  Generalized Procrustes Analysis (GPA) is the problem of bringing multiple
shapes into a common reference by estimating transformations. GPA has been
extensively studied for the Euclidean and affine transformations. We introduce
GPA with deformable transformations, which forms a much wider and difficult
problem. We specifically study a class of transformations called the Linear
Basis Warps (LBWs), which contains the affine transformation and most of the
usual deformation models, such as the Thin-Plate Spline (TPS). GPA with
deformations is a nonconvex underconstrained problem. We resolve the
fundamental ambiguities of deformable GPA using two shape constraints requiring
the eigenvalues of the shape covariance. These eigenvalues can be computed
independently as a prior or posterior. We give a closed-form and optimal
solution to deformable GPA based on an eigenvalue decomposition. This solution
handles regularization, favoring smooth deformation fields. It requires the
transformation model to satisfy a fundamental property of free-translations,
which asserts that the model can implement any translation. We show that this
property fortunately holds true for most common transformation models,
including the affine and TPS models. For the other models, we give another
closed-form solution to GPA, which agrees exactly with the first solution for
models with free-translation. We give pseudo-code for computing our solution,
leading to the proposed DefGPA method, which is fast, globally optimal and
widely applicable. We validate our method and compare it to previous work on
six diverse 2D and 3D datasets, with special care taken to choose the
hyperparameters from cross-validation.
