6,997 research outputs found
Stable Camera Motion Estimation Using Convex Programming
We study the inverse problem of estimating n locations (up to
global scale, translation and negation) in from noisy measurements of a
subset of the (unsigned) pairwise lines that connect them, that is, from noisy
measurements of for some pairs (i,j) (where the
signs are unknown). This problem is at the core of the structure from motion
(SfM) problem in computer vision, where the 's represent camera locations
in . The noiseless version of the problem, with exact line measurements,
has been considered previously under the general title of parallel rigidity
theory, mainly in order to characterize the conditions for unique realization
of locations. For noisy pairwise line measurements, current methods tend to
produce spurious solutions that are clustered around a few locations. This
sensitivity of the location estimates is a well-known problem in SfM,
especially for large, irregular collections of images.
In this paper we introduce a semidefinite programming (SDP) formulation,
specially tailored to overcome the clustering phenomenon. We further identify
the implications of parallel rigidity theory for the location estimation
problem to be well-posed, and prove exact (in the noiseless case) and stable
location recovery results. We also formulate an alternating direction method to
solve the resulting semidefinite program, and provide a distributed version of
our formulation for large numbers of locations. Specifically for the camera
location estimation problem, we formulate a pairwise line estimation method
based on robust camera orientation and subspace estimation. Lastly, we
demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts
updated, some typos correcte
Robust Camera Location Estimation by Convex Programming
D structure recovery from a collection of D images requires the
estimation of the camera locations and orientations, i.e. the camera motion.
For large, irregular collections of images, existing methods for the location
estimation part, which can be formulated as the inverse problem of estimating
locations in
from noisy measurements of a subset of the pairwise directions
, are
sensitive to outliers in direction measurements. In this paper, we firstly
provide a complete characterization of well-posed instances of the location
estimation problem, by presenting its relation to the existing theory of
parallel rigidity. For robust estimation of camera locations, we introduce a
two-step approach, comprised of a pairwise direction estimation method robust
to outliers in point correspondences between image pairs, and a convex program
to maintain robustness to outlier directions. In the presence of partially
corrupted measurements, we empirically demonstrate that our convex formulation
can even recover the locations exactly. Lastly, we demonstrate the utility of
our formulations through experiments on Internet photo collections.Comment: 10 pages, 6 figures, 3 table
Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition
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.Comment: The material contained in this paper is part of a manuscript
submitted to CVI
Disparity and Optical Flow Partitioning Using Extended Potts Priors
This paper addresses the problems of disparity and optical flow partitioning
based on the brightness invariance assumption. We investigate new variational
approaches to these problems with Potts priors and possibly box constraints.
For the optical flow partitioning, our model includes vector-valued data and an
adapted Potts regularizer. Using the notation of asymptotically level stable
functions we prove the existence of global minimizers of our functionals. We
propose a modified alternating direction method of minimizers. This iterative
algorithm requires the computation of global minimizers of classical univariate
Potts problems which can be done efficiently by dynamic programming. We prove
that the algorithm converges both for the constrained and unconstrained
problems. Numerical examples demonstrate the very good performance of our
partitioning method
Spectral Motion Synchronization in SE(3)
This paper addresses the problem of motion synchronization (or averaging) and
describes a simple, closed-form solution based on a spectral decomposition,
which does not consider rotation and translation separately but works straight
in SE(3), the manifold of rigid motions. Besides its theoretical interest,
being the first closed form solution in SE(3), experimental results show that
it compares favourably with the state of the art both in terms of precision and
speed
A closed-form solution to estimate uncertainty in non-rigid structure from motion
Semi-Definite Programming (SDP) with low-rank prior has been widely applied
in Non-Rigid Structure from Motion (NRSfM). Based on a low-rank constraint, it
avoids the inherent ambiguity of basis number selection in conventional
base-shape or base-trajectory methods. Despite the efficiency in deformable
shape reconstruction, it remains unclear how to assess the uncertainty of the
recovered shape from the SDP process. In this paper, we present a statistical
inference on the element-wise uncertainty quantification of the estimated
deforming 3D shape points in the case of the exact low-rank SDP problem. A
closed-form uncertainty quantification method is proposed and tested. Moreover,
we extend the exact low-rank uncertainty quantification to the approximate
low-rank scenario with a numerical optimal rank selection method, which enables
solving practical application in SDP based NRSfM scenario. The proposed method
provides an independent module to the SDP method and only requires the
statistic information of the input 2D tracked points. Extensive experiments
prove that the output 3D points have identical normal distribution to the 2D
trackings, the proposed method and quantify the uncertainty accurately, and
supports that it has desirable effects on routinely SDP low-rank based NRSfM
solver.Comment: 9 pages, 2 figure
Exact Camera Location Recovery by Least Unsquared Deviations
We establish exact recovery for the Least Unsquared Deviations (LUD)
algorithm of Ozyesil and Singer. More precisely, we show that for sufficiently
many cameras with given corrupted pairwise directions, where both camera
locations and pairwise directions are generated by a special probabilistic
model, the LUD algorithm exactly recovers the camera locations with high
probability. A similar exact recovery guarantee was established for the
ShapeFit algorithm by Hand, Lee and Voroninski, but with typically less
corruption
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