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Underwater imaging has been present for many decades due to its relevance in vision and navigation systems. In recent years, advances in robotics have led to the availability of autonomous or unmanned underwater vehicles (AUVs, UUVs). Despite the rapid development of new studies and promising algorithms in this field, there is currently a lack of research toward standardized, general-approach proposals. This issue has been stated in the literature as a limiting factor to be addressed in the future. The key starting point of this work is to identify a synergistic effect between professional photography and scientific fields by analyzing image acquisition issues. Subsequently, we discuss underwater image enhancement and quality assessment, image mosaicking and algorithmic concerns as the last processing step. In this line, statistics about 120 AUV articles fro recent decades have been analyzed, with a special focus on state-of-the-art papers from recent years. Therefore, the aim of this paper is to identify critical issues in autonomous underwater vehicles encompassing the entire process, starting from optical issues in image sensing and ending with some issues related to algorithmic processing. In addition, a global underwater workflow is proposed, extracting future requirements, outcome effects and new perspectives in this context.

Taking advantage of the topological and isotopic properties of binary digital images, we present here anew algorithm for connected component labeling (CLL). A local-to-global treatment of the topologicalinformation within the image, allows us to develop an inherent parallel approach. The time complexityorder for an image of m ×n pixels, under the assumption that a processing element exists for each pixel, is near O (log(m + n )) . Additionally, our method computes both the foreground and background CCL, and allows a straightforward computation of topological features like Adjacency Trees. Experiments show thatour method obtains better performance metrics than other approaches. Our work aims at generating anew class of labeling algorithms: those centered in fully parallel approaches based on computationaltopology, thus allowing a perfect concurrent execution in multiple threads and preventing the use ofcritical sections and atomic instructions. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

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)

Dissertation presented as the partial requirement for obtaining a Master's degree in Data Science and Advanced Analytics The present dissertation focuses on proposing and describing a collaborative filtering approach for Music Recommender Systems. Music Recommender Systems, which are part of a broader class of Recommender Systems, refer to the task of automatically filtering data to predict the songs that are more likely to match a particular profile. So far, academic researchers have proposed a variety of machine learning approaches for determining which tracks to recommend to users. The most sophisticated among them consist, often, on complex learning techniques which can also require considerable computational resources. However, recent research studies proved that more simplistic approaches based on nearest neighbors could lead to good results, often at much lower computational costs, representing a viable alternative solution to the Music Recommender System problem. Throughout this thesis, we conduct offline experiments on a freely-available collection of listening histories from real users, each one containing several different music tracks. We extract a subset of 10 000 songs to assess the performance of the proposed system, comparing it with a Popularity-based model approach. Furthermore, we provide a conceptual overview of the recommendation problem, describing the state-of-the-art methods, and presenting its current challenges. Finally, the last section is dedicated to summarizing the essential conclusions and presenting possible future improvements.

Fractal dimension is a powerful tool employed as a measurement of geometric aspects. In this work we propose a method of topological fractal analysis for 2D binary digital images by using a graph-based topological model of them, called Homological Spanning Forest (HSF, for short). Defined at interpixel level, this set of two trees allows to topologically describe the (black and white) connected component distribution within the image with regards to the relationship “to be surrounded by”. This distribution is condensed into a rooted tree, such that its nodes are connected components determined by some special sub-trees of the previous HSF and the levels of the tree specify the degree of nesting of each connected component. We ask for topological auto-similarity by comparing this topological description of the whole image with a regular rooted tree pattern. Such an analysis can be used to directly quantify some characteristics of biomedical images (e.g. cells samples or clinical images) that are not so noticeable when using geometrical approaches. Ministerio de Economía y Competitividad TEC2016-77785-P Ministerio de Economía y Competitividad MTM2016-81030-P

International audience; Discrete gradient vector fields are combinatorial structures that can be used for accelerating the homology computation of CW complexes, such as simplicial or cubical complexes, by reducing their number of cells. Consequently, they provide a bound for the Betti numbers (the most basic homological information). A discrete gradient vector field can eventually reduce the complex to its minimal form, having as many cells of each dimension as its corresponding Betti number, but this is not guaranteed. Moreover, finding an optimal discrete gradient vector field is an NP-hard problem. We describe here a generalization, which we call Homological Discrete Vector Field (HDVF), which can overcome these limitations by allowing cycles under a certain algebraic condition. In this work we define the HDVF and its associated reduction, we study how to efficiently compute a HDVF, we establish the relation between the HDVF and other concepts in computational homology and we estimate the average complexity of its computation. We also introduce five basic operations for modifying a HDVF, which can also be applied to discrete gradient vector fields.

In this paper, we propose an analytical rapid method to estimate the endocardial borders of the left ventricular walls on echocardiographic images for prospective clinical integration. The procedure was created as a diagnostic support tool for the clinician and it is based on the use of the anisotropic generalized Hough transform. Its application is guided by a Gabor-like filtering for the approximate delimitation of the region of interest without the need for computing further anatomical characteristics. The algorithm is applying directly a deformable template on the predetermined filtered region and therefore it is responsive and straightforward implementable. For accuracy considerations, we have employed a support vector machine classifier to determine the confidence level of the automated marking. The clinical tests were performed at the Cardiology Clinic of the County Emergency Hospital Timisoara and they improved the physicians perception in more than 50% of the cases. The report is concluded with medical discussions. European Union (UE) Ministerio de Economía y Competitividad (MINECO). España

In this work, a new method to determine and correct the linear drift for any crystalline orientation in a single-column-resolved high-resolution scanning transmission electron microscopy (HR-STEM) image, which is based on angle measurements in the Fourier space, is presented. This proposal supposes a generalization and the improvement of a previous work that needs the presence of two symmetrical planes in the crystalline orientation to be applicable. Now, a mathematical derivation of the drift effect on two families of asymmetric planes in the reciprocal space is inferred. However, though it was not possible to find an analytical solution for all conditions, a simple formula was derived to calculate the drift effect that is exact for three specific rotation angles. Taking this into account, an iterative algorithm based on successive rotation/drift correction steps is devised to remove drift distortions in HR-STEM images. The procedure has been evaluated using a simulated micrograph of a monoclinic material in an orientation where all the reciprocal lattice vectors are different. The algorithm only needs four iterations to resolve a 15° drift angle in the image. Ministerio de Economía y Competitividad MAT2016-77491-C2-2-R Ministerio de Ciencia, Innovación y Universidades TEC2017-86102-C2-2-R

In this paper we develop a new computational technique called boundary scale-space theory. This tech- nique is based on the topol1 ogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects { Ki }i ≥ 1 all of them having the same number of closed pieces than K. Each piece of Ki ( ∀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-scale-space cell complexes. Moreover, the so called homology boundary scale-space model of K ( hbss -model for short) is introduced here. Thismodel 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 scaleindices linked by ( hbss -transition maps) preserving homology classes. Various codes for each connectedsubgraph 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. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2016-77785-P

Image processing is a fundamental operation in many real time applications, where lots of parallelism can be extracted. Segmenting the image into different connected components is the most known operations, but there are many others like extracting the region adjacency graph (RAG) of these regions, or searching for features points, being invariant to rotations, scales, brilliant changes, etc. Most of these algorithms part from the basis of Tracing-type approaches or scan/raster methods. This fact necessarily implies a data dependence between the processing of one pixel and the previous one, which prevents using a pure parallel approach. In terms of time complexity, this means that linear order O(N) (N being the number of pixels) cannot be cut down. In this paper, we describe a novel approach based on the building of a pure Topological framework, which allows to implement fully parallel algorithms. Concerning topological analysis, a first stage is computed in parallel for every pixel, thus conveying the local neighboring conditions. Then, they are extended in a second parallel stage to the necessary global relations (e.g. to join all the pixels of a connected component). This combinatorial optimization process can be seen as the compression of the whole image to just one pixel. Using this final representation, every region can be related with the rest, which yields to pure topological construction of other image operations. Besides, complex data structures can be avoided: all the processing can be done using matrixes (with the same indexation as the original image) and element-wise operations. The time complexity order of our topological approach for a m×n pixel image is near O(log(m+n)), under the assumption that a processing element exists for each pixel. Results for a multicore processor show very good scalability until the memory bandwidth bottleneck is reached, both for bigger images and for much optimized implementations. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures.

In this paper, we explore the possibility of applying the anisotropic generalized Hough transform (AGHT) enhanced with a Gabor based time-frequency filtering (GTF) for the determination of the mandibular canal in digital dental panoramic radiographs. The proposed method is based on template matching us- ing the fact that the shape of the mandibular canal is usually the same, followed by a filtering of the accumulator space in the Gabor domain for a precise detection of the position. The proposed procedure consists of a detailed description of the shape of the canal in its canonical form and on preserving Gabor filtering information for sorting the hierarchy of location candidates after applying anisotropically the ex- traction algorithm. The experimental results show that the proposed procedure is robust to recognition under occlusion and under the presence of additional structures e.g. teeth, projection errors.

In [14], a topologically consistent framework to support parallel topological analysis and recognition for 2 D digital objects was introduced. Based on this theoretical work, we focus on the problem of finding efficient algorithmic solutions for topological interrogation of a 2 D digital object of interest D of a pre-segmented digital image I , using 4-adjacency between pixels of D . In order to maximize the degree of parallelization of the topological processes, we use as many elementary unit processing as pixels the image I has. The mathematical model underlying this framework is an appropriate extension of the classical concept of abstract cell complex: a primal–dual abstract cell complex (pACC for short). This versatile data structure encompasses the notion of Homological Spanning Forest fostered in [14,15]. Starting from a symmetric pACC associated with I , the modus operandi is to construct via combinatorial operations another asymmetric one presenting the maximal number of non-null primal elementary interactions between the cells of D . The fundamental topological tools have been transformed so as to promote an efficient parallel implementation in any parallel-oriented architecture (GPUs, multi-threaded computers, SIMD kernels and so on). A software prototype modeling such a parallel framework is built.

Segmentations of a digital object based on a connectivity criterion at n-xel or sub-n-xel level are useful tools in image topological analysis and recognition. Working with cell complex analogous of digital objects, an example of this kind of segmentation is that obtained from the combinatorial representation so called Homological Spanning Forest (HSF, for short) which, informally, classifies the cells of the complex as belonging to regions containing the maximal number of cells sharing the same homological (algebraic homology with coefficient in a field) information. We design here a parallel method for computing a HSF (using homology with coefficients in Z/2Z) of a 3D digital object. If this object is included in a 3D image of m1 × m2 × m3 voxels, its theoretical time complexity order is near O(log(m1 + m2 + m3)), under the assumption that a processing element is available for each voxel. A prototype implementation validating our results has been written and several synthetic, random and medical tridimensional images have been used for testing. The experiments allow us to assert that the number of iterations in which the homological information is found varies only to a small extent from the theoretical computational time. Ministerio de Economía y Competitividad MTM2016-81030-P

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

In this paper, we make use of the topological invariants of 2D images for an accelerated training and an improved recognition ability of a deep learning neural network applied to digital image objects. For our test images, we generate the associated simplicial complexes and from them we compute the Betti numbers which for a 2D object are the number of connected components and the number of holes. These information are used for training the network according to the corresponding Betti number. Experiments on the MNIST databases are presented in support of the proposed method. European Union (UE). H2020 INFRAIA- 2016-1-730897

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

Connected component labeling (CCL) of binary images is one of the fundamental operations in real time applications. The adjacency tree (AdjT) of the connected components offers a region-based representation where each node represents a region which is surrounded by another region of the opposite color. In this paper, a fully parallel algorithm for computing the CCL and AdjT of a binary digital image is described and implemented, without the need of using any geometric information. The time complexity order for an image of m × n pixels under the assumption that a processing element exists for each pixel is near O(log(m+ n)). Results for a multicore processor show a very good scalability until the so-called memory bandwidth bottleneck is reached. The inherent parallelism of our approach points to the direction that even better results will be obtained in other less classical computing architectures. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2012-37868-C04-02

En este estudio se evaluó el comportamiento de modelos tradicionales de altura dominante y área basal incorporando variables ambientales para plantaciones de Eucalyptus globulus en la zona sur de Chile

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.

The paper provides a theorem for the characteriza- tion of numerical stability of spline-type systems. These systems are generated through shifted copies of a given atom over a time lattice. Also, we reformulate the well known Gabor systems via modulated spline-type systems and we apply the corresponding numerical stability to these systems. The numerical stability is tested for consistency against deformations. Austrian Science Fund (FWF) P27516

In this paper, a homotopy-based method is employed for the recovery of speech recordings from missing or corrupted samples taken in a noisy environment. The model for the acquisition device is a compressed sensing scenario using Gabor frames. To recover an approximation of the speech file, we used the basis pursuit denoising method with the homotopy continuation algorithm. We tested the proposed method with various speech recordings.

A design and implementation of a parallel algorithm for computing the Region-Adjacency Tree of a given segmentation of a 2D digital image is given. The technique is based on a suitable distributed use of the algorithm for computing a Homological Spanning Forest (HSF) structure for each connected region of the segmentation and a classical geometric algorithm for determining inclusion between regions. The results show that this technique scales very well when executed in a multicore processor. Ministerio de Ciencia e Innovación TEC2012-37868-C04-02 Universidad de Sevilla 2014/753

Provided that a cohomological model for $$G$$G is known, we describe a method for constructing a basis for $$n$$n-cocycles over $$G$$G, from which the whole set of $$n$$n-dimensional $$n$$n-cocyclic matrices over $$G$$G may be straightforwardly calculated. Focusing in the case $$n=2$$n=2 (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative $$2$$2-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When $$n>2$$n>2, this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for $$n=2,3$$n=2,3. In particular, we give some examples of improper 3-dimensional $$3$$3-cocyclic Hadamard matrices.

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

In this work, a multiple generalized anisotropic Hough transform (AGHT) is used to detect the mandibular canal in dental panoramic radiographs. The proposed method relies on a sequential application of the Hough transform that we call double Hough transform. The recognition of the mandibular canal is based on a double template matching compared with the clinical detection using the fact that the shape of the mandibular canal is usually the same and it is situated inside the mandibular bone. The experiments performed on real orthopantomographic images shown that the risk of false detection is significantly decreased, while the recognition is not affected by occlusion and by the presence of additional structures, e.g., teeth, projection errors.

An appropriate generalization of the classical notion of abstract cell complex, called primal-dual abstract cell complex (pACC for short) is the combinatorial notion used here for modeling and analyzing the topology of nD digital objects and images. Let D ⊂ I be a set of n-xels (ROI) and I be a n-dimensional digital image.We design a theoretical parallel algorithm for constructing a topologically meaningful asymmetric pACC HSF(D), called Homological Spanning Forest of D (HSF of D, for short) starting from a canonical symmetric pACC associated to I and based on the application of elementary homotopy operations to activate the pACC processing units. From this HSF-graph representation of D, it is possible to derive complete homology and homotopy information of it. The preprocessing procedure of computing HSF(I) is thoroughly discussed. In this way, a significant advance in understanding how the efficient HSF framework for parallel topological computation of 2D digital images developed in [2] can be generalized to higher dimension is made. Ministerio de Economía y Competitividad TEC2016-77785-P Ministerio de Economía y Competitividad MTM2016-81030-P

To prevent the release of large quantities of CO2 into the atmosphere, carbon capture and storage (CCS) represents a potential means of mitigating the contribution of fossil fuel emissions to global warming and ocean acidification. Fluvial saline aquifers are favourite targeted reservoirs for CO2 storage. These reservoirs are very heterogeneous but their heterogeneities were rarely integrated into CO2 reservoir models. Moreover, contrary to petroleum reservoirs, the available dataset is very limited and not supposed to be enriched. This leads to wide uncertainties on reservoir characteristics required for CSS management (injection location, CO2 plume migration, etc.). Stochastic simulations are classical strategies in such under-constrained context. They aim at generating a wide number of models that all fit the available dataset. The generated models serve as support for computing the required reservoir characteristics and their uncertainties. A challenge is to optimize the uncertainty computations by selecting stochastic models that should have a priori very different flow behaviours. Fluid flows depend on the connectivity of reservoir rocks (channel deposits). In this paper, it is proposed to study the variability of the Betti numbers in function of different fluvial architectures. The aim is to quantify the impact of fluvial heterogeneities and their spatial distribution on reservoir rock topology and then on CO2 storage capacities. Representative models of different scenarios of channel stacking and their internal heterogeneities are generated using geostatistical simulation approaches. The Betti numbers are computed on each generated models and statistically analysed to exhibit if fluvial architecture controls reservoir topology.

Effective Homology is an algebraic-topological method based on the com- putational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important com- putability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cellofthecomplex.Inthispaper,bytakingasinputapixel-based2Dbinaryobject,we present a logarithmic-time uniform solution for describing a chain homotopy operator φ for its adjacency graph. This solution is based on Membrane Computing tech- niques applied to the spanning forest problem and it can be easily extended to higher dimensions.

A 2D topology-based digital image processing framework is presented here. This framework consists of the computation of a flexible geometric graph-based structure, starting from a raster representation of a digital image I. This structure is called Homological Spanning Forest (HSF for short), and it is built on a cell complex associated to I. The HSF framework allows an efficient and accurate topological analysis of regions of interest (ROIs) by using a four-level architecture. By topological analysis, we mean not only the computation of Euler characteristic, genus or Betti numbers, but also advanced computational algebraic topological information derived from homological classification of cycles. An initial HSF representation can be modified to obtain a different one, in which ROIs are almost isolated and ready to be topologically analyzed. The HSF framework is susceptible of being parallelized and generalized to higher dimensions.

In this paper, we present a direct computational application of Homological Perturbation Theory (HPT, for short) to computer imagery. More precisely, the formulas of the A ∞–coalgebra maps Δ 2 and Δ 3 using the notion of AT-model of a digital image, and the HPT technique are implemented. The method has been tested on some specific examples, showing the usefulness of this computational tool for distinguishing 3D digital images.

Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, $${\overline{B}(\mathsf{\textstyle Z\kern-0.4em Z}[G])}$$, to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Alvarez et al. (J Symb Comput 44:558–570, 2009; 2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Alvarez et al. in 2006).

In this paper we present a new software tool for dealing with the problem of segmentation in Digital Imagery. The implementation is inspired in the design of a tissue-like P system which solves the problem in constant time due the intrinsic parallelism of Membrane Computing devices.

Membrane Computing is a paradigm inspired from biological cellular communication. Membrane computing devices are called P systems. In this paper we calculate some algebraic-topological information of 2D and 3D images in a general and parallel manner using P systems. First, we present a new way to obtain the homology groups of 2D digital images in time logarithmic with respect to the input data involving an improvement with respect to the algorithms development by S. Peltier et al. Second, we obtain an edge-segmentation of 2D and 3D digital images in constant time with respect to the input data.

Let 𝐺 × τ 𝐺′ be the principal twisted Cartesian product with fibre 𝐺, base 𝐺 and twisting function where 𝐺 and 𝐺′ are simplicial groups as well as 𝐺 × τ 𝐺′; and 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁 (𝐺′) be the twisted tensor product associated to 𝐶𝑁 (𝐺 × τ 𝐺′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁(𝐺′), μ) is a multiplicative Cartan's construction where μ is the standard product on 𝐶𝑁(𝐺) ⊗ 𝐶𝑁(𝐺′). Furthermore, assuming that a contraction from 𝐶𝑁(𝐺′) to 𝐻𝐺′ exists and using the techniques from homological perturbation theory, we extend the former result to other “twisted” tensor products of the form 𝐶𝑁(𝐺) ⊗ 𝐻𝐺′.

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model for G is known, the homological reduction method automatically generates a full basis for 2-cocycles over G (including 2-coboundaries). From these data, either an exhaustive or a heuristic search for Hadamard cocyclic matrices is then developed. The knowledge of an explicit basis for 2-cocycles which includes 2-coboundaries is a key point for the designing of the heuristic search. It is worth noting that some Hadamard cocyclic matrices have been obtained over groups G for which the exhaustive searching techniques are not feasible. From the computational-cost point of view, even in the case that the calculation of such a homological model is also included, comparison with other methods in the literature shows that the homological reduction method drastically reduces the required computing time of the operations involved, so that even exhaustive searches succeeded at orders for which previous calculations could not be completed. With aid of an implementation of the method in MATHEMATICA, some examples are discussed, including the case of very well-known groups (finite abelian groups, dihedral groups) for clarity.

A free resolution (R, d + h) → (M, d) for a DG-module (M, d) over a DG-algebra (A, d) is constructed in the sense of a perturbation of the differential in a free bigraded resolution (R, d) → M of the underlying graded module M over an underlying graded algebra A.

Given a basis equation image for 2-cocycles equation image over a group G of order equation image, we describe a nonlinear system of 4t-1 equations and k indeterminates equation image over equation image, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (equation image) is a solution of the system if and only if the 2-cocycle equation image gives rise to a cocyclic Hadamard matrix equation image. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in equation image which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups equation image and equation image. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups equation image, we have found that it suffices to check t instead of the 4t rows of equation image, to decide the Hadamard character of the matrix (for a special class of cocycles f). Junta de Andalucía FQM-296

In [18], 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. Discrete Morse theory, Effective Homology Theory and parallel computing techniques are suitably combined for obtaining a homological encoding, called algebraic minimal model, of a Region-Of-Interest (seen as cubical complex) of a presegmented k-D digital image.

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.

Let G, G′, and G ×τ G′ be three simplicial groups (not necessarily abelian) and C N (G) ⊗ t C N (G′) be the “twisted” tensor product associated to C N (G ×τ G′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair (C N (G) ⊗ t C N (G′), μ) is a DGA-algebra where μ is the standard product of C N (G) ⊗ C N (G′). Furthermore, the injection of the twisted Eilenberg–Zilber contraction is a DGA-algebra morphism and the projection and the homotopy operator satisfy other weaker multiplicative properties. Junta de Andalucía FQM–296

In this article, in the setting of connected DG-modules, we prove that, for any A ∞-algebra (M, {m i } i≥1), there is a chain contraction from a DG-algebra A M onto the DG-module M such that the A ∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A ∞-algebra in the sense of (Boardman and Vogt, 1973). Appropiate dual results are given for A ∞-coalgebras.

International audience; This paper introduces a new kind of skeleton for binary volumes called the cellular skeleton. This skeleton is not a subset of voxels of a volume nor a subcomplex of a cubical complex: it is a chain complex together with a reduction from the original complex.Starting from the binary volume we build a cubical complex which represents it regarding 6 or 26-connectivity. Then the complex is thinned using the proposed method based on elementary collapses, which preserves significant geometric features. The final step reduces the number of cells using Discrete Morse Theory. The resulting skeleton is a reduction which preserves the homology of the original complex and the geometrical information of the output of the previous step.The result of this method, besides its skeletonization content, can be used for computing the homology of the original complex, which usually provides well shaped homology generators.

This paper analyses the topological information of a digital object $$O$$O under a combined combinatorial-algebraic point of view. Working with a topology-preserving cellularization $$K(O)$$K(O) of the object, algebraic and combinatorial tools are jointly used. The combinatorial entities used here are vector fields, $$V$$V-paths and directed graphs. In the algebraic side, chain complexes with extra $$2$$2-nilpotent operators are considered. By mixing these two perspectives we are able to explore the problems of combinatorial and homological optimality. Combinatorial optimality is understood here as the problem for constructing a discrete gradient vector field (DGVF) in the sense of Discrete Morse Theory, such that it has the least possible number of critical cells. Fixing $${\mathbb {Z}}/2{\mathbb {Z}}$$Z/2Z as field of coefficients, by homological `optimality' we mean the problem of constructing a $$2$$2-nilpotent codifferential map $$\phi :C_*(K(O)) \rightarrow C_{*+1}(K(O))$$?:C?(K(O))?C?+1(K(O)) for finite linear combinations of cells in $$K(O)$$K(O), called homology integral operator. The homology groups associated to the chain complex $$(C(K(O)), \phi )$$(C(K(O)),?) are isomorphic to those of $$(C(K(O)), \partial )$$(C(K(O)),?), being $$\partial $$? the canonical boundary or differential operator of the cell complex $$K(O)$$K(O). Relations between these two problems are tackled here by using a type of discrete graphs associated to a homology integral operator, called Homological Spanning Forests (HSF for short). Informally, an HSF for a cell complex can be seen as a kind of combinatorial compressed representation of a homology integral operator. As main result, we refine the heuristic for computing DGVFs based on the iterative Morse complex reduction technique of [1], reducing the search space for an optimal DGVF to an HSF associated to a homology integral operator.

This paper offers an algorithmic solution to the problem of obtaining “economical” formulae for some maps in Simplicial Topology, having, in principle, a high computational cost in their evaluation. In particular, maps of this kind are used for defining cohomology operations at the cochain level. As an example, we obtain explicit combinatorial descriptions of Steenrod View the MathML sourcekth powers exclusively in terms of face operators.

Let A›tC be a twisted tensor product of an algebra A and a coalgebra C, along a twisting cochain t : C ! A. By means of what is called the tensor trick and under some nice conditions, Gugenheim, Lambe and Stashe proved in the early 90s that A ›t C is homology equivalent to the objects M ›t0 C and A›t00 N, where M and N are strong deformation retracts of A and C, respectively. In this paper, we attack this problem from the point of view of contractions. We find explicit contractions from A ›t C to M ›t0 C and A ›t00 N. Applications to the comparison of resolutions which split o of the bar resolution, as well as to some homological models for central extensions are given.