Covariance Weighted Procrustes Analysis

Abstract

We revisit the popular Procrustes matching procedure of landmark shape analysis and consider the situation where the landmark coordinates have a completely general covariance matrix, extending previous approaches based on factored covariance structures. Procrustes matching is used to compute the Riemannian metric in shape space and is used more widely for carrying out inference such as estimation of mean shape and covariance structure. Rather than matching using the Euclidean distance we consider a general Mahalanobis distance. This approach allows us to consider different variances at each landmark, as well as covariance structure between the landmark coordinates, and more general covariance structures. Explicit expressions are given for the optimal translation and rotation in two dimensions and numerical procedures are used for higher dimensions. Simultaneous estimation of both mean shape and covariance structure is difficult due to the inherent non-identifiability. The method requires the specification of constraints to carry out inference, and we discuss some possible practical choices. We illustrate the methodology using data from fish silhouettes and mouse vertebra images.

Appendix

Proof of Result 2.1.

Let $v = \text{vec}(\mu -X\varGamma )$, then

$$ \displaystyle\begin{array}{rcl} D_{\mathrm{pCWP}}^{2}(X,\mu;\varSigma )& =& (v - (I_{ m} \otimes 1_{k})\gamma )^{T}\varSigma ^{-1}(v - (I_{ m} \otimes 1_{k})\gamma ) {}\\ & =& v^{T}\varSigma ^{-1}v - 2v^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma {}\\ & & +\gamma ^{T}(I_{ m} \otimes 1_{k})^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma. {}\\ \end{array} $$

The minimising translation is found by setting the first derivative equal to zero:

$$ \displaystyle\begin{array}{rcl} \frac{dD_{\mathrm{pCWP}}^{2}} {d\gamma } = -2(I_{m} \otimes 1_{k})^{T}\varSigma ^{-1}v + 2(I_{ m} \otimes 1_{k})^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma = 0.& & {}\\ \end{array} $$

The second derivative is clearly positive because $\varSigma ^{-1}$ is positive definite. Therefore, $D_{\mathrm{pCWP}}^{2}$ is minimised when $\gamma = [(I_{m} \otimes 1_{k})^{T}\varSigma ^{-1}(I_{m} \otimes 1_{k})]^{-1}(I_{m} \otimes 1_{k})\varSigma ^{-1}v$. $\square$

Proof of Result 2.2.

From Eq. (9.2) the minimising translation is $\hat{\gamma }= A\text{vec}(\mu -X\varGamma )$, so for m = 2,

$$\displaystyle\begin{array}{rcl} \left [\begin{array}{c} \hat{\gamma }_{1}\\ \hat{\gamma }_{ 2} \end{array} \right ] = \left [\begin{array}{c} \alpha _{1} +\delta _{1}\cos \theta +\zeta _{1}\sin \theta \\ \alpha _{2} +\delta _{2}\cos \theta +\zeta _{2}\sin \theta \end{array} \right ],\text{ because, }\varGamma = \left [\begin{array}{cc} \cos \theta &\sin \theta \\ -\sin \theta &\cos \theta \end{array} \right ].& & {}\\ \end{array}$$

Therefore,

$$\displaystyle\begin{array}{rcl} \text{vec}(\mu -X\varGamma - 1_{k}\gamma ^{T})&& {}\\ & & = \left [\begin{array}{c} (\mu _{1} - 1_{k}\alpha _{1}) - (X_{1} + 1_{k}\delta _{1})\cos \theta + (X_{2} - 1_{k}\zeta _{1})\sin \theta \\ (\mu _{2} - 1_{k}\alpha _{2}) - (X_{2} + 1_{k}\delta _{2})\cos \theta - (X_{1} + 1_{k}\zeta _{2})\sin \theta \end{array} \right ],{}\\ \end{array}$$

and $D_{\mathrm{pCWP}}^{2}(X,\mu;\varSigma ) = C + P\cos ^{2}\theta + Q\sin ^{2}\theta + R\cos \theta \sin \theta + S\cos \theta + T\sin \theta$ where

$$\displaystyle\begin{array}{rcl} C = \left [\begin{array}{c} (\mu _{1} - 1_{k}\alpha _{1}) \\ (\mu _{2} + 1_{k}\alpha _{2}) \end{array} \right ]^{T}\varSigma ^{-1}\left [\begin{array}{c} (\mu _{1} - 1_{k}\alpha _{1}) \\ (\mu _{2} + 1_{k}\alpha _{2}) \end{array} \right ].& & {}\\ \end{array}$$

Let $\lambda$ be the real Lagrangian multiplier to enforce the constraint $\cos ^{2}\theta +\sin ^{2}\theta = 1$ and let $L = D_{\mathrm{pCWP}}^{2}(X,\mu;\varSigma ) +\lambda (1 -\cos ^{2}\theta -\sin ^{2}\theta )$. Then,

$$\displaystyle\begin{array}{rcl} \frac{\partial L} {\partial (\cos \theta )}& =& 2(P-\lambda )\cos \theta + R\sin \theta + S = 0, {}\\ \frac{\partial L} {\partial (\sin \theta )}& =& 2(Q-\lambda )\sin \theta + R\cos \theta + T = 0, {}\\ \frac{\partial L} {\partial \lambda } & =& 1 -\cos ^{2}\theta -\sin ^{2}\theta = 0. {}\\ \end{array}$$

Solving the first two equations simultaneously and substituting the solutions in the third gives the expressions for $\cos \theta$, $\sin \theta$ and the quartic equation, respectively. To show this is a minimum of $D_{\mathrm{pCWP}}^{2}$, consider the matrix of second derivatives,

$$\displaystyle\begin{array}{rcl} S^{{\ast}} = \left [\begin{array}{cc} \frac{\partial ^{2}L} {\partial (\cos ^{2}\theta )} & \frac{\partial ^{2}L} {\partial (\cos \theta )\partial (\sin \theta )} \\ \frac{\partial ^{2}L} {\partial (\cos \theta )\partial (\sin \theta )} & \frac{\partial ^{2}L} {\partial (\sin ^{2}\theta )} \end{array} \right ] = \left [\begin{array}{cc} 2(P-\lambda )& R\\ R &2(Q-\lambda ) \end{array} \right ].& & {}\\ \end{array}$$

Let $\xi _{1} \geq \xi _{2}$ be the eigenvalues of S ^∗. Then, $\vert S^{{\ast}}-\xi _{i}I\vert = (\xi _{i} + 2\lambda )^{2} - 2(P + Q)(\xi _{i} + 2\lambda ) + 4PQ - R^{2}$, so $(\xi _{i} + 2\lambda ) = P + Q \pm \sqrt{(P - Q)^{2 } + R^{2}}$. Given $\varSigma ^{-1}$ is positive definite, P > 0 and Q > 0, then $\xi _{2}$ is strictly positive if $P + Q - 2\lambda -\sqrt{(P - Q)^{2 } + R^{2}} > 0$ which is true if the constraint on $\lambda$ is satisfied. $\square$

Proof of Result 2.3.

Let v = vec(μ) and $\xi = \text{vec}(X\varGamma )$, then

$$\displaystyle\begin{array}{rcl} D_{\mathrm{CWP}}^{2}(X,\mu;\varSigma )& =& (v -\beta \xi -(I_{ m} \otimes 1_{k})\gamma )^{T}\varSigma ^{-1}(v -\beta \xi -(I_{ m} \otimes 1_{k})\gamma ) {}\\ & =& v^{T}\varSigma ^{-1}v - 2\beta \xi ^{T}\varSigma ^{-1}v - 2v^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma +\beta ^{2}\xi ^{T}\varSigma ^{-1}\xi {}\\ & & +2\beta \xi ^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma +\gamma ^{T}(I_{ m} \otimes 1_{k})^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma. {}\\ \end{array}$$

This implies

$$\displaystyle\begin{array}{rcl} \frac{dD_{\mathrm{CWP}}^{2}} {d\gamma } & =& -2(I_{m} \otimes 1_{k})^{T}\varSigma ^{-1}v + 2\beta (I_{ m} \otimes 1_{k})^{T}\varSigma ^{-1}\xi {}\\ & & +2(I_{m} \otimes 1_{k})^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma, {}\\ \frac{dD_{\mathrm{CWP}}^{2}} {d\beta } & =& -2\xi ^{T}\varSigma ^{-1}v + 2\beta \xi ^{T}\varSigma ^{-1}\xi + 2\xi ^{T}\varSigma ^{-1}(I_{ m} \otimes 1_{k})\gamma. {}\\ \end{array}$$

Therefore, the minimum is at the solution of

$$\displaystyle\begin{array}{rcl} B\left [\begin{array}{c} \gamma \\ \beta \end{array} \right ] = \left [\begin{array}{c} (I_{m} \otimes 1_{k})^{T}\varSigma ^{-1}\mathrm{vec}(\mu ) \\ \mathrm{vec}(X\varGamma )^{T}\varSigma ^{-1}\mathrm{vec}(\mu ) \end{array} \right ].& & {}\\ \end{array}$$

The matrix of second derivatives is clearly positive because $\varSigma ^{-1}$ is positive definite. $\square$

Proof of Result 2.4.

Replacing $\cos \theta$ with $\beta \cos \theta$ and $\sin \theta$ with $\beta \sin \theta$ in the proof of Result 9.2.2 gives $D_{\mathrm{CWP}}^{2}(X,\mu;\varSigma ) = C + P\beta ^{2}\cos ^{2}\theta + Q\beta ^{2}\sin ^{2}\theta + R\beta ^{2}\cos \theta \sin \theta + S\beta \cos \theta + T\beta \sin \theta$. Let $\psi _{1} =\beta \cos \theta$ and $\psi _{2} =\beta \sin \theta$, then

$$\displaystyle\begin{array}{rcl} \frac{dD_{\mathrm{CWP}}^{2}} {d\psi _{1}} = 2P\psi _{1} + R\psi _{2} + S,\qquad \frac{dD_{\mathrm{CWP}}^{2}} {d\psi _{2}} = 2Q\psi _{2} + R\psi _{1} + T.& & {}\\ \end{array}$$

Setting these expressions equal to zero and solving them simultaneously gives the required expressions for ψ ₁ and ψ ₂. Solving $\psi _{1} =\beta \cos \theta$ and $\psi _{2} =\beta \sin \theta$ subject to the constraint that $\cos ^{2}\theta +\sin ^{2}\theta = 1$ gives the rotation and scale parameters. Given these, the translation is obtained by letting $v = \text{vec}(\mu -\beta X\varGamma )$ in the proof of Result 9.2.1. $\square$

Proof of Result 2.5.

If $\varSigma = I_{m} \otimes \varSigma _{k}$, then the similarity transformation estimates of Result 9.2.4 can be simplified. For the translation,

$$\displaystyle\begin{array}{rcl} \hat{\gamma }& =& [(I_{m} \otimes 1_{k})^{T}(I_{ m} \otimes \varSigma _{k})^{-1} {}\\ & & \times (I_{m} \otimes 1_{k})]^{-1}(I_{ m} \otimes 1_{k})^{T}(I_{ m} \otimes \varSigma _{k})^{-1}\text{vec}(\mu -\beta X\varGamma ) {}\\ & =& [I_{m} \otimes (1_{k}^{T}\varSigma _{ k}^{-1}1_{ k})]^{-1}[I_{ m} \otimes (1_{k}^{T}\varSigma _{ k}^{-1})]\text{vec}(\mu -\beta X\varGamma ) {}\\ & =& [I_{m} \otimes (1_{k}^{T}\varSigma _{ k}^{-1}1_{ k})^{-1}(1_{ k}^{T}\varSigma _{ k}^{-1})]\text{vec}(\mu -\beta X\varGamma ). {}\\ \end{array}$$

Therefore, $\hat{\gamma }^{T} = (1_{k}^{T}\varSigma _{k}^{-1}1_{k})^{-1}1_{k}^{T}\varSigma _{k}^{-1}(\mu -\beta X\varGamma )$, which is zero given $1_{k}^{T}\varSigma _{k}^{-1}X = 0 = 1_{k}^{T}\varSigma _{k}^{-1}\mu$. Referring to the notation of Result 9.2.2, if $\varSigma = I_{m} \otimes \varSigma _{k}$ then $A_{11} = A_{22} = (1_{k}^{T}\varSigma _{k}^{-1}1_{k})^{-1}1_{k}^{T}\varSigma _{k}^{-1}$ and $A_{12} = A_{21} = 0_{k}^{T}$. Then, from Eq. (9.4), if X and μ are located such that $1_{k}^{T}\varSigma _{k}^{-1}X = 0 = 1_{k}^{T}\varSigma _{k}^{-1}\mu$, then $\alpha _{i} =\delta _{i} =\zeta _{i} = 0$, for i = 1, 2, and P, Q, R, S and T simplify to

$$\displaystyle\begin{array}{rcl} P = Q& =& X_{1}^{T}\varSigma _{ k}^{-1}X_{ 1} + X_{2}^{T}\varSigma _{ k}^{-1}X_{ 2}, {}\\ R& =& -2(X_{1}^{T}\varSigma _{ k}^{-1}X_{ 2} - X_{2}^{T}\varSigma _{ k}^{-1}X_{ 1}) = 0, {}\\ S& =& -2(X_{1}^{T}\varSigma _{ k}^{-1}\mu _{ 1} + X_{2}^{T}\varSigma _{ k}^{-1}\mu _{ 2}), {}\\ T& =& 2(X_{2}^{T}\varSigma _{ k}^{-1}\mu _{ 1} - X_{1}^{T}\varSigma _{ k}^{-1}\mu _{ 2}). {}\\ \end{array}$$

The minimising rotation and scaling can then be obtained and have been derived by Brignell [2]. $\square$

Proof of Result 3.1.

$$\displaystyle\begin{array}{rcl} G_{\mathrm{CWP}}(X_{1},\ldots,X_{n};\varSigma )& =&\sum _{ i=1}^{n}\text{vec}(R_{i}\varGamma )^{T}\varSigma ^{-1}\text{vec}(R_{i}\varGamma ), {}\\ & = & \sum _{i=1}^{n}\left [\begin{array}{c} R_{i1}\cos \theta - R_{i2}\sin \theta \\ R_{i2}\cos \theta + R_{i1}\sin \theta \end{array} \right ]^{T}\varSigma ^{-1}\left [\begin{array}{c} R_{i1}\cos \theta - R_{i2}\sin \theta \\ R_{i2}\cos \theta + R_{i1}\sin \theta \end{array} \right ] {}\\ & = & p\cos ^{2}\theta + q\sin ^{2}\theta + 2r\cos \theta \sin \theta {}\\ \frac{dG_{\mathrm{CWP}}} {d\theta } & =& (q - p)\sin 2\theta + 2r\cos 2\theta. \\ \end{array}$$

Therefore, the minimum of G _CWP is when $\theta$ is a solution of this last equation. $\square$

Proof of Result 3.2.

The log-likelihood of the multivariate normal model, $\text{vec}(X_{i}) \sim N_{km}(\text{vec}(\mu ),\varSigma )$, where X _i are shapes invariant under Euclidean similarity transformations, is

$$\displaystyle\begin{array}{rcl} \log L(X_{1},\ldots,X_{n};\mu,\varSigma ) = -\frac{n} {2} \log \vert 2\pi \varSigma \vert && {}\\ & &-\frac{1} {2}\sum _{i=1}^{n}\text{vec}(\beta _{ i}X_{i}\varGamma _{i} + 1_{k}\gamma _{i}^{T}-\mu )^{T}\varSigma ^{-1}\text{vec}(\beta _{ i}X_{i}\varGamma _{i} + 1_{k}\gamma _{i}^{T}-\mu ). {}\\ \end{array}$$

Therefore, the MLE of the mean shape is the solution of

$$\displaystyle\begin{array}{rcl} \frac{d\log L} {d\mu } & =& \sum _{i=1}^{n}\varSigma ^{-1}\text{vec}(\beta _{ i}X_{i}\varGamma _{i} + 1_{k}\gamma _{i}^{T}) - n\varSigma ^{-1}\mu = 0. {}\\ \end{array}$$

Hence, $\hat{\mu }=\bar{ X} = \frac{1} {n}\sum _{i=1}^{n}(\beta _{ i}X_{i}\varGamma _{i} + 1_{k}\gamma _{i}^{T})$ and

$$\displaystyle\begin{array}{rcl} \log L& =& -\frac{n} {2} \log \vert 2\pi \varSigma \vert -\frac{1} {2}\inf _{\beta _{i},\varGamma _{i},\gamma _{i}}\sum _{i=1}^{n}\|\beta _{ i}X_{i}\varGamma _{i} + 1_{k}\gamma _{i}^{T} -\bar{ X}\|_{\varSigma }^{2} {}\\ & =& -\frac{n} {2} \log \vert 2\pi \varSigma \vert -\frac{1} {2}G_{\mathrm{CWP}}(X_{1},\ldots,X_{n};\varSigma ). {}\\ \end{array}$$

Therefore, minimising G _CWP is equivalent to maximising $L(X_{1},\ldots,X_{n};\mu,\varSigma )$. $\square$

Proof of Result 4.1.

Let the m columns of $(I_{m} \otimes 1_{k})$ be 1 _j for j = 1, …, m and let γ _ij be the jth element of the translation vector for shape X _i , then the log of the likelihood, L, for the multivariate normal model can be written:

$$\displaystyle\begin{array}{rcl} \log L =& & -\frac{n} {2} \log \vert 2\pi \varSigma \vert -\frac{1} {2}\sum _{i=1}^{n}\left (\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu )^{T}\varSigma ^{-1}\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu )\right ) {}\\ & & -\frac{1} {2}\sum _{i=1}^{n}\left (-2\sum _{ j=1}^{m}\gamma _{ ij}1_{j}^{T}\varSigma ^{-1}\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu ) +\sum _{ j=1}^{m}\gamma _{ ij}^{2}1_{ j}^{T}\varSigma ^{-1}1_{ j}\right ). {}\\ \end{array}$$

Now, $1_{j}^{T}\varSigma ^{-1} = \frac{\sigma _{j}^{-1}} {\sqrt{k}} 1_{j}^{T}1_{j}1_{j}^{T}$ as all the eigenvectors of $\varSigma$ are orthogonal to 1 _j except the one proportional to 1 _j . Therefore,

$$\displaystyle\begin{array}{rcl} \log L& =& -\frac{n} {2} \log \vert 2\pi \varSigma \vert -\frac{1} {2}\sum _{i=1}^{n}\left (\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu )^{T}\varSigma ^{-1}\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu )\right ) {}\\ & & -\frac{1} {2}\sum _{i=1}^{n}\left (-2\sum _{ j=1}^{m} \frac{\gamma _{ij}} {\sigma _{j}\sqrt{k}}1_{j}^{T}1_{ j}1_{j}^{T}\text{vec}(\beta _{ i}X_{i}\varGamma _{i}-\mu ) +\sum _{ j=1}^{m} \frac{\gamma _{ij}^{2}} {\sigma _{j}\sqrt{k}}1_{j}^{T}1_{ j}1_{j}^{T}1_{ j}\right ). {}\\ \end{array}$$

Given X _i and μ are all centred, $1_{j}^{T}\text{vec}(\beta _{i}X_{i}\varGamma _{i}-\mu ) = 0$, and the maximizing translation is clearly γ _ij  = 0 for all i = 1, …, n and j = 1, …, m. $\square$