Membrane parallelism for discrete Morse theory applied to digital images

Abstract

In this paper, we propose a bio-inspired membrane computational framework for constructing discrete Morse complexes for binary digital images. Our approach is based on the discrete Morse theory and we work with cubical complexes. As example, a parallel algorithm for computing homology groups of binary 3D digital images is designed.

Effective homology of k-D digital objects (partially) calculated in parallel

“…In this paper, the bio-inspired theoretical framework of Reina-Molina et al (2015) is optimized in order to be implemented using GPGPU computing. Taking as input a ROI K of A C C E P T E D M A N U S C R I P T a pre-segmented k-D digital image, the final output of this effective homological approach of Discrete Morse Theory is an AMmodel for a cubical complex version of the ROI.…”

“…Hence, an optimization is required to implement them in current computers. Although the algorithms developed in this paper are directly inspired by those developed in Reina-Molina et al (2015), the content of current paper can be read independently.…”

“…Within the context of Digital Imagery, computational topology has acquired a significant role in fields of applications like Computer Vision, Digital Image Processing or Medical Imaging. In Reina-Molina et al (2015), a series of massively parallel algorithms using membrane computing are developed in order to calculate a discrete Morse complex associated to the set of foreground k-xels of a k-D binary digital image. However, nowadays there is no device capable of executing such algorithms.…”

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In Reina-Molina et al (2015), a membrane parallel theoretical framework for computing (co)homology information of foreground or background of binary digital images is developed. Starting from this work, we progress here in two senses: (a) providing advanced topological information, such as (co)homology torsion and efficiently answering to any decision or classification problem for sum of k-xels related to be a (co)cycle or a (co)boundary; (b) optimizing the previous framework to be implemented in using GPGPU computing.

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Strong Discrete Morse Theory and Simplicial L–S Category: A Discrete Version of the Lusternik–Schnirelmann Theorem

“…Since its inception by Robin Forman [9], discrete Morse theory has been a powerful and versatile tool used not only in diverse fields of mathematics, but also in applications to other areas [14] as well as a computational tool [6]. Its adaptability stems in part from the fact that it is a discrete version of the beautiful and successful "smooth" Morse theory [11].…”