Appendix

1_k: Vector of ones (k × 1) vector
A^T: Transpose of matrix A
||A|| = √trace(A*A): Euclidean norm
: Complex conjugate of a
a*: Transpose of the complex conjugate of a
: Centring matrix (k × k)
D(X): Baseline size (positive scalar)
d_P(X₁, X₂): Partial Procrustes distance (0 ≤ d_P ≤ √2)
d_F(X₁, X₂): Full Procrustes distance (0 ≤ d_F ≤ 1)
ρ(X₁, X₂): Riemannian distance (0 ≤ ρ ≤ π/2)
d_S(X₁, X₂): Riemannian distance in size-and-shape space (0 ≤ d_S < ∞)
H: Helmert sub-matrix ((k − 1) × k)
H_hat = X_D(X^T_DX_D)^{− 1}X_D: Hat matrix
H_j = X_Dj(X^T_DjX_Dj)^{− 1}X_Dj: jth hat matrix
I_k: Identity matrix (k × k)
k: Number of landmarks
: Dimension of shape space
m: Real dimension of object
R: Rotation and reflection matrix ( ∈ O(m)) (m × m)
S(X): Centroid size (positive scalar)
S^l: Unit sphere in l + 1 real dimensions
S^l(r): Sphere in l + 1 real dimensions with radius r
: Coordinates of landmarks to be matched (k × m matrix)
U^B = (u^B₃, …, u_k^B, v^B₃, …, v_k^B)^T: Bookstein coordinates for 2D data ((2k − 4) × 1 vector)
U^K = (u^K₃, …, u_k^K, v^K₃, …, v_k^K)^T: Kendall coordinates for 2D data ((2k − 4) × 1 vector)
v: Tangent plane coordinates
W = X_HΓ: Size-and-shape of X (Γ ∈ SO(k))
X: Configuration matrix of landmark coordinates (k × m matrix)
[X]: Shape of X
[X]_I: Icon (representative configuration)
[X]_S: Shape of X
[X]_R: Reflection shape of X
[X]_RS: Reflection size-and-shape of X
X_H = HX: Helmertized landmark coordinates ((k − 1) × m matrix)
X_D = I_m⊗[1_k, T]: Design matrix
X_Dj: Design matrix for the jth configuration
X^P: Full Procrustes fitted configuration
: Coordinates of reference configuration (k × m matrix)
Y = [1_k, T]B: Affine transformation between configurations
vec(Y) = X_Dβ: Vectorized equation for affine/shape transformation
y = At + c: Affine transformation between points
Z = X_H/||X_H||: Pre-shape ((k − 1) × m matrix)
Z_C = H^TZ_H: Centred pre-shape
z: Complex pre-shape ((k − 1) × 1 complex vector)
z^o: Original complex landmark coordinates (k × 1 complex vector)
z_H: Helmertized complex landmarks ((k − 1) × 1 complex vector)
Γ: Rotation matrix ( ∈ SO(m)) (m × m matrix)
Φ(t) = [Φ₁(t), …, Φ_m(t)]^T: Deformation from to
Θ^B = (θ^B₃, …, θ_k^B, ϕ^B₃, …, ϕ_k^B)^T: Bookstein coordinates of population mean shape
Θ^K = (θ^K₃, …, θ_k^K, ϕ^K₃, …, ϕ_k^K)^T: Kendall coordinates of population mean shape
μ: Population average configuration (e.g., mean or mode)
Σ: Covariance matrix of Helmertized landmarks
Σ^k_m: Shape space for k points in
SΣ^k_m: Size-and-shape space
Ω: Covariance matrix of original landmarks
: Set of coincident points
: k-dimensional complex space
: l-dimensional complex projective space
: Unit complex sphere in l + 1 complex dimensions = S^{2l + 1}
: Set of less than full-rank points
₁F₁( · ): Confluent hypergeometric function
I_ν( · ): Bessel function of the first kind
: Simple Laguerre polynomial of degree j
: Legendre polynomial of degree j
: l-dimensional real space
: j-dimensional simplex
arginf: value that gives the infemum
argsup: value that gives the supremum
Arg: argument of a complex number