Homological methods for extraction and analysis of linear features in multidimensional images

Mrożek, Marian; Żelawski, Marcin; Gryglewski, Andrzej; Han, Sejin; Krajniak, Andrzej

doi:10.1016/j.patcog.2011.04.020

Cited by 21 publications

(10 citation statements)

References 31 publications

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“…The last two decades have witnessed a rapid growth in the range of applications of algebraic topology including sensor networks, image analysis, data analysis, material science and nonlinear dynamics [20,13,5,12,16,15,28,10,9,29].…”

Section: Introductionmentioning

confidence: 99%

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

et al. 2013

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We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic pre-processing framework for deriving chain maps from such set-valued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function.

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Section: Introductionmentioning

confidence: 99%

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

et al. 2013

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“…The use of certified tools able to compute homology groups will be important in the future; for instance, to recognize the structure of a neuron; a problem which seems to involve the homology group in dimension 1, see [20]. Other techniques, like the ones of persistent homology, could be applied in stacks of neurons to remove the noise of the images and help to the detection of the dendrites (the branches of the neuron).…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Towards a Certified Computation of Homology Groups for Digital Images

Heras

Dénès

Mata

et al. 2012

Computational Topology in Image Context

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“…It has been applied to dynamical systems [13,15], material science [4,18], electromagnetism [8,7], image understanding [1,14] and sensor networks [6].…”

Section: Introductionmentioning

confidence: 99%

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

Gonzalez-Lorenzo

Juda

Bac

et al. 2016

Computational Topology in Image Context

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Abstract. Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simple method for computing the Betti numbers of a three-dimensional cubical complex that takes advantage on its regular structure, which is not possible with other types of CW complexes such as simplicial or polyhedral complexes. This algorithm is also restricted to three-dimensional spaces since it exploits the Euler-Poincaré formula and the Alexander duality in order to avoid any matrix manipulation. The method runs in linear time on a single core CPU. Moreover, the regular cubical structure allows us to obtain an efficient implementation for a multi-core architecture.

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Homological methods for extraction and analysis of linear features in multidimensional images

Cited by 21 publications

References 31 publications

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

Discrete Morse Theoretic Algorithms for Computing Homology of Complexes and Maps

Towards a Certified Computation of Homology Groups for Digital Images

Fast, Simple and Separable Computation of Betti Numbers on Three-Dimensional Cubical Complexes

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