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

Topologie digitale dans un espace localement fini

Dans ce mémoire, nous présentons d'abord une introduction à la théorie classique de la topologie digitale en utilisant une approche de"graphe d'adjacence". Les concepts de cette théorie sont examinés en détail et ensuite, nous discutons des désavantages inhérents à cette approche, dus essentiellement au manque de rigueur axiomatique dans son élaboration. Par la suite, nous étudions une nouvelle approche à la théorie de topologie digitale telle que développée par V. Kovalevsky. Cette théorie, basée sur une approche axiomatique, permet de contourner la plupart des problèmes rencontrés dans la théorie classique. Elle présente des axiomes pour bien définir une topologie digitale. Après avoir présenté les axiomes, nous construisons un contre exemple qui démontre une inconsistance dans l'approche de Kovalevsky. Afin de pallier à cette difficulté, un nouvel axiome est rajouté à cet effet. Au lieu de se consacrer à l'étude des complexes cellulaires abstraits, nous faisons appel à la théorie des CW-complexes, telle que développée par J.H.C. Whitehead, et aux complexes cubiques, tels que formalisés par T. Kaczynski et al. pour établir des liens entre la toplogie digitale axiomatique et la topologie algébrique dont le formalisme puissant permet d'élargir les champs d'application de la topologie digitale. Finalement des exemples concrets sont donnés pour démontrer l'utilité de cette théorie pour l'analyse d'images digitales

Cellular Automata as Cellular Spaces

The topology of an abstract cellular complex is developed from the idea of a locally finite space consisting of countably many elements, and can provide a coordinate- free definition of both the geometric structure and discrete dynamics of cellular automata

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

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

PROPIEDADES TOPOLÓGICAS DE LA LÍNEA KHALIMSKY

   In the years 1977 and 1986, Khalimsky, and more recently Kovalevsky (1988), have shown that the digital images are associated to topological notions. Since then, the notions of general topology had been used in the processing of digital images. In this context, one of the more useful topologies is the Khalimsky’s topology. The aim of this work is to categorize the Khalimsky’s topological space taking as a starting point the properties of the spaces of Alexandroff.    En los años 1977 y 1986, Khalimsky, y más recientemente Kovalevsky (1988), propusieron que una imagen digital está asociada a un espacio topológico. Desde entonces, las nociones de topología general son usadas en el procesamiento de imágenes digitales. En este contexto, una de las topologías más utilizada es la topología de Khalimsky. El objetivo de este trabajo es categorizar el espacio topológico de Khalimsky tomando como punto de partida las propiedades de los espacios de Alexandroff.&nbsp

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

Un nuevo enfoque para la clasificación de imágenes multiespectrales basado en complejos cartesianos

El presente artículo propone un nuevo enfoque de clasificación de imágenes que utiliza como representación del espacio complejos cartesianos libre de ambigüedad en las relaciones topológicas. El enfoque propuesto comprende seis fases: (i) conversión de la imagen convencional al espacio de complejos cartesianos; (ii) transformación a niveles de grises; (iii) producción de super-píxeles basada en la transformada de cuenca; (iv) producción del espacio de textura aprovechando elementos inter-pixel 1-dimensionales; (v) clasificación mediante máquinas de soporte vectorial; y (vi) evaluación de los resultados. Aunque la exactitud temática de la clasificación a partir del nuevo enfoque mejora la exactitud obtenida con una representación convencional del espacio, la prueba de confianza indica que esa mejoría no es estadísticamente significativa. Sin embargo, el nuevo enfoque puede ser fortalecido en el futuro mediante la incorporación de técnicas para mejorar la definición de límites entre cuencas utilizando valores de probabilidad y espacios de textura multiescala.This article proposes a new approach to image classification using a space representation as a Cartesian complex free of ambiguity in the topological relationships of adjacency, connectivity, and boundary. The proposed model comprises six phases: (i) image conversion from the conventional space into the Cartesian complex space; (ii) greyscale transformation, (iii) super-pixel space production based on watershed transform, (iv) texture space production taking advantage of 1-dimensional interpixel elements, (v) classification using support vector machines and (vi) results quality assessment. Although global accuracy of the proposed classification improves accuracy of results obtained with a conventional representation of space, the confidence test shows that this improvement is not statistically significant. However, the new approach can be further strengthened by incorporating techniques to improve boundaries definition between watersheds based on probability values as well as using spaces of multiscale texture