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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., Published on International journal of computer vision (IJCV) 2022

Registration is the problem of bringing together two or more 3D shapes, either of the same object or of two different but similar objects. This chapter first introduces the classical Iterative Closest Point (ICP) algorithm, which represents the gold standard registration method. Current limitations of ICP are addressed and the most popular variants are described to improve the basic implementation in several ways. Challenging registration scenarios are analyzed and a taxonomy of recent and promising alternative registration techniques is introduced. Three case studies are then described with an increasing level of problem difficulty. The first case study describes a simple but effective technique to detect outliers. The second case study uses the Levenberg-Marquardt optimization procedure to solve standard pairwise registration. The third case study focuses on the challenging problem of deformable object registration. Finally, open issues and directions for future work are discussed and conclusions are drawn.