165 research outputs found
Homological spanning forest framework for 2D image analysis
A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions
Homological Spanning Forests for Discrete Objects
Computing and representing topological information form an important
part in many applications such as image representation and compression,
classification, pattern recognition, geometric modelling, etc. The homology
of digital objects is an algebraic notion that provides a concise description
of their topology in terms of connected components, tunnels and cavities.
The purpose of this work is to develop a theoretical and practical frame-
work for efficiently extracting and exploiting useful homological information
in the context of nD digital images. To achieve this goal, we intend to
combine known techniques in algebraic topology, and image processing.
The main notion created for this purpose consists of a combinatorial
representation called Homological Spanning Forest (or HSF, for short) of a
digital object or a digital image. This new model is composed of a set of
directed forests, which can be constructed under an underlying cell complex
format of the image. HSF’s are based on the algebraic concept of chain
homotopies and they can be considered as a suitable generalization to higher
dimensional cell complexes of the topological meaning of a spanning tree of
a geometric graph.
Based on the HSF representation, we present here a 2D homology-based
framework for sequential and parallel digital image processing.Premio Extraordinario de Doctorado U
Algebraic topological analysis of time-sequence of digital images
This paper introduces an algebraic framework for a topological analysis of time-varying 2D digital binary–valued images, each of them defined as 2D arrays of pixels. Our answer is based on an algebraic-topological coding, called AT–model, for a nD (n=2,3) digital binary-valued image I consisting simply in taking I together with an algebraic object depending on it. Considering AT–models for all the 2D digital images in a time sequence, it is possible to get an AT–model for the 3D digital image consisting in concatenating the successive 2D digital images in the sequence. If the frames are represented in a quadtree format, a similar positive result can be derived
Polynomiality of surface braid and mapping class group representations
We study a wide range of homologically-defined representations of surface
braid groups and of mapping class groups of surfaces, extending the
Lawrence-Bigelow representations of the classical braid groups. These
representations naturally come in families, defined either on all surface braid
groups as the number of strands varies or on all mapping class groups as the
genus varies. We prove that each of these families of representations is
polynomial. This has applications to twisted homological stability as well as
to understanding the structure of the representation theory of these families
of groups. Our polynomiality result is a consequence of a more fundamental
result establishing relations amongst the families of representations that we
consider via short exact sequences of functors. As well as polynomiality, these
short exact sequences also have applications to understanding the kernels of
the homological representations under consideration.Comment: 47 pages, 12 figures. The exposition has been reorganised and
streamlined in v2. Comments welcome
- …