65,062 research outputs found
Engineering Art Galleries
The Art Gallery Problem is one of the most well-known problems in
Computational Geometry, with a rich history in the study of algorithms,
complexity, and variants. Recently there has been a surge in experimental work
on the problem. In this survey, we describe this work, show the chronology of
developments, and compare current algorithms, including two unpublished
versions, in an exhaustive experiment. Furthermore, we show what core
algorithmic ingredients have led to recent successes
A Novel Method for the Absolute Pose Problem with Pairwise Constraints
Absolute pose estimation is a fundamental problem in computer vision, and it
is a typical parameter estimation problem, meaning that efforts to solve it
will always suffer from outlier-contaminated data. Conventionally, for a fixed
dimensionality d and the number of measurements N, a robust estimation problem
cannot be solved faster than O(N^d). Furthermore, it is almost impossible to
remove d from the exponent of the runtime of a globally optimal algorithm.
However, absolute pose estimation is a geometric parameter estimation problem,
and thus has special constraints. In this paper, we consider pairwise
constraints and propose a globally optimal algorithm for solving the absolute
pose estimation problem. The proposed algorithm has a linear complexity in the
number of correspondences at a given outlier ratio. Concretely, we first
decouple the rotation and the translation subproblems by utilizing the pairwise
constraints, and then we solve the rotation subproblem using the
branch-and-bound algorithm. Lastly, we estimate the translation based on the
known rotation by using another branch-and-bound algorithm. The advantages of
our method are demonstrated via thorough testing on both synthetic and
real-world dataComment: 10 pages, 7figure
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
A Minimalist Approach to Type-Agnostic Detection of Quadrics in Point Clouds
This paper proposes a segmentation-free, automatic and efficient procedure to
detect general geometric quadric forms in point clouds, where clutter and
occlusions are inevitable. Our everyday world is dominated by man-made objects
which are designed using 3D primitives (such as planes, cones, spheres,
cylinders, etc.). These objects are also omnipresent in industrial
environments. This gives rise to the possibility of abstracting 3D scenes
through primitives, thereby positions these geometric forms as an integral part
of perception and high level 3D scene understanding.
As opposed to state-of-the-art, where a tailored algorithm treats each
primitive type separately, we propose to encapsulate all types in a single
robust detection procedure. At the center of our approach lies a closed form 3D
quadric fit, operating in both primal & dual spaces and requiring as low as 4
oriented-points. Around this fit, we design a novel, local null-space voting
strategy to reduce the 4-point case to 3. Voting is coupled with the famous
RANSAC and makes our algorithm orders of magnitude faster than its conventional
counterparts. This is the first method capable of performing a generic
cross-type multi-object primitive detection in difficult scenes. Results on
synthetic and real datasets support the validity of our method.Comment: Accepted for publication at CVPR 201
On the matrix square root via geometric optimization
This paper is triggered by the preprint "\emph{Computing Matrix Squareroot
via Non Convex Local Search}" by Jain et al.
(\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent
for computing the square root of a positive definite matrix. Contrary to claims
of~\citet{jain2015}, our experiments reveal that Newton-like methods compute
matrix square roots rapidly and reliably, even for highly ill-conditioned
matrices and without requiring commutativity. We observe that gradient-descent
converges very slowly primarily due to tiny step-sizes and ill-conditioning. We
derive an alternative first-order method based on geodesic convexity: our
method admits a transparent convergence analysis ( page), attains linear
rate, and displays reliable convergence even for rank deficient problems.
Though superior to gradient-descent, ultimately our method is also outperformed
by a well-known scaled Newton method. Nevertheless, the primary value of our
work is its conceptual value: it shows that for deriving gradient based methods
for the matrix square root, \emph{the manifold geometric view of positive
definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and
more words about the rank-deficient cas
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