Chain Homotopies for Object Topological Representations

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here. A concept of generators which are "nicely" representative cycles is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse)

Axiomatic Digital Topology

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested

Topological challenges in multispectral image segmentation

Land cover classification from remote sensing multispectral images has been traditionallyconducted by using mainly spectral information associated with discrete spatial units (i.e. pixels).Geometric and topological characteristics of the spatial context close to every pixel have been either not fully treated or completely ignored.This article provides a review of the strategies used by a number of researchers in order to include spatial and topological properties in image segmentation.­­­It is shown how most of researchers have proposed to perform -previous to classification- a grouping or segmentation of nearby pixels by modeling neighborhood relationships as 4-connected, 8-connected and (a, b) – connected graphs.In this object-oriented approach, however, topological concepts such as neighborhood, contiguity, connectivity and boundary suffer from ambiguity since image elements (pixels) are two-dimensional entities composing a spatially uniform grid cell (i.e. there are not uni-dimensional nor zero-dimensional elements to build boundaries). In order to solve such topological paradoxes, a few proposals have been proposed. This review discusses how the alternative of digital images representation based on Cartesian complexes suggested by Kovalevsky (2008) for image segmentation in computer vision, does not present topological flaws, typical of conventional solutions based on grid cells. However, such a proposal has not been yet applied to multispectral image segmentation in remote sensing. This review is part of the PhD in Engineering research conducted by the first author under guidance of the second one. This review concludes suggesting the need to research on the potential of using Cartesian complexes for multispectral image segmentation

On the Cohomology of 3D Digital Images

We propose a method for computing the cohomology ring of three--dimensional (3D) digital binary-valued pictures. We obtain the cohomology ring of a 3D digital binary--valued picture $I$, via a simplicial complex K(I)topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the "digital" cohomology ring of 3D digital binary-valued pictures is tested by means of a small program visualizing the different steps of the method. Some examples concerning topological thinning, the visualization of representative (co)cycles of (co)homology generators and the computation of the cup product on the cohomology of simple pictures are showed.Comment: Special Issue: Advances in Discrete Geometry and Topolog

Inteligencia computacional aplicada a la segmentación de imágenes de resonancia magnética cerebral para la diagnosis y tratamiento médico

Segmentation of medical magneticresonance images present in most of themethod described will be developed insome kind of methodology or related clustering data design to classifiers models, as an introduction to the basic ideas under lying fuzzy pattern recognition, topological properties inreconstructing anatomical tissue andquality representation such medical images features. This is way and approach involving techniques working with modeling the vagueness oruncertainty particularly, fuzzy clustering models as the tool used on the classification systems combined with anatomical reconstruction model evidencing the pathological tissuei dentification process. Finally, this groupof structured techniques algorithms transfers knowledge of the medical domain for use in the reconstruction of volumetric surfaces retain the anatomy ofthe object of interest, in thiscase thepossibility to locate a tumor or lesion.La segmentación de imágenes médicas de resonancia magnética, presente en los métodos que se describen, y que serán desarrollados con algún tipo de tecnología relacionada a los modelos de clasificación en agrupamiento de datos,están basados en las teorías básicas subyacentes al reconocimiento de patrones difusos, a las propiedades topológicas en la reconstrucción de tejido anatómico y a la calidad de representación de características de la imagen. Así se involucran técnicas de trabajo con el modelado de vaguedad o incertidumbre utilizando modelos de agrupamiento difuso como herramienta principal del sistema de clasificación,combinados con el modelado de reconstrucción anatómica que evidencie el proceso de identificación de tejido patológico. Finalmente, este grupo de técnicas estructuradas en forma de algoritmos, transfiere conocimiento del dominio médico para ser utilizados en la reconstrucción de superficies volumétricas que conserven la anatomía del objeto de interés; en nuestro caso, la posibilidad de localizar o representar un tumor o lesión

Multispectral image classification from axiomatic locally finite spaces-based segmentation

Geographical object-based image analysis (GEOBIA) usually starts defining coarse geometric space elements, i.e. image-objects, by grouping near pixels based on (a, b)-connected graphs as neighbourhood definitions. In such an approach, however, topological axioms needed to ensure a correct representation of connectedness relationships can not be satisfied. Thus, conventional image-object boundaries definition presents ambiguities because one-dimensional contours are represented by two-dimensional pixels. In this paper, segmentation is conducted using a novel approach based on axiomatic locally finite spaces (provided by Cartesian complexes) and their linked oriented matroids. For the test, the ALFS-based image segments were classified using the support vector machine (SVM) algorithm using directional filter response as an additional channel. The proposed approach uses a multi-scale approach for the segmentation, which includes multi-scale texture and spectral affinity analysis in boundary definition. The proposed approach was evaluated comparatively with conventional pixel representation on a small subset of GEOBIA2016 benchmark dataset. Results show that classification accuracy is increased in comparison to a conventional pixel segmentation.El análisis de imagenes basado en objetos geográficos (GEOBIA por su sigla en inglés) comienza generalmente definiendo elementos más gruesos del espacio geométrico u objetos de imagen, agrupando píxeles cercanos con base en grafos (a, b)-conectados como definiciones de vecindario. En este enfoque, sin embargo, pueden no cumplirse algunos axiomas topológicos requeridos para garantizar una correcta representación de las relaciones de conexión. Por lo tanto, la definición convencional de límites de objetos de imagen, presenta ambigüedades debido a que los contornos unidimensionales están representados por píxeles bidimensionales. En este trabajo, la segmentación se lleva a cabo mediante un nuevo enfoque basado en espacios axiomáticos localmente finitos (proporcionados por complejos cartesianos) y sus matroides orientados asociados. Para probar el enfoque propuesto, los segmentos de la imagen basada en ALFS fueron clasificados usando el algoritmo de máquina de soporte vectorial (SVM por su sigla en inglés) usando la respuesta a filtros direccionales como un canal adicional. El enfoque propuesto utiliza un enfoque multiescala para la segmentación, que incluye análisis de textura y de afinidad espectral en la definición de límite. La propuesta se evaluó comparativamente con la representación de píxeles convencionales en un pequeño subconjunto del conjunto de datos de referencia GEOBIA2016. Los resultados muestran que la exactitud de la clasificación se incrementa en comparación con la segmentación convencional de pixeles

Retos topológicos en la segmentación de imágenes multiespectrales

Land cover classification from remote sensing multispectral images has been traditionallyconducted by using mainly spectral information associated with discrete spatial units (i.e. pixels).Geometric and topological characteristics of the spatial context close to every pixel have been either not fully treated or completely ignored.This article provides a review of the strategies used by a number of researchers in order to include spatial and topological properties in image segmentation.­­­It is shown how most of researchers have proposed to perform -previous to classification- a grouping or segmentation of nearby pixels by modeling neighborhood relationships as 4-connected, 8-connected and (a, b) – connected graphs.In this object-oriented approach, however, topological concepts such as neighborhood, contiguity, connectivity and boundary suffer from ambiguity since image elements (pixels) are two-dimensional entities composing a spatially uniform grid cell (i.e. there are not uni-dimensional nor zero-dimensional elements to build boundaries). In order to solve such topological paradoxes, a few proposals have been proposed. This review discusses how the alternative of digital images representation based on Cartesian complexes suggested by Kovalevsky (2008) for image segmentation in computer vision, does not present topological flaws, typical of conventional solutions based on grid cells. However, such a proposal has not been yet applied to multispectral image segmentation in remote sensing.  This review is part of the PhD in Engineering research conducted by the first author under guidance of the second one. This review concludes suggesting the need to research on the potential of using Cartesian complexes for multispectral image segmentation.La clasificación de la cobertura terrestre a partir de imágenes multiespectrales de teledetección se ha llevado a cabo tradicionalmente utilizando información principalmente espectral asociada a unidades espaciales discretas (es decir, píxeles). Las características geométricas y topológicas del contexto espacial cercanas a cada píxel no se han tratado del todo o se han ignorado por completo. proporciona una revisión de las estrategias utilizadas por un número de investigadores para incluir propiedades espaciales y topológicas en la segmentación de imágenes. Se muestra cómo la mayoría de los investigadores han propuesto realizar, antes de la clasificación, una agrupación o segmentación de píxeles cercanos modelando el vecindario relaciones como 4 conectadas, 8 conectadas y (a, b) conectadas. Sin embargo, en este enfoque orientado a objetos, los conceptos topológicos como vecindad, contigüidad, conectividad y límite sufren de ambigüedad ya que los elementos de imagen (píxeles) son dos entidades tridimensionales que componen una celda de cuadrícula espacialmente uniforme (es decir, no hay uni-di elementos mensionales o de cero dimensiones para construir límites). Para resolver tales paradojas topológicas, se han propuesto algunas propuestas. Esta revisión discute cómo la alternativa de representación de imágenes digitales basada en complejos cartesianos sugerida por Kovalevsky (2008) para la segmentación de imágenes en visión artificial, no presenta fallas topológicas, típicas de soluciones convencionales basadas en celdas de grillas. Sin embargo, tal propuesta aún no se ha aplicado a la segmentación de imágenes multiespectrales en teledetección. Esta revisión es parte del doctorado en investigación de ingeniería conducida por el primer autor bajo la dirección del segundo. Esta revisión concluye sugiriendo la necesidad de investigar sobre el potencial del uso de complejos cartesianos para la segmentación de imágenes multiespectrales