4,107 research outputs found
Semi-algebraic geometry of common lines
Cryo-electron microscopy is a technique in structural biology for
discovering/determining the 3D structure of small molecules. A key step in this
process is detecting common lines of intersection between unknown embedded
image planes. We intrinsically characterize such common lines in terms of the
unembedded geometric data detected in experiments. We show these common lines
form a semi-algebraic set, i.e., they are defined by polynomial equalities and
inequalities. These polynomials are low degree and, using techniques from
spherical geometry, we explicitly derive them in this paper.Comment: Pre-print, comments welcom
Ramified optimal transportation in geodesic metric spaces
An optimal transport path may be viewed as a geodesic in the space of
probability measures under a suitable family of metrics. This geodesic may
exhibit a tree-shaped branching structure in many applications such as trees,
blood vessels, draining and irrigation systems. Here, we extend the study of
ramified optimal transportation between probability measures from Euclidean
spaces to a geodesic metric space. We investigate the existence as well as the
behavior of optimal transport paths under various properties of the metric such
as completeness, doubling, or curvature upper boundedness. We also introduce
the transport dimension of a probability measure on a complete geodesic metric
space, and show that the transport dimension of a probability measure is
bounded above by the Minkowski dimension and below by the Hausdorff dimension
of the measure. Moreover, we introduce a metric, called "the dimensional
distance", on the space of probability measures. This metric gives a geometric
meaning to the transport dimension: with respect to this metric, the transport
dimension of a probability measure equals to the distance from it to any finite
atomic probability measure.Comment: 22 pages, 4 figure
On a family of strong geometric spanners that admit local routing strategies
We introduce a family of directed geometric graphs, denoted \paz, that
depend on two parameters and . For and , the \paz graph is a strong
-spanner, with . The out-degree of a node
in the \paz graph is at most . Moreover, we show that routing can be
achieved locally on \paz. Next, we show that all strong -spanners are also
-spanners of the unit disk graph. Simulations for various values of the
parameters and indicate that for random point sets, the
spanning ratio of \paz is better than the proven theoretical bounds
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