10,909 research outputs found
Combinatorial Continuous Maximal Flows
Maximum flow (and minimum cut) algorithms have had a strong impact on
computer vision. In particular, graph cuts algorithms provide a mechanism for
the discrete optimization of an energy functional which has been used in a
variety of applications such as image segmentation, stereo, image stitching and
texture synthesis. Algorithms based on the classical formulation of max-flow
defined on a graph are known to exhibit metrication artefacts in the solution.
Therefore, a recent trend has been to instead employ a spatially continuous
maximum flow (or the dual min-cut problem) in these same applications to
produce solutions with no metrication errors. However, known fast continuous
max-flow algorithms have no stopping criteria or have not been proved to
converge. In this work, we revisit the continuous max-flow problem and show
that the analogous discrete formulation is different from the classical
max-flow problem. We then apply an appropriate combinatorial optimization
technique to this combinatorial continuous max-flow CCMF problem to find a
null-divergence solution that exhibits no metrication artefacts and may be
solved exactly by a fast, efficient algorithm with provable convergence.
Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the
fact, already proved by Nozawa in the continuous setting, that the max-flow and
the total variation problems are not always equivalent.Comment: 26 page
Recurrence and transience for suspension flows
We study the thermodynamic formalism for suspension flows over countable
Markov shifts with roof functions not necessarily bounded away from zero. We
establish conditions to ensure the existence and uniqueness of equilibrium
measures for regular potentials. We define the notions of recurrence and
transience of a potential in this setting. We define the "renewal flow", which
is a symbolic model for a class of flows with diverse recurrence features. We
study the corresponding thermodynamic formalism, establishing conditions for
the existence of equilibrium measures and phase transitions. Applications are
given to suspension flows defined over interval maps having parabolic fixed
points.Comment: In this version of the paper some typos have been corrected and some
references updated. Note that the former title of this paper was "Parabolic
suspension flows
Dynamics of piecewise contractions of the interval
We study the asymptotical behaviour of iterates of piecewise contractive maps
of the interval. It is known that Poincar\'e first return maps induced by some
Cherry flows on transverse intervals are, up to topological conjugacy,
piecewise contractions. These maps also appear in discretely controlled
dynamical systems, describing the time evolution of manufacturing process
adopting some decision-making policies. An injective map is
a {\it piecewise contraction of intervals}, if there exists a partition of
the interval into intervals ,..., such that for every
, the restriction is -Lipschitz for some
. We prove that every piecewise contraction of
intervals has at most periodic orbits. Moreover, we show that every
piecewise contraction is topologically conjugate to a piecewise linear
contraction
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