Shape spaces

KendallsPreShapeSpace{T} <: 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 [Ken84][Ken89].

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; parameter::Symbol=:type)

See also

KendallsShapeSpace, esp. for the references

KendallsShapeSpace{T} <: AbstractDecoratorManifold{ℝ}

Kendall's shape space, defined as quotient of a KendallsPreShapeSpace (represented by n×k matrices) modulo column wise application of a single rotation.

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

Constructor

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

References

check_point(M::KendallsPreShapeSpace, p; atol=sqrt(max_eps(X, Y)), kwargs...)

Check whether p is a valid point on KendallsPreShapeSpace, i.e. whether each row has zero mean. Other conditions are checked via embedding in ArraySphere.

check_vector(M::KendallsPreShapeSpace, p, X; kwargs... )

Check whether X is a valid tangent vector on KendallsPreShapeSpace, i.e. whether each row has zero mean. Other conditions are checked via embedding in ArraySphere.

get_embedding(M::KendallsPreShapeSpace)

Return the space KendallsPreShapeSpace M is embedded in, i.e. ArraySphere of matrices of the same shape.

manifold_dimension(M::KendallsPreShapeSpace)

Return the dimension of the KendallsPreShapeSpace manifold M. The dimension is given by $n(k - 1) - 1$.

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.

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.

exp(M::KendallsShapeSpace, p, X)

Compute the exponential map on KendallsShapeSpace M. See [GMTP21] for discussion about its computation.

log(M::KendallsShapeSpace, p, q)

Compute the logarithmic map on KendallsShapeSpace M. See the [exp](@ref exp(::KendallsShapeSpace, ::Any, ::Any)onential map for more details

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.

get_embedding(M::KendallsShapeSpace)

Get the manifold in which KendallsShapeSpace M is embedded, i.e. KendallsPreShapeSpace of matrices of the same shape.

get_total_space(::KendallsShapeSpace)

Return the total space of the KendallsShapeSpace manifold, which is the KendallsPreShapeSpace manifold.

horizontal_component(::KendallsShapeSpace, p, X)

Compute the horizontal component of tangent vector X at p on KendallsShapeSpace M. See [GMTP21], Section 2.3 for details.

is_flat(::KendallsShapeSpace)

Return false. KendallsShapeSpace is not a flat manifold.

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.