386 research outputs found
Convex Relaxations for Pose Graph Optimization with Outliers
Pose Graph Optimization involves the estimation of a set of poses from
pairwise measurements and provides a formalization for many problems arising in
mobile robotics and geometric computer vision. In this paper, we consider the
case in which a subset of the measurements fed to pose graph optimization is
spurious. Our first contribution is to develop robust estimators that can cope
with heavy-tailed measurement noise, hence increasing robustness to the
presence of outliers. Since the resulting estimators require solving nonconvex
optimization problems, we further develop convex relaxations that approximately
solve those problems via semidefinite programming. We then provide conditions
under which the proposed relaxations are exact. Contrarily to existing
approaches, our convex relaxations do not rely on the availability of an
initial guess for the unknown poses, hence they are more suitable for setups in
which such guess is not available (e.g., multi-robot localization, recovery
after localization failure). We tested the proposed techniques in extensive
simulations, and we show that some of the proposed relaxations are indeed tight
(i.e., they solve the original nonconvex problem 10 exactly) and ensure
accurate estimation in the face of a large number of outliers.Comment: 10 pages, 5 figures, accepted for publication in the IEEE Robotics
and Automation Letters, 201
A Decoupled and Linear Framework for Global Outlier Rejection over Planar Pose Graph
We propose a robust framework for the planar pose graph optimization
contaminated by loop closure outliers. Our framework rejects outliers by first
decoupling the robust PGO problem wrapped by a Truncated Least Squares kernel
into two subproblems. Then, the framework introduces a linear angle
representation to rewrite the first subproblem that is originally formulated
with rotation matrices. The framework is configured with the Graduated
Non-Convexity (GNC) algorithm to solve the two non-convex subproblems in
succession without initial guesses. Thanks to the linearity properties of both
the subproblems, our framework requires only linear solvers to optimally solve
the optimization problems encountered in GNC. We extensively validate the
proposed framework, named DEGNC-LAF (DEcoupled Graduated Non-Convexity with
Linear Angle Formulation) in planar PGO benchmarks. It turns out that it runs
significantly (sometimes up to over 30 times) faster than the standard and
general-purpose GNC while resulting in high-quality estimates.Comment: 7 pages, 4 figures. Submitted to the IEEE International Conference on
Robotics and Automation (ICRA
In Perfect Shape: Certifiably Optimal 3D Shape Reconstruction from 2D Landmarks
We study the problem of 3D shape reconstruction from 2D landmarks extracted
in a single image. We adopt the 3D deformable shape model and formulate the
reconstruction as a joint optimization of the camera pose and the linear shape
parameters. Our first contribution is to apply Lasserre's hierarchy of convex
Sums-of-Squares (SOS) relaxations to solve the shape reconstruction problem and
show that the SOS relaxation of minimum order 2 empirically solves the original
non-convex problem exactly. Our second contribution is to exploit the structure
of the polynomial in the objective function and find a reduced set of basis
monomials for the SOS relaxation that significantly decreases the size of the
resulting semidefinite program (SDP) without compromising its accuracy. These
two contributions, to the best of our knowledge, lead to the first certifiably
optimal solver for 3D shape reconstruction, that we name Shape*. Our third
contribution is to add an outlier rejection layer to Shape* using a truncated
least squares (TLS) robust cost function and leveraging graduated non-convexity
to solve TLS without initialization. The result is a robust reconstruction
algorithm, named Shape#, that tolerates a large amount of outlier measurements.
We evaluate the performance of Shape* and Shape# in both simulated and real
experiments, showing that Shape* outperforms local optimization and previous
convex relaxation techniques, while Shape# achieves state-of-the-art
performance and is robust against 70% outliers in the FG3DCar dataset.Comment: Camera-ready, CVPR 2020. 18 pages, 5 figures, 1 tabl
On Semidefinite Relaxations for Matrix-Weighted State-Estimation Problems in Robotics
In recent years, there has been remarkable progress in the development of
so-called certifiable perception methods, which leverage semidefinite, convex
relaxations to find global optima of perception problems in robotics. However,
many of these relaxations rely on simplifying assumptions that facilitate the
problem formulation, such as an isotropic measurement noise distribution. In
this paper, we explore the tightness of the semidefinite relaxations of
matrix-weighted (anisotropic) state-estimation problems and reveal the
limitations lurking therein: matrix-weighted factors can cause convex
relaxations to lose tightness. In particular, we show that the semidefinite
relaxations of localization problems with matrix weights may be tight only for
low noise levels. We empirically explore the factors that contribute to this
loss of tightness and demonstrate that redundant constraints can be used to
regain tightness, albeit at the expense of real-time performance. As a second
technical contribution of this paper, we show that the state-of-the-art
relaxation of scalar-weighted SLAM cannot be used when matrix weights are
considered. We provide an alternate formulation and show that its SDP
relaxation is not tight (even for very low noise levels) unless specific
redundant constraints are used. We demonstrate the tightness of our
formulations on both simulated and real-world data
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