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

[Ken84], [Ken89]

helmert_submatrix(::Type{T}, k::Int)

Return the (k-1)×k matrix below the first row of the Helmert matrix of order k, whose j-th row is $(h_j, …, h_j, -jh_j, 0, …, 0)$ with $h_j = (j(j+1))^{-1/2}$. With the first row $(1/\sqrt{k}, …, 1/\sqrt{k})$ the Helmert matrix is orthogonal, see Eq. (8.87) in [Gen17, Section 8.8.1].

check_point(M::KendallsPreShapeSpace, p; atol=sqrt(eps(float(eltype(p)))), 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; atol=sqrt(eps(float(eltype(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_coordinates(M::KendallsPreShapeSpace, p, X, ::DefaultOrthonormalBasis)

Compute the coordinates of the tangent vector X at p on the KendallsPreShapeSpace M in an orthonormal basis. With the helmert_submatrix $H$, $p ↦ pH^{\mathrm{T}}$ maps M isometrically onto the ArraySphere(n, k-1), see [Ken89, Section 2], and the coordinates are those of $XH^{\mathrm{T}}$ at $pH^{\mathrm{T}}$ on that sphere.

get_embedding(M::KendallsPreShapeSpace)

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

get_vector(M::KendallsPreShapeSpace, p, c, ::DefaultOrthonormalBasis)

Compute the tangent vector at p on the KendallsPreShapeSpace M with the coordinates c in an orthonormal basis. It is $YH$, where $H$ is the helmert_submatrix and $Y$ is the tangent vector with the coordinates c at $pH^{\mathrm{T}}$ on the ArraySphere(n, k-1), see [Ken89, Section 2].

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.

isapprox(M::KendallsShapeSpace, p, X, Y; kwargs...)

Test whether two tangent vectors X, Y at p on the KendallsShapeSpace M are (approximately) the same, that is whether $\lVert X - Y \rVert_p \approx 0$ with the classical isapprox. The kwargs... are passed on to this; atol defaults to sqrt(max_eps(X, Y)).

isapprox(M::KendallsShapeSpace, p, q; kwargs...)

Test whether two points p, q on the KendallsShapeSpace M are (approximately) the same. Since this is a quotient manifold, the test is performed by checking their distance if they are not equal, that is $d_{\mathcal M}(p,q) \approx 0$ with the classical isapprox. The kwargs... are passed on to this; atol defaults to sqrt(max_eps(p, q)).

log(M::KendallsShapeSpace, p, q)

Compute the logarithmic map on KendallsShapeSpace M. See the exponential 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. See [Ken84] for a discussion of the over-dimensioned case.