Homological spanning forest framework for 2D image analysis

et al. 2015

AAECC

Self Cite

“…We are also interested in the representational power of our framework, as a parallel implementation of the works of Real and Molina-Abril [21,28].…”

Section: Discussionmentioning

confidence: 99%

Membrane parallelism for discrete Morse theory applied to digital images

Reina-Molina

Díaz-Pernil

et al. 2015

AAECC

Self Cite

“…These complex homotopy operations are geometric realizations of elementary algebraic operations appearing in the construction of a homology integral operator and they can be expressed in graphical terms by elementary directed graphs. Some preliminary efforts to clarify this aspect have been the introduction of the chain-integral complex algebraic notion in [1] and the Homological Spanning Forest concept in [12,13].…”

Section: Combinatorial and Homological Optimalitymentioning

confidence: 99%

“…In order to compute numerical or algebraic information related to homology generators and their relations between them at both homology and cycle levels, the importance of the notion of homology integral operator is great. In order to use this tool for progressing in computing advanced topological invariants, a successful homological algebra framework provided by integral-chain complexes is proposed in [12].…”

Section: Sdr-condition Of M With Regards To the Boundary Differentialmentioning

confidence: 99%

See 1 more Smart Citation

Searching combinatorial optimality using graph-based homology information

Molina‐Abril

Gonzalez-Lorenzo

et al. 2015

AAECC

Self Cite

This paper analyses the topological information of a digital object O under a combined combinatorial-algebraic point of view. Working with a topologypreserving cellularization K (O) of the object, algebraic and combinatorial tools are jointly used. The combinatorial entities used here are vector fields, V -paths and directed graphs. In the algebraic side, chain complexes with extra 2-nilpotent operators are considered. By mixing these two perspectives we are able to explore the problems of combinatorial and homological optimality. Combinatorial optimality is understood here as the problem for constructing a discrete gradient vector field (DGVF) in the sense of Discrete Morse Theory, such that it has the least possible number of critical cells. Fixing Z/2Z as field of coefficients, by homological 'optimality' we mean the problem of constructing a 2-nilpotent codifferential map φ : C(K (O)), φ) are isomorphic to those of (C(K (O)), ∂), being ∂ the canonical boundary or differential operator of the cell complex K (O).Relations between these two problems are tackled here by using a type of discrete graphs associated to a homology integral operator, called Homological Spanning Forests (HSF for short). Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator.

“…These hierarchical tree-like structure gives a positive and efficient answer to the problem of codifying and computing classical algebraic topological information (Euler characteristic, Betti numbers, classification and relations between cycles, etc.). A detailed explanation about the topological information that the HSF codifies, and its formal definition can be found in [5]. Relations between the HSF and Morse Theory, in [4].…”

Section: Homological Spanning Forest Representationmentioning

confidence: 99%

Triangle Mesh Compression and Homological Spanning Forests

Carnero

Molina‐Abril

Computational Topology in Image Context

2012

Self Cite

Abstract. Triangle three-dimensional meshes have been widely used to represent 3D objects in several applications. These meshes are usually surfaces that require a huge amount of resources when they are stored, processed or transmitted. Therefore, many algorithms proposing an efficient compression of these meshes have been developed since the early 1990s. In this paper we propose a lossless method that compresses the connectivity of the mesh by using a valence-driven approach. Our algorithm introduces an improvement over the currently available valencedriven methods, being able to deal with triangular surfaces of arbitrary topology and encoding, at the same time, the topological information of the mesh by using Homological Spanning Forests. We plan to develop in the future (geo-topological) image analysis and processing algorithms, that directly work with the compressed data.