Procrustes Analysis with Deformations: A Closed-Form Solution by Eigenvalue Decomposition

Abstract

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 freetranslations, 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.

KernelGPA: A Deformable SLAM Back-end

“…One of the major barricades of extending SLAM from the rigid to nonrigid environment is the shortage of principled global registration techniques that handle deformations. At this stage, there are only several options available in this regards, i.e., the low-rank shape basis decomposition [14], the isometric nonrigid structure-frommotion [40], and more recently the generalized Procrustes analysis (GPA) with the linear basis warps (LBWs) [4].…”

“…We consider the case where sensor readings are interpreted as point-clouds. The closest work to this purpose is the DefGPA developed in [4], where the globally optimal map and the deformable transformations are estimated in closedform using the LBW models. In the language of SLAM, the results in [4] show that the principal axes of the environment map can be estimated globally, up to a global scale ambiguity along each axis.…”

“…The closest work to this purpose is the DefGPA developed in [4], where the globally optimal map and the deformable transformations are estimated in closedform using the LBW models. In the language of SLAM, the results in [4] show that the principal axes of the environment map can be estimated globally, up to a global scale ambiguity along each axis. The authors proposed a method to estimate the global scale ambiguity using pairwise rigid Procrustes analysis, by assuming sufficient co-visibility across sensor measurements at different poses.…”

“…We show that KernelGPA can be solved globally in closed-form, up to an unknown global scale ambiguity. KernelGPA provides extra modeling power compared to the DefGPA framework [4] based on the LBWs. • Second, we give insights that the global scale ambiguity and the rigid transformations are correlated, by assuming the transformation to be as-rigid-as-possible (ARAP).…”

“…Therefore we can solve both in a unified optimization framework. We do not require good co-visibility as in [4], and the new cost term provides a natural extension to the sensor fusion framework.…”