64 research outputs found
A QCQP Approach to Triangulation
Triangulation of a three-dimensional point from at least two noisy 2-D images
can be formulated as a quadratically constrained quadratic program. We propose
an algorithm to extract candidate solutions to this problem from its
semidefinite programming relaxations. We then describe a sufficient condition
and a polynomial time test for certifying when such a solution is optimal. This
test has no false positives. Experiments indicate that false negatives are
rare, and the algorithm has excellent performance in practice. We explain this
phenomenon in terms of the geometry of the triangulation problem.Comment: 14 pages, to appear in the proceedings of the 12th European
Conference of Computer Vision, Firenze, Italy, 7-13 October 201
On the local stability of semidefinite relaxations
We consider a parametric family of quadratically constrained quadratic
programs (QCQP) and their associated semidefinite programming (SDP)
relaxations. Given a nominal value of the parameter at which the SDP relaxation
is exact, we study conditions (and quantitative bounds) under which the
relaxation will continue to be exact as the parameter moves in a neighborhood
around the nominal value. Our framework captures a wide array of statistical
estimation problems including tensor principal component analysis, rotation
synchronization, orthogonal Procrustes, camera triangulation and resectioning,
essential matrix estimation, system identification, and approximate GCD. Our
results can also be used to analyze the stability of SOS relaxations of general
polynomial optimization problems.Comment: 23 pages, 3 figure
Certifiably Correct Range-Aided SLAM
We present the first algorithm to efficiently compute certifiably optimal
solutions to range-aided simultaneous localization and mapping (RA-SLAM)
problems. Robotic navigation systems increasingly incorporate point-to-point
ranging sensors, leading to state estimation problems in the form of RA-SLAM.
However, the RA-SLAM problem is significantly more difficult to solve than
traditional pose-graph SLAM: ranging sensor models introduce non-convexity and
single range measurements do not uniquely determine the transform between the
involved sensors. As a result, RA-SLAM inference is sensitive to initial
estimates yet lacks reliable initialization techniques. Our approach,
certifiably correct RA-SLAM (CORA), leverages a novel quadratically constrained
quadratic programming (QCQP) formulation of RA-SLAM to relax the RA-SLAM
problem to a semidefinite program (SDP). CORA solves the SDP efficiently using
the Riemannian Staircase methodology; the SDP solution provides both (i) a
lower bound on the RA-SLAM problem's optimal value, and (ii) an approximate
solution of the RA-SLAM problem, which can be subsequently refined using local
optimization. CORA applies to problems with arbitrary pose-pose, pose-landmark,
and ranging measurements and, due to using convex relaxation, is insensitive to
initialization. We evaluate CORA on several real-world problems. In contrast to
state-of-the-art approaches, CORA is able to obtain high-quality solutions on
all problems despite being initialized with random values. Additionally, we
study the tightness of the SDP relaxation with respect to important problem
parameters: the number of (i) robots, (ii) landmarks, and (iii) range
measurements. These experiments demonstrate that the SDP relaxation is often
tight and reveal relationships between graph rigidity and the tightness of the
SDP relaxation.Comment: 17 pages, 9 figures, submitted to T-R
Carving out OPE space and precise O(2) model critical exponents
We develop new tools for isolating CFTs using the numerical bootstrap. A “cutting surface” algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d O(2) model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old 8σ discrepancy between theory and experiment
Carving out OPE space and precise O(2) model critical exponents
We develop new tools for isolating CFTs using the numerical bootstrap. A “cutting surface” algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d O(2) model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old 8σ discrepancy between theory and experiment
Convex Global 3D Registration with Lagrangian Duality
The registration of 3D models by a Euclidean transformation is a fundamental task at the core of many application in computer vision. This problem is non-convex due to the presence of rotational constraints, making traditional local optimization methods prone to getting stuck in local minima. This paper addresses finding the globally optimal transformation in various 3D registration problems by a unified formulation that integrates common geometric registration modalities (namely point-to-point, point-to-line and point-to-plane). This formulation renders the optimization problem independent of both the number and nature of the correspondences.
The main novelty of our proposal is the introduction of a strengthened Lagrangian dual relaxation for this problem, which surpasses previous similar approaches [32] in effectiveness.
In fact, even though with no theoretical guarantees, exhaustive empirical evaluation in both synthetic and real experiments always resulted on a tight relaxation that allowed to recover a guaranteed globally optimal solution by exploiting duality theory.
Thus, our approach allows for effectively solving the 3D registration with global optimality guarantees while running at a fraction of the time for the state-of-the-art alternative [34], based on a more computationally intensive Branch and Bound method.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
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