Robust Absolute Rotation Estimation via Low-Rank and Sparse Matrix Decomposition

Abstract

This paper deals with the rotation synchronization problem, which arises in global registration of 3D point-sets and in structure from motion. The problem is formulated in an unprecedented way as a "low-rank and sparse" matrix decomposition that handles both outliers and missing data. A minimization strategy, dubbed R-GoDec, is also proposed and evaluated experimentally against state-of-the-art algorithms on simulated and real data. The results show that R-GoDec is the fastest among the robust algorithms. of structure from motion, or local coordinates where 3D points are represented, in which case we are dealing with a 3D point-set registration problem.More abstractly, the goal of the group synchronization problem [1, 2] is to recover elements of a group from noisy measures of their ratios. In our case, absolute angular attitudes R 1 , . . . , R n are elements of the Special Orthogonal Group SO (3), and relative attitudes R ij = R i R T j are their ratios. The same problem is analysed in depth in [3], under the name "multiple rotation averaging". Our solution to the rotation synchronization problem is inspired by recent advances in the fields of robust principal component analysis and matrix completion. The main and original contribution of this paper is the formulation of the rotation synchronization problem as a "low-rank and sparse matrix decomposition", by conceiving a novel cost function that naturally includes missing data and outliers in its definition. Secondly, we develop a minimization scheme for that cost function -called R-GoDec -that leverages on the GoDec algorithm [4]. The resulting method is robust, by construction, fast, thanks to the use of Bilateral Random Projections (BRP) in place of Singular Value Decomposition (SVD), and compact, as it consist of a single fixed point iteration that can be coded in a few lines of MATLAB. Most of all, the framework is modular, as -in principle -any low-rank and sparse decomposition method able to deal with outliers and missing data can replace R-GoDec.This paper is organized as follows. Applications of the rotation synchronization problem are presented in Section 2 while existing solutions are described in Section 3. Section 4 is an overview of the theoretical background required to define our algorithm, i.e. lowrank and sparse matrix decomposition. Section 5 defines the rotation synchronization problem. Section 6 provides a detailed description of our robust solution. The method proposed in this section is supported by experimental results on both synthetic and real data, shown in Section 7. The conclusions are presented in Section 8. This paper is an extended version of [5].

Robust Global Motion Estimation with Matrix Completion

“…As an alternative, we are developing a matrix completion algorithm which is robust both with respect to noise and outliers, avoiding the need of a demanding preliminary outlier rejection step (Arrigoni et al, 2014). Finally, we plan to investigate sufficient conditions under which the position estimation problem admits a unique non-trivial solution.…”

Learning Transformation Synchronization

“…The first category of methods [27,22,49,36,52] uses combinatorial optimization to select a subgraph that only contains consistent cycles. The second category of methods [47,31,25,23,24,13,53,42,33,26,7,39,38,2,9,4,5,41,19,46,6,21] can be viewed from the perspective that there is an equivalence between cycleconsistent transformations and the fact that the map collection matrix that stores relative transformations in blocks is semidefinite and/or low-rank (c.f. [23]).…”

“…Our approach is motivated from iteratively reweighted least squares (or IRLS) [18], which has been applied to transformation synchronization (e.g. [11,3,15,26]). The key idea of IRLS is to maintain an edge weight w ij , (i, j) ∈ E for each input transformation T in ij so that the objective function becomes quadratic in the variables, and transformation synchronization admits a closed-form solution.…”

“…This cycle-consistency constraint allows us to jointly improve relative transformations by either detecting inconsistent cycles [14,36] or performing low-rank matrix recovery [23,47,39,7,9]. However, the success of existing transformation synchronization [47,11,3,26] and more general map synchronization [23,39,38,13,42,26] techniques heavily depends on the compatibility between the loss function and the noise pattern of the input data. For example, approaches based on robust norms (e.g., L1 [23,13]) can tolerate either a constant fraction of adversarial noise (c.f.…”

“…Our approach is motivated by reweighted least squares and their application in transformation synchronization (c.f. [11,3,15,26]), where the loss function dictates how we update the weight associated with each input relative transformation during the synchronization process. Specifically, we design a recurrent neural network that reflects this reweighted scheme.…”