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A new algorithm for computing the α-tree hierarchical repre sentation of a grey-scale digital image is presented here. The technique is based on an efficient simplified version of the Homological Spanning For est (HSF) for encoding homological and homotopy-based information of binary digital images. We create one Adjacency Tree (AdjT) for each intensity contrast in a fully parallel manner. These trees, which define a Contrast Adjacency Forest (CAdjF), are in turn transversely intercon nected by another couple of trees: the classical α-tree, and a new one complementing it, called here the α∗-tree. They convey the information of the contours and the flat regions of the original color image, plus the relations between them. Using both the α and α∗-trees, this new topolog ical representation prevents some classical drawbacks that appear when working with a single tree. An implementation in OCTAVE/MATLAB validates the correctness of our algorithm. Ministerio de Ciencia e Innovación PID2019-110455GB-I00 (Par-HoT)

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

In this paper, the concept of homogeneity is defined, from a topological perspective, in order to analyze how uniform is the material composition in 2D electron microscopy images. Topological multiresolution parameters are taken into account to obtain better results than classical techniques. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

Given a 3D binary digital image I, we define and compute an edge-weighted tree, called Homological Region Tree (or Hom-Tree, for short). It coincides, as unweighted graph, with the classical Region Adjacency Tree of black 6-connected components (CCs) and white 26- connected components of I. In addition, we define the weight of an edge (R, S) as the number of tunnels that the CCs R and S “share”. The Hom-Tree structure is still an isotopic invariant of I. Thus, it provides information about how the different homology groups interact between them, while preserving the duality of black and white CCs. An experimentation with a set of synthetic images showing different shapes and different complexity of connected component nesting is performed for numerically validating the method. Ministerio de Economía y Competitividad MTM2016-81030-P

This is an experimental paper in which we introduce the possibility to analyze and to synthesize 3D medical images by using multivariate Gabor frames with Gaussian windows. Our purpose is to apply a space-variant filter-like operation in the space-frequency domain to correct medical images corrupted by different types of acquisitions errors. The Gabor frames are constructed with Gaussian windows sampled on non-separable lattices for a better packing of the space-frequency plane. An implementable solution for 3D-Gabor frames with non-separable lattice is given and numerical tests on simulated data are presented. Austrian Science Fund (FWF) P27516

Given a binary object (2D or 3D), its Betti numbers characterize the number of holes in each dimension. They are obtained algebraically, and even though they are perfectly defined, there is no unique way to display these holes. We propose two geometric measures for the holes, which are uniquely defined and try to compensate the loss of geometric information during the homology computation: the thickness and the breadth. They are obtained by filtering the information of the persistent homology computation of a filtration defined through the signed distance transform of the binary object.

This special issue comprises thirteen recent and relevant works in different aspects of the computer analysis of images and patterns. We cover important topics like stereo image processing, image and shape representation, color texture classification, parallel systems and membrane computing, face recognition, pedestrian classification, estimation of photometry and analysis of medical images. The papers have been divided into four groups—three are based on similarities in the addressed problems and the last focuses on six different applications. The first section presents stereo image processing from a variety of perspectives. “Tuning of Adaptive Weight Depth Map Generation Algorithms” by Acosta et al. shows a systematic statistical approach for the parameter setting problem in disparity map estimation within the context of depth map generation algorithms. The next paper, “P-System Implementation of Dynamic Programming Stereo” by Gimel’farb et al. discusses the capabilities of the main tools of Membrane Computing, the P-systems, and how to implement the symmetric dynamic programming stereo (SDPS) matching algorithm. The authors propose a novel method of stereo matching based on a combination of local edge detection, cost aggregation computation and belief propagation optimization techniques in “Stereo Matching Based on Dissimilar Intensity Support and Belief Propagation” by Da et al. The next section studies representation and estimation of shape from different mathematical points of view.

Membrane Computing is a biologically inspired computational model. Its devices are called P systems and they perform computations by applying a finite set of rules in a synchronous, maximally parallel way. In this paper, we follow a new research line using tissue-like P systems to do a parallel color segmentation of images using a thresholding to look for edge pixels. We have chosen this variant of P systems because it uses a less number of computational ingredients with respect to classical variants. Ministerio de Educación y Ciencia MTM2006-03722 Junta de Andalucía PO6-TIC-02268

In [3] “small” 1-homological model H of a commutative differential graded algebra is described. Homological Perturbation Theory (HPT) [7-9] provides an explicit description of an A∞-coalgebra structure (Δ1,Δ2,Δ3,...) of H. In this paper, we are mainly interested in the determination of the map Δ2 : H - H H ⊗ H as a first step in the study of this structure. Developing the techniques given in [20] (inversion theory), we get an important improvement in the computation of Δ2 with regard to the first formula given by HPT. In the case of purely quadratic algebras, we sketch a procedure for giving the complete Hopf algebra structure of its 1-homology. Ministerio de Educación y Ciencia PB98-1621-C02-02 Junta de Andalucía FQM-296