A solution for multi-alignment by transformation synchronisation

Abstract

The alignment of a set of objects by means of transformations plays an important role in computer vision. Whilst the case for only two objects can be solved globally, when multiple objects are considered usually iterative methods are used. In practice the iterative methods perform well if the relative transformations between any pair of objects are free of noise. However, if only noisy relative transformations are available (e.g. due to missing data or wrong correspondences) the iterative methods may fail.Based on the observation that the underlying noise-free transformations can be retrieved from the null space of a matrix that can directly be obtained from pairwise alignments, this paper presents a novel method for the synchronisation of pairwise transformations such that they are transitively consistent.Simulations demonstrate that for noisy transformations, a large proportion of missing data and even for wrong correspondence assignments the method delivers encouraging results.

Distributed synchronization of Euclidean transformations with guaranteed convergence

“…Examples of such include the 3D localization problem, where rigid transformations are calculated from camera measurements [1]- [3]; the problem of registering multiple images [4]; the Generalized Procrustes Problem, where rotations, translations and scales are calculated from multiple point-clouds [5]- [9]. Several more applications are enlisted in [10].…”

Synchronization over Cartan Motion Groups via Contraction

“…Second is the spectral method as done in [2,5]; however, we further accommodate it by allowing a varying scale on the translations as we do in the contraction method. The third is a separation based method implemented similarly to the approach in [15] (solving first the rotational parts of synchronization and then using the result for estimating the translational parts).…”

“…The matrix L can also be considered as an extension of the "twisted Laplacian," as defined in [34]. This cost functions bring us back to the spectral method, as the operator L is related to the Laplacian operator in [2] and [5]. Note that both papers [2,5] focus on the case d = 3 and relax the constraint µ i ∈ ρ (G) by forcing the solution µ to coincide with (0, 0, 0, 1) in each fourth row and then project the upper left block of each µ i on SO(3).…”

“…This cost functions bring us back to the spectral method, as the operator L is related to the Laplacian operator in [2] and [5]. Note that both papers [2,5] focus on the case d = 3 and relax the constraint µ i ∈ ρ (G) by forcing the solution µ to coincide with (0, 0, 0, 1) in each fourth row and then project the upper left block of each µ i on SO(3). Another interesting relation, this time with the "connection Laplacian," is established in [9], where a maximum likelihood estimation (MLE) is constructed for synchronization over SO(d).…”

“…Nevertheless, there is a way to address this difficulty, by considering a Laplacian operator and then to reveal its null space using the singular value decomposition (SVD). This spectral method is studied in both [2] and [5], where it was shown to be an efficient and accurate method.…”