8,008 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
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
QuickCSG: Fast Arbitrary Boolean Combinations of N Solids
QuickCSG computes the result for general N-polyhedron boolean expressions
without an intermediate tree of solids. We propose a vertex-centric view of the
problem, which simplifies the identification of final geometric contributions,
and facilitates its spatial decomposition. The problem is then cast in a single
KD-tree exploration, geared toward the result by early pruning of any region of
space not contributing to the final surface. We assume strong regularity
properties on the input meshes and that they are in general position. This
simplifying assumption, in combination with our vertex-centric approach,
improves the speed of the approach. Complemented with a task-stealing
parallelization, the algorithm achieves breakthrough performance, one to two
orders of magnitude speedups with respect to state-of-the-art CPU algorithms,
on boolean operations over two to dozens of polyhedra. The algorithm also
outperforms GPU implementations with approximate discretizations, while
producing an output without redundant facets. Despite the restrictive
assumptions on the input, we show the usefulness of QuickCSG for applications
with large CSG problems and strong temporal constraints, e.g. modeling for 3D
printers, reconstruction from visual hulls and collision detection
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