Stratified Generalized Procrustes Analysis

Abstract

Generalized procrustes analysis computes the best set of transformations that relate matched shape data. In shape analysis the transformations are usually chosen as similarities, while in general statistical data analysis other types of transformation groups such as the affine group may be used. Generalized procrustes analysis has a nonlinear and nonconvex formulation. The classical approach alternates the computation of a so-called reference shape and the computation of transformations relating this reference shape to each shape datum in turn.

We propose the stratified approach to generalized procrustes analysis. It first uses the affine transformation group to analyze the data and then upgrades the solution to the sought after group, whether Euclidean or similarity. We derive a convex formulation for each of these two steps, and efficient practical algorithms that gracefully handle missing data (incomplete shapes).

Extensive experimental results show that our approaches perform well on simulated and real data. In particular our closed-form solution gives very accurate results for generalized procrustes analysis of Euclidean data.

Appendices

Appendix A: Demonstrations

1.1 A.1 Solution to $\min_{\mathsf {S}} \| \mathsf {L} \mathsf {S} \|_{\mathcal{F}}$ Such That S ^⊤ S=I

Let L∈ℝ^q×d we want to show that:

$$\arg\min_{\mathsf {S}\in O(d)} \| \mathsf {L} \mathsf {S} \|_{\mathcal{F}}^2 = \bar {\mathsf {V}}, $$

with $\bar {\mathsf {V}}$ the last d columns of matrix V in the SVD . We write s _k , k=1,…,d the columns of S (similarly for V with v _k ). We rewrite the above problem as a set of nested problems:

The last column s _d is found first by computing the partial derivatives of the lagrangian:

$$\mathcal{L}_d(\mathbf {s}_d) \stackrel{\mathrm{def}}{=}\| \mathsf {L} \mathbf {s}_d \|_2^2 + \lambda\bigl(1 - \| \mathbf {s}_d \|^2_2\bigr), $$

where λ is a Lagrange multiplier, giving:

$${ \mathsf {L}}^{\top }\mathsf {L} \mathbf {s}_d = \lambda \mathbf {s}_d. $$

This shows that s _d is a right singular vector of L, and so s _d =v _k for some k∈{1,…,d}. The cost for v _k is $\| \mathsf {L}\mathbf {v}_{k}\|_{2}^{2} = \sigma_{k}^{2}$ and⁸ we therefore choose s _d ←v _d to minimize the cost.

The other columns are then found in turn, as follows. Consider s _d−1. The lagrangian is:

$$\mathcal{L}_{d-1}(\mathbf {s}_{d-1}) \stackrel{\mathrm{def}}{=}\| \mathsf {L} \mathbf {s}_{d-1} \|_2^2 + \lambda\bigl( 1-\| \mathbf {s}_{d-1} \|_2^2\bigr) + \mu\bigl( { \mathbf {s}}^{\top }_{d-1} \mathbf {s}_d \bigr)^2, $$

and its partial derivatives are:

$$\frac{1}{2} \frac{\partial \mathcal{L}_{d-1}}{\partial \mathbf {s}_{d-1}} = { \mathsf {L}}^{\top }\mathsf {L} \mathbf {s}_{d-1} - \lambda \mathbf {s}_{d-1} + \mu { \mathbf {s}}^{\top }_d \mathbf {s}_d \mathbf {s}_{d-1} = 0. $$

Since ${ \mathbf {s}}^{\top }_{d-1} \mathbf {s}_{d-1} = \| \mathbf {s}_{d-1} \|^{2}_{2} = 1$ this gives:

$${ \mathsf {L}}^{\top }\mathsf {L} \mathbf {s}_{d-1} = (\lambda-\mu) \mathbf {s}_{d-1}. $$

Similarly to s _d we finally find that s _d−1←v _k for some k∈{1,…,d}, and choose s _d−1←v _d−1 to minimize the cost and satisfy the constraints. Doing the same reasoning for all the remaining columns leads to $\mathsf {S} \leftarrow \bar {\mathsf {V}}$.

1.2 A.2 Translational Gauge Constraints for the Reference-Space Cost

The reference-space cost is translation invariant, which means that we can simply fix the translation part of the gauge by adding a gauge constraint to the cost, as if it were a simple penalty. We chose to center the reference shape using S ^⊤ 1=0. Below, we prove that the original cost function is invariant to translations of the reference shape. Note that a translation represented by vector g∈ℝ^d applies to the reference shape S as S   →   S+1g ^⊤ with 1∈ℝ^m.

Consider the minimization problem (15), and plug in the translational gauge transformation of the reference shape mentioned directly above:

$$\big\| ( {\mathrm{I}}- \hat {\mathsf {K}}_B ) \mathsf {K}_S \mathsf {S} \big\|^2_{\mathcal{F}} \,\,\,\rightarrow\,\,\, \big\| ( {\mathrm{I}}- \hat {\mathsf {K}}_B ) \mathsf {K}_S \bigl( \mathsf {S} + \mathbf {1} {\mathbf {g}}^{\top } \bigr) \big\|^2_{\mathcal{F}}. $$

This is expanded to $\| ( {\mathrm{I}}- \hat {\mathsf {K}}_{B} ) \mathsf {K}_{S} \mathsf {S} + ( {\mathrm{I}}- \hat {\mathsf {K}}_{B} ) \mathsf {K}_{S} \mathbf {1} {\mathbf {g}}^{\top } \|^{2}_{\mathcal{F}}$. Consider the second factor. By construction, matrix K _S ∈ℝ^α×m is a stack of identity matrices to which some rows were removed, and thus K _S 1=1∈ℝ^α, leading to:

$$ ( {\mathrm{I}}- \hat {\mathsf {K}}_B ) \mathsf {K}_S \mathbf {1} {\mathbf {g}}^{\top } = \mathbf {1} {\mathbf {g}}^{\top } - \hat {\mathsf {K}}_B \mathbf {1} {\mathbf {g}}^{\top }. $$

(33)

We have that $\hat {\mathsf {K}}_{B} = \mathrm{diag}(\hat {\mathsf {K}}_{B,i})$ and K _B,i =(D _i     1). Denoting $\mathbf {d}_{i} = { \mathsf {D}}^{\top }_{i} \mathbf {1}$, we get:

from which, defining $\mathsf {H}_{i} \stackrel{\mathrm{def}}{=}{ ( { \mathsf {D}}^{\top }_{i} \mathsf {D}_{i} - \frac{1}{m} \mathbf {d}_{i} { \mathbf {d}}^{\top }_{i} ) }^{-1}$ we get:

This shows from (33) that $( {\mathrm{I}}- \hat {\mathsf {K}}_{B} ) \mathsf {K}_{S} \mathbf {1} {\mathbf {g}}^{\top } = \mathbf {0}$ and that the reference-space cost is thus translational gauge invariant: $\| ( {\mathrm{I}}- \hat {\mathsf {K}}_{B} ) \mathsf {K}_{S} \mathsf {S} \| ^{2}_{\mathcal{F}} = \| ( {\mathrm{I}}- \hat {\mathsf {K}}_{B} ) \mathsf {K}_{S} ( \mathsf {S} + \mathbf {1} {\mathbf {g}}^{\top } ) \| ^{2}_{\mathcal{F}}$.

1.3 A.3 The Closest Special Orthonormal Matrix

We give an simple demonstration to find the closest special orthonormal E∈SO(d) matrix to a matrix M with det(M)>0 minimizing the cost $\mathcal{C}\stackrel{\mathrm{def}}{=}\| \mathsf {E} - \mathsf {M} \|^{2}_{\mathcal{F}}$. Different solutions and proofs exist in the literature (Horn et al. 1988; Schönemann and Carroll 1970); they are more involved than ours. We first plug the SVD $\mathsf {M} \stackrel{\mathtt{SVD}}{\rightarrow} \mathsf {U}\mathsf {\Sigma} { \mathsf {V}}^{\top }$ in the cost, and since U∈O(d) and V∈O(d) we get with $\mathsf {H} \stackrel{\mathrm{def}}{=}{ \mathsf {U}}^{\top } \mathsf {E} \mathsf {V}$ and . We are left with the problem of finding the closest orthonormal matrix to a diagonal matrix. Expanding the norm in the cost, we get that we rewrite as:

$$\mathcal{C}= d + \sigma_1 + \cdots+ \sigma_d - 2 ( H_{1,1} \sigma_1 + \cdots+ H_{d,d} \sigma_d). $$

Given that H∈O(d) we have that H _k,k ≤1, k=1,…,d. Noting that σ _k ≥0, k=1,…,d, the solution that minimizes the expression of $\mathcal{C}$ directly above is given by choosing H _k,k =1, k=1,…,d, leading to H=I. We finally get the solution:

$$\mathsf {E} = \mathsf {U} { \mathsf {V}}^{\top }. $$

It is to be noted that σ _k >0, k=1,…,d then . Since det(M)>0, U∈O(d) and V∈O(d) this implies that det(E)=det(U)det(V)=1 and thus E∈SO(d) as sought.

We now look into the demonstration of the scaled case, where we have to find both matrix E∈SO(d) and a scale factor ζ>0 such that the following cost $\mathcal{S}\stackrel{\mathrm{def}}{=}\| \zeta \mathsf {E} - \mathsf {M}\|^{2}_{\mathcal{F}}$ is minimized. As in the above case, we use the SVD of M to get where H is defined as above. We expand the norm in the cost as above and get that we rewrite as:

$$\mathcal{S}= d \zeta^2 + \sigma_1 + \cdots+ \sigma_d - 2 \zeta( H_{1,1} \sigma_1 + \cdots+ H_{d,d} \sigma_d). $$

The same argument as above directly leads to H=I (and thus E=UV ^⊤). This leaves us with:

$$\mathcal{S}= d\zeta^2 + \sigma_1 + \cdots+ \sigma_d - 2 \zeta( \sigma _1 + \cdots+ \sigma_d). $$

Setting the derivatives of $\mathcal{S}$ with respect to ζ to as to find the minimizer gives:

$$\frac{1}{2} \frac{\partial \mathcal{S}}{\partial\zeta} = d\zeta- ( \sigma _1 + \cdots+ \sigma_d) = 0, $$

and readily leads to:

$$\zeta = \frac{1}{d} \sum_{k=1}^d \sigma_k. $$

It is easily verified that ζ>0 as required.

Appendix B: Affine Generalized Procrustes Analysis with the Data-Space Model and PCA

We here show that affine generalized procrustes analysis with the data-space model is equivalent to PCA. We instanciate the data-space cost (4) using affine transformations T _A,i =(A _i ;a _i ):

$$\mathcal{R}(\mathcal{T}_A,\mathsf {S}) = \sum_{i=1}^n v_{i,j} \big\| \mathsf {D}_i^{\top} - \mathsf {A}_i \mathsf {S}^{\top} - \mathbf {a}_i { \mathbf {1}}^{\top } \big\|^2_{\mathcal{F}}. $$

Minimizing $\mathcal{R}$ in that case is a nonconvex problem, equivalent to an L ₂ norm bilinear factorization of a data matrix containing all the shape points. The first factor contains the affine transformations while the second factor contains the reference shape. This makes the problem easy to solve by simple rank-d matrix factorization when there is no missing data (for v _i,j =1, i=1,…,n, j=1,…,m, from Eckart-Young/Schmidt’s approximation theorem (Stewart 1993)). In the presence of missing data however, the problem does not have such a simple solution. We solve it using Orthogonal Distance Regression (ODR) (Boggs et al. 1989) using the reference-space solution as initialization but techniques from PCA with missing data could also be used (Jacobs 2001).

Appendix C: Implementation of Initialization and Upgrading

We here give details of our affine reference-space initialization and similarly-Euclidean upgrading procedures. These are respectively given in Tables 4 and 5.

Table 4 Affine registration with the reference-space model. Implementation of our initial closed-form affine registration algorithm in the reference-space model. The mathematical derivation is in Sect. 5

Table 5 Similarity-Euclidean registration by upgrading an affine registration. Implementation of our closed-form upgrading algorithm. The mathematical derivation is in Sect. 6

Appendix D: The Generalized Procrustes Analysis Toolbox

We created a Matlab toolbox that implements our proposed algorithms. We called it the generalized procrustes analysis toolbox; it will be released under the GPL license. Our toolbox offers a ready-to-use interface for registering shape data using any of the described methods for affine, Euclidean and similarity transformations. It also includes the classical alternation approach for similarity and Euclidean transformations and copes with incomplete shape data.

4.1 D.1 Main Function

The whole set of algorithms is bundled in a single function called gpa. It has the following syntax:

$$\mbox{\texttt{T=gpa(D,V,options);}} $$

where D contains the shape data, V indicates the missing points and options specifies which algorithm to use, the level of verbosity and the maximum number of iterations. The output T is a structure that contains the computed transformations, the error for the different trials and the computational time for various steps of the algorithms. Of course, a detailed help can be obtained by typing:

$$\mbox{\texttt{help gpa;}} $$

4.2 D.2 Simulating Data and Testing

All the experiments with simulated data that we reported in this paper can be replicated using an experimental front-end function. It has a compact interface that generates data and then calls the gpa function. This function is called genExperiment and has the following syntax:

$$\mbox{\texttt{genExperiment(options,name);}} $$

There is no output since the results will be saved in a file. options is a structure that defines the method or methods that will be tested, the number of shape data, the dimension, the ratio of missing data, the amount of added noise or of deformation between the shapes, the parameter that will be varied, the number of random trials, the name of the file to save the results and the type of data (transformations) to be generated. The second input parameter called name specifies the name of a file containing the results of an experiment which is yet to be completed and from which the current experiment will be started.

As an example of usage of genExperiment, we give the piece of code that replicates experiment the top-row noise graph shown in Fig. 4:

The results will be saved in a file called exp1RMSvssigma.mat. If the experiment could not complete for some reason it can be restarted from where it stopped by typing:

$$\mbox{\texttt{genExperiment(options,'exp1RMSvssigma');}} $$

Of course, a detailed help for this function can be obtained by typing:

$$\mbox{\texttt{help genExperiment;}} $$