# Shape spaces

Shape spaces are spaces of $k$ points in $\mathbb{R}^n$ up to simultaneous action of a group on all points. The most commonly encountered are Kendall's pre-shape and shape spaces. In the case of the Kendall's pre-shape spaces the action is translation and scaling. In the case of the Kendall's shape spaces the action is translation, scaling and rotation.

using Manifolds, Plots

M = KendallsShapeSpace(2, 3)
# two random point on the shape space
p = [
0.4385117672460505 -0.6877826444042382 0.24927087715818771
-0.3830259932279294 0.35347460720654283 0.029551386021386548
]
q = [
-0.42693314765896473 -0.3268567431952937 0.7537898908542584
0.3054740561061169 -0.18962848284149897 -0.11584557326461796
]
# let's plot them as triples of points on a plane
fig = scatter(p[1,:], p[2,:], label="p", aspect_ratio=:equal)
scatter!(fig, q[1,:], q[2,:], label="q")

# aligning q to p
A = get_orbit_action(M)
a = optimal_alignment(A, p, q)
rot_q = apply(A, a, q)
scatter!(fig, rot_q[1,:], rot_q[2,:], label="q aligned to p")

A more extensive usage example is available in the hand_gestures.jl tutorial.

Manifolds.KendallsPreShapeSpaceType
KendallsPreShapeSpace{n,k} <: AbstractSphere{ℝ}

Kendall's pre-shape space of $k$ landmarks in $ℝ^n$ represented by n×k matrices. In each row the sum of elements of a matrix is equal to 0. The Frobenius norm of the matrix is equal to 1 [Kendall1984][Kendall1989].

The space can be interpreted as tuples of $k$ points in $ℝ^n$ up to simultaneous translation and scaling of all points, so this can be thought of as a quotient manifold.

Constructor

KendallsPreShapeSpace(n::Int, k::Int)

KendallsShapeSpace, esp. for the references

source
Manifolds.KendallsShapeSpaceType
KendallsShapeSpace{n,k} <: AbstractDecoratorManifold{ℝ}

Kendall's shape space, defined as quotient of a KendallsPreShapeSpace (represented by n×k matrices) by the action ColumnwiseMultiplicationAction.

The space can be interpreted as tuples of $k$ points in $ℝ^n$ up to simultaneous translation and scaling and rotation of all points [Kendall1984][Kendall1989].

This manifold possesses the IsQuotientManifold trait.

Constructor

KendallsShapeSpace(n::Int, k::Int)

References

source

## Provided functions

ManifoldsBase.projectMethod
project(M::KendallsPreShapeSpace, p)

Project point p from the embedding to KendallsPreShapeSpace by selecting the right element from the orthogonal section representing the quotient manifold M. See Section 3.7 of [Srivastava2016] for details.

The method computes the mean of the landmarks and moves them to make their mean zero; afterwards the Frobenius norm of the landmarks (as a matrix) is normalised to fix the scaling.

source
Base.randMethod
rand(::KendallsShapeSpace; vector_at=nothing)

When vector_at is nothing, return a random point x on the KendallsShapeSpace manifold M by generating a random point in the embedding.

When vector_at is not nothing, return a random vector from the tangent space with mean zero and standard deviation σ.

source
• Kendall1989

D. G. Kendall, “A Survey of the Statistical Theory of Shape,” Statist. Sci., vol. 4, no. 2, pp. 87–99, May 1989 doi: 10.1214/ss/1177012582.

• Kendall1984

D. G. Kendall, “Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces,” Bull. London Math. Soc., vol. 16, no. 2, pp. 81–121, Mar. 1984 doi: 10.1112/blms/16.2.81.

• Srivastava2016

A. Srivastava and E. P. Klassen, Functional and Shape Data Analysis. Springer New York, 2016. ISBN: 978-1-4939-4018-9. doi: 10.1007/978-1-4939-4020-2.

• Guigui2021

N. Guigui, E. Maignant, A. Trouvé, and X. Pennec, “Parallel Transport on Kendall Shape Spaces,” in Geometric Science of Information, Cham, 2021, pp. 103–110. doi: 10.1007/978-3-030-80209-7_12.