Digital Topology Research Articles

Detection of the non-topology preservation of Ma and Sonka’s algorithm, by the use of P-simple points

In 1996, Ma and Sonka proposed a thinning algorithm which yields curve skeletons for 3D binary images [C. Ma, M. Sonka, A fully parallel 3D thinning algorithm and its applications, Comput. Vis. Image Underst. 64 (3) (1996) 420–433]. This algorithm is one of the most referred thinning algorithms in the context of digital topology: either by its use in medical applications or for comparisons with other thinning algorithms. In 2007, Wang and Basu [T. Wang, A. Basu, A note on ‘a fully parallel 3D thinning algorithm and its applications’, Pattern Recognit. Lett. 28 (4) (2007) 501–506] wrote a paper in which they claim that Ma and Sonka’s 3D thinning algorithm does not preserve topology. As they highlight in their paper, a counter-example was given in 2001, in Lohou’s thesis [C. Lohou, Contribution à l’analyse topologique des images: étude d’algorithmes de squelettisation pour images 2D et 3D selon une approche topologie digitale ou topologie discrète. Ph.D. thesis, University of Marne-la-Vallée, France, 2001]. In this paper, it is shown how P-simple points have guided the author towards a proof that Ma and Sonka’s algorithm does not always preserve topology. Moreover, the reasoning being very general, it could be reused for such a purpose, i.e., to simplify the proof on the non-topology preservation.

Interactive characterization of granulated materials

This paper illustrates an interactive approach to digital characterization of granulated materials, based on mathematical morphology, digital topology and non-linear differential equations. The digital characterization is achieved using a three-step sequential process, namely: denoising, interactive segmentation and quantification, with emphasis in the interactive segmentation step, since it has significant impact on the characterization results. Denoising is used to extract foreground and background markers. Image segmentation results from an anisotropic region-growing and merging process based on immersion simulation. Region-growing evolves as a fully automated process, initiated at local image minima, while merging requires manual insertion of one single marker per region-of-interest. Non-significant regions are merged according to a hierarchy established by a set of markers and merging rules, which represent a robust solution to the over-segmentation problem inherent to immersion simulation algorithms. The sensitivity of the segmentation algorithm to local intensity variations allows the detection of the edges with high accuracy. Quantification is achieved using a suitable stereometric method, which computes the three-dimensional size distribution from the two-dimensional size distribution observed on the image. It is shown that this interactive approach represents a viable solution to digital characterization of granulated materials.

Adaptive generation of hexahedral element meshes for finite element analysis of metal plastic forming process

Abstract A method for the adaptive generation of hexahedral element mesh based on the geometric features of solid model is proposed. The first step is to construct the refinement information fields of source points and the corresponding ones of elements according to the surface curvature of the analyzed solid model. A thickness refinement criterion is then used to construct the thickness-based refinement information field of elements from digital topology. The second step is to generate a core mesh through removing all the undesired elements using even and odd parity rules. Then the core mesh is magnified in an inside–out manner method through a surface node projection process using the closest position approach. Finally, in order to match the mesh to the characteristic boundary of the solid model, a threading method is proposed and applied. The present method was applied in the mesh construction of different engineering problems. The resulting meshes are well-shaped and capture all the geometric features of the original solid models.

New Characterizations of Simple Points in 2D, 3D, and 4D Discrete Spaces

A point of a discrete object is called simple if it can be deleted from this object without altering topology. In this article, we present new characterizations of simple points which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting such points. In order to prove these characterizations, we establish two confluence properties of the collapse operation which hold in the neighborhood of a point in spaces of low dimension. This work is settled in the framework of cubical complexes, which provides a sound topological basis for image analysis, and allows to retrieve the main notions and results of digital topology, in particular the notion of simple point.

Digital Topology on Adaptive Octree Grids

The theory of digital topology is used in many different image processing and computer graphics algorithms. Most of the existing theories apply to uniform cartesian grids, and they are not readily extensible to new algorithms targeting at adaptive cartesian grids. This article provides a rigorous extension of the classical digital topology framework for adaptive octree grids, including the characterization of adjacency, connected components, and simple points. Motivating examples, proofs of the major propositions, and algorithm pseudocodes are provided.

On Concise 3-D Simple Point Characterizations: A Marching Cubes Paradigm

The centerlines of tubular structures are useful for medical image visualization and computer-aided diagnosis applications. They can be effectively extracted by using a thinning algorithm that erodes an object layer by layer until only a skeleton is left. An object point is "simple" and can be safely deleted only if the resultant image is topologically equivalent to the original. Numerous characterizations of 3-D simple points based on digital topology already exist. However, little work has been done in the context of marching cubes (MC). This paper reviews several concise 3-D simple point characterizations in a MC paradigm. By using the Euler characteristic and a few newly observed properties in the context of connectivity-consistent MC, we present concise and more self-explanatory proofs. We also present an efficient method for computing the Euler characteristic locally for MC surfaces. Performance evaluations on different implementations are conducted on synthetic data and multidetector computed tomography examination of virtual colonoscopy and angiography.

Another Approach to Connectedness with Respect to a Closure Operator

We introduce a new concept of connectedness with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject semilattices, naturally generalizes the classical connectedness of topological spaces and we show that it also behaves accordingly. Moreover, as the main result, we prove that the connectedness introduced is preserved, under some natural conditions, by inverse images of subobjects under quotient morphisms. An application of this result in digital topology is discussed too.

3D reconstruction and quantification of porous structures

In this paper, we describe the methodology that we have designed to quantify the pores distribution in bone implants and the empirical results that we have obtained with BioCAD designed scaffolds, microCT and confocal microscopy data. Our method is based on 3D digital topology properties of the porous structure. We segment the 3D images into three regions: exterior, bone and pore space. Next, we divide the pore space into pores and connection paths. We compute a graph of the pore space such that each node of the graph represents a pore, and an arc between two nodes indicates that the two pores are path-connected. On the basis of the graph and the segmented model, we are able to compute several properties of the material such as global porosity, effective porosity and radial pore distribution.

A quotient-universal digital topology

We study a special topology on Z 2 and show that both the Marcus topology and the Khalimsky topology and also one more digital topology on Z 2 may be obtained as three of its quotient topologies. A quotient closure operator of the topology studied is discussed, too, and an analogue of the Jordan curve theorem for this closure operator is proved.

Two-Dimensional Parallel Thinning Algorithms Based on Critical Kernels

Critical kernels constitute a general framework in category of abstract complexes for study of parallel thinning in any dimension. The most fundamental result in this framework is that, if a subset Y of X contains critical kernel of X, then Y is guaranteed to have the same topology as X. Here, we focus on 2D structures in spaces of two and three dimensions. We introduce notion of crucial pixel, which permits to link this work with framework of digital topology. We prove simple local characterizations, which allow us to express thinning algorithms by way of sets of masks. We propose several new parallel algorithms, which are both fast and simple to implement, that yield symmetrical or non-symmetrical skeletons of 2D objects in 2D or 3D grids. We prove some properties of these skeletons, related to topology preservation, to minimality, and to inclusion of topological axis. The latter may be seen as a generalization of medial axis. We also show how to use critical kernels in order to provide simple proofs of topological soundness of existing thinning schemes. Finally, we clarify link between critical kernels, minimal non-simple sets, and P-simple points.

Adaptive hexahedral mesh generation based on local domain curvature and thickness using a modified grid-based method

This paper presents an adaptive hexahedral mesh generation algorithm for three-dimensional models. It is based on the conventional grid-based method and geometry-adaptive technique to generate uniformly sized or locally refined hexahedral meshes. The first step is to construct the refinement information fields of the source points and the corresponding elements according to the surface curvature of the solid model. A thickness refinement criterion is then used to construct the thickness-based refinement information field of elements from digital topology. Secondly, the element refinement fields are integrated and modified according to the conformal refinement templates proposed in this paper, and used as a metric to generate an initial grid structure enveloping the solid model completely. In order to ensure the conformity of the generated mesh, a set of new refinement templates are proposed, and the corresponding modes for converting arbitrary refinement fields to the templates is established. Especially two kinds of new transition templates, which avoid the propagation of the refinement field, are added. Hexahedral element meshes capturing all the geometry characters of the solid models are generated. The effectiveness and robustness of the algorithm are tested through the applications in several complex three-dimensional models.

Integrating digital topology in image-processing libraries

This paper describes a method to integrate digital topology informations in image-processing libraries. This additional information allows a library user to write algorithms respecting topological constraints, for example, a seed fill or a skeletonization algorithm. As digital topology is absent from most image-processing libraries, such constraints cannot be fulfilled. We describe and give code samples for all the structures necessary for this integration, and show a use case in the form of a homotopic thinning filter inside ITK. The obtained filter can be up to a hundred times as fast as ITK's thinning filter and works for any image dimension. This paper mainly deals of integration within ITK, but can be adapted with only minor modifications to other image-processing libraries.

Local characterization of a maximum set of digital (26, 6)-surfaces

We introduce a new family S of surfaces in the discrete space Z 3 for the usual (26, 6)-adjacency that strictly contains the family of simplicity 26-surfaces and other objects considered as surfaces in the literature. Actually, S characterizes the strongly 6-separating objects of a family S U of digital surfaces defined by means of continuous analogues. The family S U consists of all objects whose continuous analogue is a surface in some homogeneous (26, 6)-connected digital space as defined in the approach to Digital Topology introduced in [R. Ayala, E. Domínguez, A.R. Francés, A. Quintero, Weak Lighting Functions and Strong 26-surfaces. Theoretical Computer Science 283 (2002) 29–66.]. Therefore, S is the largest possible set of surfaces in Z 3 in that setting.

Digital Topology

This documents describes a set of classes to integrate digital topology in the Insight Toolkit, with an application to homotopic thinning. The skeletonization filter implemented using our set of classes is twice as fast as BinaryThinningImageFilter and can be used on images of any dimension.

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− 1 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.

Compatible topologies on graphs: An application to graph isomorphism problem complexity

On one hand the graph isomorphism problem ( GI) has received considerable attention due to its unresolved complexity status and its many practical applications. On the other hand a notion of compatible topologies on graphs has emerged from digital topology (see [A. Bretto, Comparability graphs and digital topology, Comput. Vision Graphic Image Process. (Image Understanding), 82 (2001) 33–41; J.M. Chassery, Connectivity and consecutivity in digital pictures, Comput. Vision Graphic Image Process. 9 (1979) 294–300; L.J. Latecki, Topological Connectedness and 8-connectness in digital pictures, CVGIP Image Understanding 57(2) (1993) 261–262; U. Eckhardt, L.J. Latecki, Topologies for digital spaces Z 2 and Z 3 , Comput. Vision Image Understanding 95 (2003) 261–262; T.Y. Kong, R. Kopperman, P.R. Meyer, A topological approach to digital topology, Amer. Math. Monthly Archive 98(12) (1991) 901–917; R. Kopperman, Topological digital topology, Discrete geometry for computer imagery, 11th International Conference, Lecture Notes in Computer Science, Vol. 2886, DGCI 2003, Naples, Italy, November 19–21, pp. 1–15]). In this article we study GI from the topological point of view. Firstly, we explore the poset of compatible topologies on graphs and in particular on bipartite graphs. Then, from a graph we construct a particular compatible Alexandroff topological space said homeomorphic-equivalent to the graph. Conversely, from any Alexandroff topology we construct an isomorphic-equivalent graph on which the topology is compatible. Finally, using these constructions, we show that GI is polynomial-time equivalent to the topological homeomorphism problem ( TopHomeo). Hence GI and TopHomeo are in the same class of complexity.

Digital Jordan curves

A digital Jordan curve theorem is proved for a new topology defined on Z 2 . This topology is compared with the classical Khalimsky and Marcus topologies used in digital topology. We show that the Jordan curves with respect to the topology defined, unlike the Jordan curves with respect to any of the two classical topologies mentioned, may turn at the acute angle π 4 . We also discuss a quotient topology of the new topology.

Finite approximation of measure and integration

Digital topology is an extreme approach to constructive spatial representation in that a classical space is replaced or represented by a finite combinatorial space. This has led to a popular research area in which theory and applications are very closely related, but the question remains as to ultimately how mathematically viable this approach is, and what the formal relationship between a space and its finite representations is. Several researchers have attempted to answer this by showing that a space can be constructed as the inverse limit of its finite representations, and this construction has been developed independently in computer science and physics. Therefore, in this formal sense, finite representations of classical spaces can be arbitrarily good. The relationship between classical and finite topology is by now fairly well understood, and in this work we extend the theory by showing that measures on the classical space are approximated by measures on finite spaces.

Properties of Digital Homotopy

Several recent papers have adapted notions of geometric topology to the emerging field of `digital topology.' An important notion is that of digital homotopy. In this paper, we study a variety of digitally-continuous functions that preserve homotopy types or homotopy-related properties such as the digital fundamental group.

Topology preserving alternating sequential filter for smoothing two-dimensional and three-dimensional objects

We introduce the homotopic alternating sequential filter as a new method for smoothing two-dimensional (2D) and three-dimensional (3D) objects in binary images. Unlike existing methods, our method offers a strict guarantee of topology preservation. This property is ensured by the exclusive use of homotopic transformations defined in the framework of digital topology. Smoothness is obtained by the use of morphological openings and closings by metric disks or balls of increasing radius, in the manner of an alternating sequential filter. The homotopic alternating sequential filter operates both on the object and on the background, in an equilibrated way. It takes an original image X and a control image C as input, and smooths X "as much as possible" while respecting the topology of X and geometrical constraints implicitly represented by C. Based on this filter, we introduce a general smoothing procedure with a single parameter which allows to control the degree of smoothing. Furthermore, the result of this procedure presents small variations in response to small variations of the parameter value. We also propose a method with no parameter for smoothing zoomed binary images in 2D or 3D while preserving topology.