horizontal tangent space to the pre-shape sphere.

Constructing the Procrustes tangent space involves first choosing a pole p, a k × m matrix, which is assumed to be non-degenerate (Kent and Mardia 2001) which means that the eigenvalues of p^Tp satisfy λ_{m − 1} > 0 (i.e. the rank of p is at least m − 1). At degenerate p the shape space has a singularity (Kendall 1984), and so we assume that p is away from such singularities. For m = 2 all pre-shapes are non-degenerate (Kent and Mardia 2001). After choosing a non-degenerate pole p we carry out Procrustes rotation of X onto p (where p and X have been centred or Helmertized) to give , where is given in Equation (4.6). The Procrustes tangent matrix is then (Kent and Mardia 2001):

where α = cos ρ(X, p) > 0, with ρ(X, p) the Riemannian distance between X and p, and X and p are non-degenerate [see Equation (4.7)]. Note that we do need to restrict ourselves to α > 0, and so the projection cannot be used for two shapes which are maximally remote at distance ρ = π/2 apart. We can write:

where V₀ = V/sin ρ, and 0 ≤ ρ < π/2. From Kent and Mardia (2001) the Procrustes tangent matrix satisfies the following properties:

1. V is centred V^T1 = 0.
2. V is orthogonal to p: trace(p^TV) = 0.
3. p^TV is symmetric: p^TV = V^Tp.
4. V is bounded: ||V|| < 1.
5. The eigenvalues of the symmetric matrix p^T[(1 − ||V||²)^1/2p + V] are optimally signed and non-degenerate, see Equation (4.4) and Equation (4.7).

Note that the number of linear constraints on V from (i)–(iii) is m + 1 + m(m − 1)/2. Property (iii) ensures horizontality. Property (iv) ensures that α > 0. The Procrustes tangent coordinates are also known as Kent’s partial Procrustes tangent coordinates (Kent 1995), as the matching is only done over the ‘partial’ transformation group of rotations and not scale.

4.4.3 Planar Procrustes tangent coordinates

We now provide explicit expressions for the Procrustes tangent coordinates for planar data. Consider complex landmarks z^o = (z^o₁, …, z_k^o)^T with pre-shape

Let γ be a complex pole on the complex pre-shape sphere usually chosen to correspond to an average shape [e.g. corresponding to the full Procrustes mean shape from Equation (6.11)].

Let us rotate the configuration by an angle θ to be as close as possible to the pole and then project onto the tangent plane at γ, denoted by T_γ. Note that minimizes ||γ − ze^iθ||² (see Result 4.4).

Definition 4.4 The Procrustes tangent coordinates for a planar shape are given by:

(4.28)

where . Procrustes tangent coordinates involve only rotation (and not scaling) to match the pre-shapes.

Note that v*_Pγ = 0 and so the complex constraint means we can regard the tangent space as a real subspace of of dimension 2k − 4. The matrix I_{k − 1} − γγ* is the matrix for complex projection into the space orthogonal to γ. In Figure 4.8 we see a section of the pre-shape sphere showing the tangent plane coordinates. Note that the inverse projection from v_P to is given by:

(4.29)

Hence an icon for Procrustes tangent coordinates is given by X_I = H^Tz.

Example 4.3 For the T2 mouse vertebral data of Section 1.4.1 we consider in Figure 4.9 a plot of the icons for the Procrustes tangent coordinates when using a pole corresponding to the full Procrustes mean shape defined later in Equation (6.11). One must remember that the dimension is 2k − 4, as each configuration is centred, is of unit centroid size and is rotated to be as close as possible to the pole. There is clearly non-circular variability in landmarks 1, 2 and 4. Pairwise plots of the icon coordinates and centroid size are given in Figure 4.10 and we see strong structure in the plot – there are strong correlations between some of the icon coordinates, for example x₁ and x₂ with y₄; and centroid size S with x₁, x₂, y₁, y₂ and y₄.   □

Result 4.8 The Euclidean norm of a point v in the partial Procrustes tangent space is equal to the full Procrustes distance from the original configuration z^o corresponding to v to an icon of the pole H^Tγ, that is

Proof: It can be seen that

(4.30)

   □

Important point: This result means that multivariate methods in tangent space which involve calculating distances to the pole γ will be equivalent to non-Euclidean shape methods which require the full Procrustes distance to the icon H^Tγ. Also, if X₁ and X₂ are close in shape, and v₁ and v₂ are the tangent plane coordinates, then

(4.31)

For practical purposes this means that standard multivariate statistical techniques in tangent space can be good approximations to non-Euclidean shape methods, provided the data are not too highly dispersed.

4.4.4 Higher dimensional Procrustes tangent coordinates

The Procrustes tangent coordinates can be extended simply into m ≥ 3 dimensions, as proposed by Dryden and Mardia (1993). We first need to define the vectorize operator and its inverse.

Definition 4.5 The vectorize operator vec(X) of an l × m matrix X stacks the columns of X to give an lm-vector, i.e. if X has columns x₁, x₂, …, x_m, then

(4.32)

The inverse vectorize operator vec^{− 1}_m( · ) is the inverse operation of vec( · ), forming a matrix of m columns, that is if vec(Y) = X, then vec^{− 1}_m(X) = Y, where Y is an l × m matrix.

To obtain the tangent coordinates, a pole γ [a (k − 1) × m matrix] is chosen on the pre-shape sphere; for example, this could correspond to the full Procrustes mean shape of Equation (6.11). Given a pre-shape Z this is rotated to the pole to be as close as possible, and we write the rotated pre-shape as , where . We choose to minimize

which is obtained using Equation (4.6). Then we project onto the tangent plane at γ,

(4.33)

where vec(X) is the vectorize operator. The inverse transformation is given by

(4.34)

The tangent space is an q = (k − 1)m − m(m − 1)/2 − 1 dimensional subspace of and the dimensionality of the tangent space can be seen from the fact there are m location constraints, one size constraint and m(m − 1)/2 rotational constraints, after choosing . For m = 2 the complex Procrustes tangent coordinates of Equation (4.28) are identical to those in Equation (4.33).

On a historical note, Goodall (1991) provided one of the earliest discussions of tangent coordinates, although he concentrated on the Procrustes residuals which are approximate tangent coordinates (see Section 4.4.6). Kent (1992) gave the first treatment of Procrustes tangent coordinates to the pre-shape sphere, which was extended to higher dimensions by Dryden and Mardia (1993).

4.4.5 Inverse exponential map tangent coordinates

In Riemannian geometry a natural projection from the metric space to a tangent space is via the inverse of the exponential map (or logarithmic map) as discussed in Section 3.1. The exponential map provides a projection from the tangent space to the Riemannian manifold which preserves the Riemannian distance from each point to the pole [see Equation (3.1)]. Let γ(t) be the unique geodesic starting at p with initial tangent vector v_E = γ′(0) passing through the shape corresponding to X at γ(1). If v_P are the partial tangent space coordinates corresponding to pre-shape X and pole p then the inverse exponential map tangent coordinates of X with pole p are given by:

(4.35)

These coordinates are also known as the logarithmic map coordinates. Note

and so the Riemannian distance ρ to the pole is preserved, whereas for partial Procrustes tangent coordinates the full Procrustes distance d_F = sin ρ to the pole is preserved. The main practical difference between the partial Procrustes and inverse exponential map tangent coordinates are that the more extreme observations are pulled in more towards the pole with the partial Procrustes tangent coordinates.

4.4.6 Procrustes residuals

We now give an approximation to the Procrustes tangent space (Goodall 1991; Cootes et al. 1992), which can be formulated in a straightforward manner for practitioners when carrying out Procrustes analysis (see Chapter 7). The Procrustes residuals are:

where is the optimal rotation, and the optimal scale for matching z to , and is the full Procrustes mean shape of Equation (6.11). The Procrustes residuals are only approximate tangent coordinates, as they are not orthogonal to the pole, that is . For m = 2 dimensions for pre-shape z and pole the Procrustes residual is:

which do not satisfy in general, but approximately hold for z close to the pole.

If a shape is close to the pole (which we will usually have in practice), then the differences between the choices of tangent coordinates will be very small.

4.4.7 Other tangent coordinates

Other choices of tangent projection can be used. For example, the full Procrustes tangent coordinates allow additional scaling to the pole and are given by:

(4.36)

where v_P are the partial tangent coordinates. However, these coordinates are only sensible to use for 0 ≤ ρ ≤ π/4 as the distance to the pole becomes monotonic decreasing for larger ρ. Another possibility are the coordinates

which for triangle shapes reduces to coordinates from a stereographic projection on S²(1/2), which is a conformal mapping which preserves angles.

Tangent spaces are particularly useful for practical data analysis, provided the data are fairly concentrated. This approach to data analysis is considered in Chapter 9 where standard multivariate methods can be carried out on the tangent plane coordinates in this linear space. As a summary the different types of tangent space coordinates are given in Table 4.2.

4.4.8 Tangent space coordinates in R

In the shapes package in R we can choose which tangent coordinates to work with. In the function procGPA which carries out Procrustes analysis there is an option tangentcoords in which the user specifies the type of tangent coordinates and the pole is a mean shape. If the option scale=TRUE (which is used for shape analysis as opposed to size-and-shape analysis of Chapter 5) then the options are for tangentcoords are partial (Kent’s partial tangent coordinates), expomap (tangent coordinates from the inverse of the exponential map) and residual (Procrustes residuals). For example the partial tangent coordinates for the small group of mouse vertebrae that are used in the example in Section 4.4.2 are obtained from:

ans<-procGPA(qset2.dat,tangentcoords="partial",eigen2d=TRUE)}

and ans$tan is a 10 × 23 matrix of tangent coordinates, for the 2k − 2 dimensional tangent coordinates of the n = 23 mouse vertebrae. To plot the icons of the tangent coordinates:

x<-array(0,c(6,2,23))
H<-defh(5)
pole<-ans$mshape/centroid.size(ans$mshape)
x[,1,]<-t(H)%*%ans$tan[1:5,]+pole[,1]
x[,2,]<-t(H)%*%ans$tan[6:10,]+pole[,2]
plotshapes(x,symbol=3)

and the resulting plot is identical to Figure 4.9.