# Shape spaces

Shape spaces are spaces of $k$ points in $โ^n$ 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.KendallsPreShapeSpace โ Type
KendallsPreShapeSpace{T} <: AbstractSphere{โ}

Kendall's pre-shape space of $k$ landmarks in $โ^n$ represented by nร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 [Ken84][Ken89].

The space can be interpreted as tuples of $k$ points in $โ^n$ 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; parameter::Symbol=:type)

KendallsShapeSpace, esp. for the references

source
Manifolds.KendallsShapeSpace โ Type
KendallsShapeSpace{T} <: AbstractDecoratorManifold{โ}

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

The space can be interpreted as tuples of $k$ points in $โ^n$ up to simultaneous translation and scaling and rotation of all points [Ken84][Ken89].

This manifold possesses the IsQuotientManifold trait.

Constructor

KendallsShapeSpace(n::Int, k::Int; parameter::Symbol=:type)

References

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## Provided functions

ManifoldsBase.project โ Method
project(M::KendallsPreShapeSpace, p, X)

Project tangent vector X at point p from the embedding to KendallsPreShapeSpace by selecting the right element from the tangent space to orthogonal section representing the quotient manifold M. See Section 3.7 of [SK16] for details.

source
ManifoldsBase.project โ Method
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 [SK16] 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.rand โ Method
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 ฯ.

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ManifoldsBase.manifold_dimension โ Method
manifold_dimension(M::KendallsShapeSpace)

Return the dimension of the KendallsShapeSpace manifold M. The dimension is given by $n(k - 1) - 1 - n(n - 1)/2$ in the typical case where $k \geq n+1$, and $(k + 1)(k - 2) / 2$ otherwise, unless $k$ is equal to 1, in which case the dimension is 0. See [Ken84] for a discussion of the over-dimensioned case.

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