- 1k
- Vector of ones (k × 1) vector
- AT
- 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)
- dP(X1, X2)
- Partial Procrustes distance (0 ≤ dP ≤ √2)
- dF(X1, X2)
- Full Procrustes distance (0 ≤ dF ≤ 1)
- ρ(X1, X2)
- Riemannian distance (0 ≤ ρ ≤ π/2)
- dS(X1, X2)
- Riemannian distance in size-and-shape space (0 ≤ dS < ∞)
- H
- Helmert sub-matrix ((k − 1) × k)
- Hhat = XD(XTDXD)− 1XD
- Hat matrix
- Hj = XDj(XTDjXDj)− 1XDj
- jth hat matrix
- Ik
- 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)
- Sl
- Unit sphere in l + 1 real dimensions
- Sl(r)
- Sphere in l + 1 real dimensions with radius r
- Coordinates of landmarks to be matched (k × m matrix)
- UB = (uB3, …, ukB, vB3, …, vkB)T
- Bookstein coordinates for 2D data ((2k − 4) × 1 vector)
- UK = (uK3, …, ukK, vK3, …, vkK)T
- Kendall coordinates for 2D data ((2k − 4) × 1 vector)
- v
- Tangent plane coordinates
- W = XHΓ
- 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
- XH = HX
- Helmertized landmark coordinates ((k − 1) × m matrix)
- XD = Im⊗[1k, T]
- Design matrix
- XDj
- Design matrix for the jth configuration
- XP
- Full Procrustes fitted configuration
- Coordinates of reference configuration (k × m matrix)
- Y = [1k, T]B
- Affine transformation between configurations
- vec(Y) = XDβ
- Vectorized equation for affine/shape transformation
- y = At + c
- Affine transformation between points
- Z = XH/||XH||
- Pre-shape ((k − 1) × m matrix)
- ZC = HTZH
- Centred pre-shape
- z
- Complex pre-shape ((k − 1) × 1 complex vector)
- zo
- Original complex landmark coordinates (k × 1 complex vector)
- zH
- Helmertized complex landmarks ((k − 1) × 1 complex vector)
- Γ
- Rotation matrix ( ∈ SO(m)) (m × m matrix)
- Φ(t) = [Φ1(t), …, Φm(t)]T
- Deformation from to
- ΘB = (θB3, …, θkB, ϕB3, …, ϕkB)T
- Bookstein coordinates of population mean shape
- ΘK = (θK3, …, θkK, ϕK3, …, ϕkK)T
- Kendall coordinates of population mean shape
- μ
- Population average configuration (e.g., mean or mode)
- Σ
- Covariance matrix of Helmertized landmarks
- Σkm
- Shape space for k points in
- SΣkm
- 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 = S2l + 1
- Set of less than full-rank points
- 1F1( · )
- Confluent hypergeometric function
- Iν( · )
- 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
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