136 research outputs found
Inner-Cheeger Opening and Applications
International audienceThe aim of this paper is to study an optimal opening in the sense of minimize the relationship perimeter over area. We analyze theoretical properties of this opening by means of classical results from variational calculus. Firstly, we explore the optimal radius as attribute in morphological attribute filtering for grey scale images. Secondly, an application of this optimal opening that yields a decomposition into meaningful parts in the case of binary image is explored. We provide different examples of 2D, 3D images and mesh-points datasets
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Continuum limit of total variation on point clouds
We consider point clouds obtained as random samples of a measure on a
Euclidean domain. A graph representing the point cloud is obtained by assigning
weights to edges based on the distance between the points they connect. Our
goal is to develop mathematical tools needed to study the consistency, as the
number of available data points increases, of graph-based machine learning
algorithms for tasks such as clustering. In particular, we study when is the
cut capacity, and more generally total variation, on these graphs a good
approximation of the perimeter (total variation) in the continuum setting. We
address this question in the setting of -convergence. We obtain almost
optimal conditions on the scaling, as number of points increases, of the size
of the neighborhood over which the points are connected by an edge for the
-convergence to hold. Taking the limit is enabled by a transportation
based metric which allows to suitably compare functionals defined on different
point clouds
Integration over connections in the discretized gravitational functional integrals
The result of performing integrations over connection type variables in the
path integral for the discrete field theory may be poorly defined in the case
of non-compact gauge group with the Haar measure exponentially growing in some
directions. This point is studied in the case of the discrete form of the first
order formulation of the Einstein gravity theory. Here the result of interest
can be defined as generalized function (of the rest of variables of the type of
tetrad or elementary areas) i. e. a functional on a set of probe functions. To
define this functional, we calculate its values on the products of components
of the area tensors, the so-called moments. The resulting distribution (in
fact, probability distribution) has singular (-function-like) part with
support in the nonphysical region of the complex plane of area tensors and
regular part (usual function) which decays exponentially at large areas. As we
discuss, this also provides suppression of large edge lengths which is
important for internal consistency, if one asks whether gravity on short
distances can be discrete. Some another features of the obtained probability
distribution including occurrence of the local maxima at a number of the
approximately equidistant values of area are also considered.Comment: 22 page
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