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Conditional Toggle Mappings: Principles and Applications

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Abstract

We study a class of mathematical morphology filters to operate conditionally according to a set of pixels marked by a binary mask. The main contribution of this paper is to provide a general framework for several applications including edge enhancement and image denoising, when it is affected by salt-and-pepper noise. We achieve this goal by revisiting shock filters based on erosions and dilations and extending their definition to take into account the prior definition of a mask of pixels that should not be altered. New definitions for conditional erosions and dilations leading to the concept of conditional toggle mapping. We also investigate algebraic properties as well as the convergence of the associate shock filter. Experiments show how the selection of appropriate methods to generate the masks lead to either edge enhancement or salt-and-pepper denoising. A quantitative evaluation of the results demonstrates the effectiveness of the proposed methods. Additionally, we analyse the application of conditional toggle mapping in remote sensing as pre-filtering for hierarchical segmentation.

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Notes

  1. An operator ϕ:𝔼→𝔽 is idempotent if ∀x∈𝔼,ϕ 2(x)=ϕ(ϕ(x))=ϕ(x).

  2. An operator ϕ:𝔼→𝔽 is extensive (resp. anti-extensive) if ∀x∈𝔼,ϕ(x)≥x (resp. ≤).

  3. Flat zones of I are the connected component where the intensity does not change.

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Authors and Affiliations

  1. ITWM, Fraunhofer Institute, Kaiserslautern, Germany

    Santiago Velasco-Forero

  2. CMM-Centre de Morphologie Mathematique, Mathématiques et Systèmes, MINES ParisTech, 35, rue Saint-Honore, 77305, Fontainebleau cedex, France

    Jesús Angulo

  3. Joint Research Center, European Commission, Ispra, Italy

    Pierre Soille

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  1. Santiago Velasco-Forero
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  2. Jesús Angulo
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  3. Pierre Soille
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Correspondence to Santiago Velasco-Forero.

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Velasco-Forero, S., Angulo, J. & Soille, P. Conditional Toggle Mappings: Principles and Applications. J Math Imaging Vis 48, 544–565 (2014). https://doi.org/10.1007/s10851-013-0429-4

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