2,734 research outputs found
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
A stochastic evaluation of the contour strength
International audienceIf one considers only local neighborhoods for segmenting an image, one gets contours whose strength is often poorly estimated. A method for reevaluating the contour strength by taking into account non local features is presented: one generates a fixed number of random germs which serve as markers for the watershed segmentation. For each new population of markers, another set of contours is generated. "Important" contours are selected more often. The present paper shows that the probability that a contour is selected can be estimated without performing the effective simulations. Copyright Springer-Verlag 2010. The original publication is available at www.springerlink.com/content/y057x103475301r2
Tree-based Coarsening and Partitioning of Complex Networks
Many applications produce massive complex networks whose analysis would
benefit from parallel processing. Parallel algorithms, in turn, often require a
suitable network partition. For solving optimization tasks such as graph
partitioning on large networks, multilevel methods are preferred in practice.
Yet, complex networks pose challenges to established multilevel algorithms, in
particular to their coarsening phase.
One way to specify a (recursive) coarsening of a graph is to rate its edges
and then contract the edges as prioritized by the rating. In this paper we (i)
define weights for the edges of a network that express the edges' importance
for connectivity, (ii) compute a minimum weight spanning tree with
respect to these weights, and (iii) rate the network edges based on the
conductance values of 's fundamental cuts. To this end, we also (iv)
develop the first optimal linear-time algorithm to compute the conductance
values of \emph{all} fundamental cuts of a given spanning tree. We integrate
the new edge rating into a leading multilevel graph partitioner and equip the
latter with a new greedy postprocessing for optimizing the maximum
communication volume (MCV). Experiments on bipartitioning frequently used
benchmark networks show that the postprocessing already reduces MCV by 11.3%.
Our new edge rating further reduces MCV by 10.3% compared to the previously
best rating with the postprocessing in place for both ratings. In total, with a
modest increase in running time, our new approach reduces the MCV of complex
network partitions by 20.4%
Watersheds, waterfalls, on edge or node weighted graphs
We present an algebraic approach to the watershed adapted to edge or node
weighted graphs. Starting with the flooding adjunction, we introduce the
flooding graphs, for which node and edge weights may be deduced one from the
other. Each node weighted or edge weighted graph may be transformed in a
flooding graph, showing that there is no superiority in using one or the other,
both being equivalent. We then introduce pruning operators extract subgraphs of
increasing steepness. For an increasing steepness, the number of never
ascending paths becomes smaller and smaller. This reduces the watershed zone,
where catchment basins overlap. A last pruning operator called scissor
associates to each node outside the regional minima one and only one edge. The
catchment basins of this new graph do not overlap and form a watershed
partition. Again, with an increasing steepness, the number of distinct
watershed partitions contained in a graph becomes smaller and smaller.
Ultimately, for natural image, an infinite steepness leads to a unique
solution, as it is not likely that two absolutely identical non ascending paths
of infinite steepness connect a node with two distinct minima. It happens that
non ascending paths of a given steepness are the geodesics of lexicographic
distance functions of a given depth. This permits to extract the watershed
partitions as skeletons by zone of influence of the minima for such
lexicographic distances. The waterfall hierarchy is obtained by a sequence of
operations. The first constructs the minimum spanning forest which spans an
initial watershed partition. The contraction of the trees into one node
produces a reduced graph which may be submitted to the same treatment. The
process is iterated until only one region remains. The union of the edges of
all forests produced constitutes a minimum spanning tree of the initial graph
A study of observation scales based on Felzenswalb-Huttenlocher dissimilarity measure for hierarchical segmentation
International audienceHierarchical image segmentation provides a region-oriented scale-space, i.e., a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. GuimarĂŁes et al. proposed a hierarchical graph based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissimilarity. This HGB method computes, for each edge of a graph, the minimum scale in a hierarchy at which two regions linked by this edge should merge according to the dissimilarity. In order to generalize this method, we first propose an algorithm to compute the intervals which contain all the observation scales at which the associated regions should merge. Then, following the current trend in mathematical morphology to study criteria which are not increasing on a hierarchy, we present various strategies to select a significant observation scale in these intervals. We use the BSDS dataset to assess our observation scale selection methods. The experiments show that some of these strategies lead to better segmentation results than the ones obtained with the original HGB method
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