Effective homology of filtered digital images

Romero, Ana; Rubio, Julio; Sergeraert, Francis

doi:10.1016/j.patrec.2016.01.023

Cited by 6 publications

(2 citation statements)

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“…The classical method is based on the diagonalization of cell-incidence matrices to Smith normal form (SNF) (see [27] ). In the following decades, some advances in the computation of the SNF have been achieved (see [28] ), but the most successful approaches consist of reducing the number of cells in the complex using discrete-vector-field dynamics (Discrete Morse theory [29] ) before computing the SNF for the small resulting cell complex (see, for instance, [30][31][32][33][34][35] ). In this paper we go beyond homological computation and design an algorithm for computing a new representation based on homology.…”

Section: Related Workmentioning

confidence: 99%

Generating (co)homological information using boundary scale

Molina‐Abril

Real

Díaz-del-Río

2020

Pattern Recognition Letters

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In this paper we develop a new computational technique called boundary scale-space theory. This technique is based on the topol 1 ogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects { K i } i ≥ 1 all of them having the same number of closed pieces than K.Each piece of K i ( ∀i ≥ 1) presents the same interior part than the corresponding one in K, and a different boundary part depending on the scale i. Working with coefficients in a field, a scale is installed for the algebraic boundary of each piece and a new invariant for cell complex isomorphisms is given in terms of the Betti numbers of the generated boundary-scalespace cell complexes. Moreover, the so called homology boundary scale-space model of K ( hbss -model for short) is introduced here. This model consists of a hierarchical graph whose nodes are the homology generators of the different bound-ary scale levels and whose edges are specified by homology generators of consecutive boundary scale indices linked by ( hbss -transition maps) preserving homology classes. Various codes for each connected subgraph of an hbss -model are defined, which besides being fast and efficient similarity measures for cel-lular structures, they are as well relevant interpretive tools for the hbss -model. Finally, experimentations mainly aimed at clarifying and understanding the notion of hbss -model, as well as conjecturing about new graph isomorphism invariants (seeing graphs as a 1-dimensional cell complexes), are performed.

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Section: Related Workmentioning

confidence: 99%

Generating (co)homological information using boundary scale

Molina‐Abril

Real

Díaz-del-Río

2020

Pattern Recognition Letters

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“…Effective homology [131,141,152] is a algebraic-topological theory mainly based on the computational notion of chain homotopy equivalence, a concept which algebraically connects a cell complex or subdivided object with its homology groups. Roughly speaking, a chain homotopy equivalence can be specified by an operator, called chain homotopy operator, working at the level of linear combinations of cells which represents an efficient and non-redundant way of connecting cells.…”

Section: Effective Homologymentioning

confidence: 99%

Membrane computing and image processing: a short survey

2019

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Membrane computing is a well-known research area in computer science inspired by the organization and behavior of live cells and tissues. Their computational devices, called P systems, work in parallel and distributed mode and the information is encoded by multisets in a localized manner. All these features make P systems appropriate for dealing with digital images. In this paper, some of the open research lines in the area are presented, focusing on segmentation problems, skeletonization and algebraic-topological aspects of the images. An extensive bibliography about the application of membrane computing to the study of digital images is also provided.

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