1,984 research outputs found
SIM-Sync: From Certifiably Optimal Synchronization over the 3D Similarity Group to Scene Reconstruction with Learned Depth
This paper presents SIM-Sync, a certifiably optimal algorithm that estimates
camera trajectory and 3D scene structure directly from multiview image
keypoints. SIM-Sync fills the gap between pose graph optimization and bundle
adjustment; the former admits efficient global optimization but requires
relative pose measurements and the latter directly consumes image keypoints but
is difficult to optimize globally (due to camera projective geometry). The
bridge to this gap is a pretrained depth prediction network. Given a graph with
nodes representing monocular images taken at unknown camera poses and edges
containing pairwise image keypoint correspondences, SIM-Sync first uses a
pretrained depth prediction network to lift the 2D keypoints into 3D scaled
point clouds, where the scaling of the per-image point cloud is unknown due to
the scale ambiguity in monocular depth prediction. SIM-Sync then seeks to
synchronize jointly the unknown camera poses and scaling factors (i.e., over
the 3D similarity group). The SIM-Sync formulation, despite nonconvex, allows
designing an efficient certifiably optimal solver that is almost identical to
the SE-Sync algorithm. We demonstrate the tightness, robustness, and practical
usefulness of SIM-Sync in both simulated and real experiments. In simulation,
we show (i) SIM-Sync compares favorably with SE-Sync in scale-free
synchronization, and (ii) SIM-Sync can be used together with robust estimators
to tolerate a high amount of outliers. In real experiments, we show (a)
SIM-Sync achieves similar performance as Ceres on bundle adjustment datasets,
and (b) SIM-Sync performs on par with ORB-SLAM3 on the TUM dataset with
zero-shot depth prediction.Comment: 28 page
A compact formula for the derivative of a 3-D rotation in exponential coordinates
We present a compact formula for the derivative of a 3-D rotation matrix with
respect to its exponential coordinates. A geometric interpretation of the
resulting expression is provided, as well as its agreement with other
less-compact but better-known formulas. To the best of our knowledge, this
simpler formula does not appear anywhere in the literature. We hope by
providing this more compact expression to alleviate the common pressure to
reluctantly resort to alternative representations in various computational
applications simply as a means to avoid the complexity of differential analysis
in exponential coordinates.Comment: 6 page
Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality
This paper studies a problem of maximizing the sum of traces of matrix
quadratic forms on a product of Stiefel manifolds. This orthogonal trace-sum
maximization (OTSM) problem generalizes many interesting problems such as
generalized canonical correlation analysis (CCA), Procrustes analysis, and
cryo-electron microscopy of the Nobel prize fame. For these applications
finding global solutions is highly desirable but has been out of reach for a
long time. For example, generalizations of CCA do not possess obvious global
solutions unlike their classical counterpart to which a global solution is
readily obtained through singular value decomposition; it is also not clear how
to test global optimality. We provide a simple method to certify global
optimality of a given local solution. This method only requires testing the
sign of the smallest eigenvalue of a symmetric matrix, and does not rely on a
particular algorithm as long as it converges to a stationary point. Our
certificate result relies on a semidefinite programming (SDP) relaxation of
OTSM, but avoids solving an SDP of lifted dimensions. Surprisingly, a popular
algorithm for generalized CCA and Procrustes analysis may generate oscillating
iterates. We propose a simple modification of this standard algorithm and prove
that it reliably converges. Our notion of convergence is stronger than
conventional objective value convergence or subsequence convergence.The
convergence result utilizes the Kurdyka-Lojasiewicz property of the problem.Comment: 22 pages, 1 figur
SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group
Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques
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