Two Sample Test for Extrinsic Antimeans on Planar Kendall Shape Spaces with an Application to Medical Imaging

In preparation, we are using the large sample distribution for extrinsic sample antimeans given in Patrangenaru et al (2016 [12]).

Assume $j$ is an embedding of a $d$-dimensional manifold $\mathcal{M}$ such that $j(\mathcal{M})$ is closed in $\mathbb{R}^{N}$, and $Q=P_{X}$ is a $\alpha j$-nonfocal probability measure on $\mathcal{M}$ such that $j(Q)$ has finite moments of order 2. Let $\mu$ and $\Sigma$ be the mean and covariance matrix of $j(Q)$ regarded as a probability measure on $\mathbb{R}^{N}$. Let $\mathcal{F}$ be the set of $\alpha j$-focal points of $j(\mathcal{M})$, and let $P_{F,j}:{\mathcal{F}}^{c}\to j(\mathcal{M})$ be the farthest projection on $j(\mathcal{M})$. $P_{F,j}$ is differentiable at $\mu$ and has the differentiability class of $j(\mathcal{M})$ around any $\alpha j$ nonfocal point. In order to evaluate the differential $d_{\mu}P_{F,j}$ we consider a special orthonormal frame field that will ease the computations.

A local frame field $p\to(e_{1}(p),\dots,e_{k}(p))$, defined on an open neighborhood $U\subseteq\mathbb{R}^{N}$ is adapted to the embedding $j$ if it is an orhonormal frame field and $\forall x\in j^{-1}(U),e_{r}(j(x))=d_{x}j(f_{r}(x)),r\in\{1,\ldots,d\},$ where $(f_{1},\dots,f_{d})$ is a local frame field on $\mathcal{M},$ and $f_{r}(x)$ is the value of the local vector field $f_{r}$ at $x.$

Let $e_{1},\dots,e_{N}$ be the canonical basis of $\mathbb{R}^{N}$ and assume $(e_{1}(p),\dots,e_{N}(p))$ is an adapted frame field around $P_{F,j}(\mu)=j(\alpha\mu_{E})$. Then $d_{\mu}P_{F,j}(e_{b})\in T_{P_{F,j}(\mu)}j(\mathcal{M})$ is a linear combination of $e_{1}(P_{F,j}(\mu)),\dots,e_{d}(P_{F,j}(\mu))$:

where $d_{\mu}P_{F,j}$ is the differential of $P_{F,j}$ at $\mu.$ By the delta method, $n^{1/2}(P_{F,j}(\overline{j(X)})-P_{F,j}(\mu))$ converges weakly to $N_{N}(0_{N},\alpha\Sigma_{\mu})$, where $\overline{j(X)}=\frac{1}{n}\sum^{n}_{i=1}j(X_{i})$ and

Here $\Sigma$ is the covariance matrix of $j(X_{1})$ w.r.t the canonical basis $e_{1},\dots,e_{N}.$

The asymptotic distribution $N_{N}(0_{N},\alpha\Sigma_{\mu})$ is degenerate and the support of this distribution is on $T_{P_{F,j}}j(\mathcal{M})$, since the range of $d_{\mu}P_{F,j}$ is $T_{P_{F,j(\mu)}}j(\mathcal{M})$. Note that $d_{\mu}P_{F,j}(e_{b})\cdot e_{a}(P_{F,j}(\mu))=0$ for $a\in\{d+1,\dots,N\}$.

The tangential component $tan(v)$ of $v\in\mathbb{R}^{N}$, w.r.t the basis $e_{a}(P_{F,j}(\mu))\in T_{P_{F,j(\mu)}}j(\mathcal{M}),a\in\{1,\dots,d\}$ is given by

Then, the random vector ${(d_{\alpha\mu_{E}}j)}^{-1}(tan(P_{F,j}(\overline{(j(X))})-P_{F,j}(\mu)))=\sum^{d}_{a=1}{\overline{X}}^{a}_{j}f_{a}$ has the following anticovariance matrix w.r.t the basis $f_{1}(\alpha\mu_{E}),\dots,f_{d}(\alpha\mu_{E})$:

which is the anticovariance matrix of the random object $X$. To simplify the notation, we set

Similarly, given i.i.d.r.o.’s $X_{1},\cdots,X_{n}$ from $Q$, we define the sample anticovariance matrix $aS_{j,E,n}$ as the anticovariance matrix associated with the empirical distribution $\hat{Q}_{n}.$

If, in addition, rank $\alpha\Sigma_{\mu}=d$, then $\alpha\Sigma_{j,E}$ is invertible and if we define the $j$-standardized antimean vector

using basic large sample theory results, including a generalized Slutsky’s lemma ( see Patrangenaru and Ellingson(2015)[6], p.65), one has:

We consider the case of a probability measure $Q$ on the complex projective space $\mathcal{M}=\mathbb{C}P^{q}$. If we consider the action of the multiplicative group $\mathbb{C}^{*}=\mathbb{C}\backslash\{0\}$ on $\mathbb{C}^{q+1}\backslash\{0\}$, given by scalar multiplication

the quotient space is the $q$-dimensional complex projective space $\mathbb{C}P^{q}$, set of all complex lines in $\mathbb{C}^{q+1}$ going through $0\in\mathbb{C}^{q+1}$. One can show that $\mathbb{C}P^{q}$ is a 2$q$ dimensional real analytic manifold, using transition maps, similar to those in the case of $\mathbb{R}P^{q}$  (see Patrangenaru and Ellingson (2015) [20], chapter 3).

Here we are concerned with a landmark based non-parametric analysis of similarity shape data (see Bhattacharya and Patrangenaru (2014)[23]). Our analysis is for extrinsic antimeans. For landmark based shape data, one considers a $k$-ad $x=(x^{1},\dots,x^{k})\in(\mathbb{R}^{m})^{k}$, which consists of $k$ labeled points in $\mathbb{R}^{m}$ that represent coordinates of k-labeled landmarks.

In this subsection $\mathcal{G}$ is the group direct similarities of $\mathbb{R}^{m}$. A similarity is a function $f:\mathbb{R}^{m}\to\mathbb{R}^{m}$, that uniformly scale the Euclidean distances, that is, for which there is $K>0$, such that $\lVert f(x)-f(y)\rVert=K\lVert x-y\rVert,\forall x,y\in\mathbb{R}^{m}$. Using the fundamental theorem of Euclidean geometry, one can show that a similarity is given by $f(x)=Ax+b,A^{T}A=cI_{m},c>0$. A direct similarity is a similarity of this form, where $A$ has a positive determinant. Under composition direct similarities from a group. The object space considered here consist in orbits of the group action of $\mathcal{G}$ on $k$-ads, and is called direct similarities shape space of k-ads in $\mathbb{R}^{m}$.

For $m=2$, or $m=3$, a direct similarity (Kendall) shape is that the geometrical information that remains when location, scale and rotational effects are filtered out from a $k$-ads. Two $k$-ads $(z_{1},z_{2},\dots,z_{k})$ and $(z^{{}^{\prime}}_{1},z^{{}^{\prime}}_{2},\dots,z^{{}^{\prime}}_{k})$ are said to have the same shape if there is a direct similarity $T$ in the plane, that is, a composition of a rotation, a translation and a homothety such that $T(z_{j})=z^{{}^{\prime}}_{j}$ for $j=1,\dots,k$. Having the same Kendall shape is an equivalence relationship in the space of planar $k$-ads, and the set of all equivalence classes of nontrivial $k$-ads is called the Kendall planar shape space of $k$-ads, which is denoted $\Sigma_{2}^{k}$ (see Balan and Patrangenaru(2005) [15]).

Kendall (1984) [2] introduced that planar direct similarity shapes of k-ads; which is a set of k labeled points at least two of which are distinct, can be represented as points on a complex projective space $\mathbb{C}P^{k-2}$. A standard shape analysis method was also introduced by Kent(1992)[4] and is called Veronese-Whitney(VW) embedding of $\mathbb{C}P^{k-2}$ in the space of $(k-1)\times(k-1)$ self adjoint complex matrices, to represent shape data in an Euclidean space. This Veronese–Whitney embedding $j:\mathbb{C}P^{k-2}\to S(k-1,\mathbb{C})$, where $S(k-1,\mathbb{C})$ is the space of $(k-1)\times(k-1)$ Hermitian matrices, is given by

This embedding is a $SU(k-1)$ equivariant embedding and $SU(k-1)$ is called the special unitary group $(k-1)\times(k-1)$ matrices of determinant 1, since $j([Az])=Aj([z])A^{*}$, $\forall A\in SU(k-1)$. The corresponding extrinsic mean (set) of a random shape X on $\mathbb{C}P^{k-2}$ is called the VW mean (set) (see Patrangenaru and Ellingson (2015)[6], Ch.3), and the VW mean,when it exists,and is labeled $\mu_{VW}(X),\mu_{VW}$ or simply $\mu_{E}$.The corresponding extrinsic antimean (set) of a random shape X,is called the VW antimean (set) and is labeled $\alpha\mu_{VW}(X),\alpha\mu_{VW}$ or $\alpha\mu_{E}$.

For the VW-embedding of the complex projective space, we have the following theorem for sample VW-means from Bhattacharya and Patrangenaru(2003) [10]:

We also have a similar result for sample VW-antimeans (see Wang, Patrangenaru and Guo [17]):

We are concerned with a landmark based nonparametric analysis of similarity shape data. For landmark based shape data , we will denote a $k$-ad by $k$ complex numbers: $z_{j}=x_{j}+iy_{j},~{}1\leq j\leq k.$ We center the $k$-ad at $\langle z\rangle={1\over k}\sum\limits_{j=1}^{k}z_{j}$ via a translation; next we rotate it by an angle $\theta$ and scale it, operations that are achieved by multiplying $z-\langle z\rangle$ by the complex number $\lambda=re^{i\theta}.$ We can represent the shape of the $k$-ad as the complex one dimensional vector subspace passing through $0$ of the linear subspace $L^{k-1}=\{\zeta\in\mathbb{C}^{k},1_{k}^{T}\zeta=0\}.$ Thus, the space of $k$-ads is the set of all complex lines on the hyperplane, $L^{k-1}={w\in{\mathbb{C}}^{k}\backslash\{0\}}:~{}\sum\limits_{1}^{k}w_{j}=0.$ Therefore the shape space $\Sigma_{2}^{k}$ of nontrivial planar $k$-ads can be represented as the complex projective space $\mathbb{C}P^{k-2},$ the space of all complex lines through the origin in $\mathbb{C}^{k-1}$ using an isomorphism of $L^{k-1}$ and $\mathbb{C}^{k-1}.$ As in the case of $\mathbb{C}P^{k-2}$, it is convenient to represent the element of $\Sigma_{2}^{k}$ corresponding to a $k$-ad $z$ by the curve $\gamma(z)=[z]={e^{i\theta}((z-\langle z\rangle)/\|z-\langle z\rangle\|):~{}0\leq\theta\leq 2\pi}$ on the unit sphere in $L^{k-1}\approx\mathbb{C}^{k-1}.$

Let $Q_{1}$ and $Q_{2}$ be two probability measures on the shape space $\Sigma_{2}^{k},$ and let $\alpha\mu_{1}$ and $\alpha\mu_{2}$ denote the antimeans of $Q_{1}\circ j^{-1}$ and $Q_{2}\circ j^{-1},$ where $j$ is the VW-embedding that $j([z])=uu^{*},$ where $u={1\over{\|z\|}}z,$ thus $u^{*}u=1$. Suppose $[W_{1}],\cdots,[W_{n}]$ and $[Z_{1}],\cdots,[Z_{m}]$ are i.i.d. random objects from $Q_{1}$ and $Q_{2}$ respectively. Let $X_{i}=j([W_{i}]),$ $Y_{j}=j([Z_{j}])$ be their images in $j(\mathbb{C}P^{k-2})$ which are random samples from $Q_{1}\circ j^{-1}$ and $Q_{2}\circ j^{-1},$ respectively. Suppose we are to test if the VW antimeans of $Q_{1}$ and $Q_{2}$ are equal, i.e.

It is known that $\alpha\mu_{1}=j^{-1}(P_{F}(v_{1}))$, where $v_{1}=E(X_{1})$ and similarly, $\alpha\mu_{2}=j^{-1}(P_{F}(v_{2}))$, where $v_{2}=E(Y_{1})$. We assume that both $v_{1}$ and $v_{2}$ have simple smallest eigenvalues. Then under $H_{0},$ their unit corresponding eigenvectors differ by a rotation.

Choose $v\in S(k-1,\mathbb{C}),$ with the same farthest projection as $v_{1}$ and $v_{2}$. Suppose $v=u\Lambda u^{*},$ and $\Lambda=Diag(\lambda_{1}<\lambda_{2}\leq\cdots\leq\lambda_{k-1}),$ where $\lambda_{a},a=1,\dots,k-1,$ are the eigenvalues of $v,$ and $u=[u_{1},u_{2},\cdots,u_{k-1}]$ are the corresponding eigenvectors. Also, we obtain an orthonormal basis for $S(k-1,\mathbb{C})$, which is given by $\{\upsilon_{b}^{a}:1\leq a\leq b\leq k-1\}$ and $\{\omega_{b}^{a}:1\leq a\leq b\leq k-1\}$. Where $\upsilon_{b}^{a}$ has all entries zero except for those in the positions (a, b) and (b, a) that are equal to 1 and $\omega_{b}^{a}$ is a matrix with all entries zero except for those in the positions (a, b) and (b, a) that are equal to i, respectively -i.

It is defined as,

where $\{e_{a}:1\leq a\leq k-1\}$ is the standard canonical basis for $\mathbb{C}^{N}$. For any $u\in SU(k-1)(uu^{*}=u^{*}u=1,det(u)=+1)$ where SU is special unitary group of all $(k-1)\times(k-1)$ complex matrices. $\{u\upsilon_{b}^{a}u^{*}:1\leq a\leq b\leq k-1\}$ and $\{u\omega_{b}^{a}u^{*}:1\leq a\leq b\leq k-1\}$ are an orthonormal frame for $S(k-1,\mathbb{C})$. Now, we will choose the orthonormal basis frame $\{u\upsilon_{b}^{a}u^{*},u\omega_{b}^{a}u^{*}\}$ for $S(k-1,\mathbb{C})$. Then it can be shown that

Then, we write

Then from equations (5.3) ,(5.4) and (5.5), it follows that