Helena Molina-Abril

Integral Operators for Computing Homology Generators at Any Dimension

Starting from an nDgeometrical object, a cellular subdivision of such an object provides an algebraic counterpart from which homology information can be computed. In this paper, we develop a process to drastically reduce the amount of data that represent the original object, with the purpose of a subsequent homology computation. The technique applied is based on the construction of a sequence of elementary chain homotopies (integral operators) which algebraically connect the initial object with a simplified one with the same homological information than the former.

Homological spanning forest framework for 2D image analysis

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.

A bio-inspired software for segmenting digital images

Segmentation in computer vision refers to the process of partitioning a digital image into multiple segments (sets of pixels). It has several features which make it suitable for techniques inspired by nature. It can be parallelized, locally solved and the input data can be easily encoded by bio-inspired representations. In this paper, we present a new software for performing a segmentation of 2D digital images based on Membrane Computing techniques.

Homological optimality in Discrete Morse Theory through chain homotopies

Highlights? We present a heuristic to find optimal gradient vector fields on finite cell complexes. ? A new representation (Homological Spanning Forest, HSF) based in homological algebra. ? The HSF is a suitable framework for efficiently compute advanced topo-logical information. Morse theory is a fundamental tool for analyzing the geometry and topology of smooth manifolds. This tool was translated by Forman to discrete structures such as cell complexes, by using discrete Morse functions or equivalently gradient vector fields. Once a discrete gradient vector field has been defined on a finite cell complex, information about its homology can be directly deduced from it. In this paper we introduce the foundations of a homology-based heuristic for finding optimal discrete gradient vector fields on a general finite cell complex K. The method is based on a computational homological algebra representation (called homological spanning forest or HSF, for short) that is an useful framework to design fast and efficient algorithms for computing advanced algebraic-topological information (classification of cycles, cohomology algebra, homology A(∞)-coalgebra, cohomology operations, homotopy groups, ?). Our approach is to consider the optimality problem as a homology computation process for a chain complex endowed with an extra chain homotopy operator.

Advanced Homology Computation of Digital Volumes Via Cell Complexes

Given a 3D binary voxel-based digital object V , an algorithm for computing homological information for V via a polyhedral cell complex is designed. By homological information we understand not only Betti numbers, representative cycles of homology classes and homological classification of cycles but also the computation of homology numbers related additional algebraic structures defined on homology (coproduct in homology, product in cohomology, (co)homology operations,...). The algorithm is mainly based on the following facts: a) a local 3D-polyhedrization of any 2×2×2 configuration of mutually 26-adjacent black voxels providing a coherent cell complex at global level; b) a description of the homology of a digital volume as an algebraic-gradient vector field on the cell complex (see Discrete Morse Theory [5], AT-model method [7,5]). Saving this vector field, we go further obtaining homological information at no extra time processing cost.

Homological Tree-Based Strategies for Image Analysis

Homological characteristics of digital objects can be obtained in a straightforward manner computing an algebraic map ? over a finite cell complex K (with coefficients in the finite field

Searching high order invariants in computer imagery

\textbf{F}_2=\{0,1\}

Learning Biomarker Models for Progression Estimation of Alzheimer’s Disease

) which represents the digital object [9]. Computable homological information includes the Euler characteristic, homology generators and representative cycles, higher (co)homology operations, etc. This algebraic map ? is described in combinatorial terms using a mixed three-level forest. Different strategies changing only two parameters of this algorithm for computing ? are presented. Each one of those strategies gives rise to different maps, although all of them provides the same homological information for K. For example, tree-based structures useful in image analysis like topological skeletons and pyramids can be obtained as subgraphs of this forest.

Multi-stage Biomarker Models for Progression Estimation in Alzheimer’s Disease

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 Δ2 and Δ3 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.

Designing a new software tool for Digital Imagery based on P systems

Being able to estimate a patient’s progress in the course of Alzheimer’s disease and predicting future progression based on a number of observed biomarker values is of great interest for patients, clinicians and researchers alike. In this work, an approach for disease progress estimation is presented. Based on a set of subjects that convert to a more severe disease stage during the study, models that describe typical trajectories of biomarker values in the course of disease are learned using quantile regression. A novel probabilistic method is then derived to estimate the current disease progress as well as the rate of progression of an individual by fitting acquired biomarkers to the models. A particular strength of the method is its ability to naturally handle missing data. This means, it is applicable even if individual biomarker measurements are missing for a subject without requiring a retraining of the model. The functionality of the presented method is demonstrated using synthetic and—employing cognitive scores and image-based biomarkers—real data from the ADNI study. Further, three possible applications for progress estimation are demonstrated to underline the versatility of the approach: classification, construction of a spatio-temporal disease progression atlas and prediction of future disease progression.