Many problems in computer vision can be formulated as geometric estimation
problems, i.e. given a collection of measurements (e.g. point correspondences)
we wish to fit a model (e.g. an essential matrix) that agrees with our
observations. This necessitates some measure of how much an observation
``agrees" with a given model. A natural choice is to consider the smallest
perturbation that makes the observation exactly satisfy the constraints.
However, for many problems, this metric is expensive or otherwise intractable
to compute. The so-called Sampson error approximates this geometric error
through a linearization scheme. For epipolar geometry, the Sampson error is a
popular choice and in practice known to yield very tight approximations of the
corresponding geometric residual (the reprojection error).
In this paper we revisit the Sampson approximation and provide new
theoretical insights as to why and when this approximation works, as well as
provide explicit bounds on the tightness under some mild assumptions. Our
theoretical results are validated in several experiments on real data and in
the context of different geometric estimation tasks.

研究纸试图重新审视 Sampson 近似，并提供新的理论洞察力，解释了这一近似在何时何地有效，并在一些温和的假设下提供了明确的收敛边界。通过对真实数据和不同的几何估计任务进行验证，我们得出了一些实验证明了我们的理论结果。

重访几何估计问题的 Sampson 近似

Revisiting Sampson Approximations for Geometric Estimation Problems

Dense 3D reconstruction and ego-motion estimation are key challenges in
autonomous driving and robotics. Compared to the complex, multi-modal systems
deployed today, multi-camera systems provide a simpler, low-cost alternative.
However, camera-based 3D reconstruction of complex dynamic scenes has proven
extremely difficult, as existing solutions often produce incomplete or
incoherent results. We propose R3D3, a multi-camera system for dense 3D
reconstruction and ego-motion estimation. Our approach iterates between
geometric estimation that exploits spatial-temporal information from multiple
cameras, and monocular depth refinement. We integrate multi-camera feature
correlation and dense bundle adjustment operators that yield robust geometric
depth and pose estimates. To improve reconstruction where geometric depth is
unreliable, e.g. for moving objects or low-textured regions, we introduce
learnable scene priors via a depth refinement network. We show that this design
enables a dense, consistent 3D reconstruction of challenging, dynamic outdoor
environments. Consequently, we achieve state-of-the-art dense depth prediction
on the DDAD and NuScenes benchmarks.

提出了一种多摄像机系统来实现密集的三维重建和自我运动估计，通过多摄像机间的几何估计和单目深度优化来获得鲁棒的几何深度和姿态估计，同时通过深度优化网络引入可学习的场景先验，从而在具有挑战性的动态室外环境中实现了稠密、一致的三维重建。

R3D3: 多相机下动态场景的稠密三维重建

R3D3: Dense 3D Reconstruction of Dynamic Scenes from Multiple Cameras

Many geometric estimation problems take the form of synchronization over the
special Euclidean group: estimate the values of a set of poses given noisy
measurements of a subset of their pairwise relative transforms. This problem is
typically formulated as a maximum-likelihood estimation that requires solving a
nonconvex nonlinear program, which is computationally intractable in general.
Nevertheless, in this paper we present an algorithm that is able to efficiently
recover certifiably globally optimal solutions of this estimation 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 the 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 on
a low-dimensional Riemannian manifold, and then design a Riemannian
truncated-Newton trust-region method to solve this reduction efficiently. 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 globally optimal solutions when the available measurements are
corrupted by noise up to an order of magnitude greater than that typically
encountered in robotics applications, and does so at a computational cost that
scales comparably with that of direct Newton-type local search techniques.

本文介绍一种解决几何估计中同步问题的算法，通过基于半定松弛的最大似然估计得出全局最优解，该算法可以在非对抗噪声情况下高效恢复可认证的全局最优解，并在大规模实例上使用低维黎曼曲面与特殊优化方案实现了优化问题的求解。