Algorithms in Digital Geometry Based on Cellular Topology

Abstract. The paper presents some algorithms in digital geometry based on the topology of cell complexes. The paper contains an axiomatic justification of the necessity of using cell complexes in digital geometry. Algorithms for solving the following problems are presented: tracing of curves and surfaces, recognition of digital straight line segments (DSS), segmentation of digital curves into longest DSS, recognition of digital plane segments, computing the curvature of digital curves, filling of interiors of n-dimensional regions (n=2,3,4), labeling of components (n=2,3), computing of skeletons (n=2, 3).

Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model

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-

Generating Second Order (Co)homological Information within AT-Model Context

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-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

Computing the Component-Labeling and the Adjacency Tree of a Binary Digital Image in Near Logarithmic-Time

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-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

Building Hierarchical Tree Representations Using Homological-Based Tools

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

On the Topological Disparity Characterization of Square-Pixel Binary Image Data by a Labeled Bipartite Graph

Given an nD digital image I based on cubical n-xel, to fully characterize the degree of internal topological dissimilarity existing in I when using different adjacency relations (mainly, comparing 2n or 2n −1 adjacency relations) is a relevant issue in current problems of digital image processing relative to shape detection or identification. In this paper, we design and implement a new self-dual representation for a binary 2D image I, called {4, 8}-region adjacency forest of I ({4, 8}-RAF, for short), that allows a thorough analysis of the differences between the topology of the 4-regions and that of the 8-regions of I. This model can be straightforwardly obtained from the classical region adjacency tree of I and its binary complement image Ic, by a suitable region label identification. With these two labeled rooted trees, it is possible: (a) to compute Euler number of the set of foreground (resp. background) pixels with regard to 4-adjacency or 8-adjacency; (b) to identify new local and global measures and descriptors of topological dissimilarity not only for one image but also between two or more images. The parallelization of the algorithms to extract and manipulate these structures is complete, thus producing efficient and unsophisticated codes with a theoretical computing time near the logarithm of the width plus the height of an image. Some toy examples serve to explain the representation and some experiments with gray real images shows the influence of the topological dissimilarity when detecting feature regions, like those returned by the MSER (maximally stable extremal regions) method.Ministerio de Economía, Industria y Competitividad PID2019-110455GB-I00 (Par-HoT)Junta de Andalucía US-138107

Parallel Image Processing Using a Pure Topological Framework

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.1Ministerio de Economía y Competitividad (España) TEC2012-37868-C04-02AEI/FEDER (UE) MTM2016-81030-PVPPI of the University of Sevill

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

Polyhedral Surface Approximation of Non-Convex Voxel Sets and Improvements to the Convex Hull Computing Method

In this paper we introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some improvements to our convex hull algorithm to reduce computation time

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