6,660 research outputs found
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
Deterministic Sparse Pattern Matching via the Baur-Strassen Theorem
How fast can you test whether a constellation of stars appears in the night
sky? This question can be modeled as the computational problem of testing
whether a set of points can be moved into (or close to) another set
under some prescribed group of transformations.
Consider, as a simple representative, the following problem: Given two sets
of at most integers , determine whether there is some
shift such that shifted by is a subset of , i.e.,
. This problem, to which we refer as the
Constellation problem, can be solved in near-linear time by a
Monte Carlo randomized algorithm [Cardoze, Schulman; FOCS'98] and time
by a Las Vegas randomized algorithm [Cole, Hariharan; STOC'02].
Moreover, there is a deterministic algorithm running in time
[Chan, Lewenstein; STOC'15]. An
interesting question left open by these previous works is whether Constellation
is in deterministic near-linear time (i.e., with only polylogarithmic
overhead).
We answer this question positively by giving an -time
deterministic algorithm for the Constellation problem. Our algorithm extends to
various more complex Point Pattern Matching problems in higher dimensions,
under translations and rigid motions, and possibly with mismatches, and also to
a near-linear-time derandomization of the Sparse Wildcard Matching problem on
strings.
We find it particularly interesting how we obtain our deterministic
algorithm. All previous algorithms are based on the same baseline idea, using
additive hashing and the Fast Fourier Transform. In contrast, our algorithms
are based on new ideas, involving a surprising blend of combinatorial and
algebraic techniques. At the heart lies an innovative application of the
Baur-Strassen theorem from algebraic complexity theory.Comment: Abstract shortened to fit arxiv requirement
Computational Aspects of the Hausdorff Distance in Unbounded Dimension
We study the computational complexity of determining the Hausdorff distance
of two polytopes given in halfspace- or vertex-presentation in arbitrary
dimension. Subsequently, a matching problem is investigated where a convex body
is allowed to be homothetically transformed in order to minimize its Hausdorff
distance to another one. For this problem, we characterize optimal solutions,
deduce a Helly-type theorem and give polynomial time (approximation) algorithms
for polytopes
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