Digital Topology Research Articles

Remarks on fixed point assertions in digital topology, 8

This paper continues a series in which we study deficiencies in previously published works concerning fixed point assertions for digital images.

Applications of semi-pre-c-open sets

Semi-pre-c-open sets are a subtle development in topology that has a substantial impact on both theoretical investigations and real-world applications. This study explores the several uses of semi-pre-c-open sets in diverse fields. First, we look into how they may be used to improve the structure of generalized topological spaces and give a more comprehensive framework for studying compactness, continuity, and convergence. A more precise categorization of functions is made possible by semi-pre-c-open sets, opening up new function spaces and expanding on previously established findings. Furthermore, their significant applications in digital topology help to advance digital continuity notions and more efficient picture processing methods. We develop and study semi-pre-c-open sets, a novel class of generalized open sets in a topological space. The semi-pre-open sets class includes this class. It is demonstrated that the topology generated by semi-pre-c-open sets is the same. Semi-pre-c-derived set, semi-pre-c-interior points, semi-pre-c-closure, semi-pre-c-closure, semi-pre-c-frontier, and semi-pre-c-exterior are among the topological aspects of these sets that we present and examine. Using examples and counterexamples, the existence of their relationship is also examined.

Deep Closing: Enhancing Topological Connectivity in Medical Tubular Segmentation.

Accurately segmenting tubular structures, such as blood vessels or nerves, holds significant clinical implications across various medical applications. However, existing methods often exhibit limitations in achieving satisfactory topological performance, particularly in terms of preserving connectivity. To address this challenge, we propose a novel deep-learning approach, termed Deep Closing, inspired by the well-established classic closing operation. Deep Closing first leverages an AutoEncoder trained in the Masked Image Modeling (MIM) paradigm, enhanced with digital topology knowledge, to effectively learn the inherent shape prior of tubular structures and indicate potential disconnected regions. Subsequently, a Simple Components Erosion module is employed to generate topology-focused outcomes, which refines the preceding segmentation results, ensuring all the generated regions are topologically significant. To evaluate the efficacy of Deep Closing, we conduct comprehensive experiments on 4 datasets: DRIVE, CHASE DB1, DCA1, and CREMI. The results demonstrate that our approach yields considerable improvements in topological performance compared with existing methods. Furthermore, Deep Closing exhibits the ability to generalize and transfer knowledge from external datasets, showcasing its robustness and adaptability. The code for this paper has been available at: https://github.com/5k5000/DeepClosing.

Remarks on fixed point assertions in digital topology, 7

This paper continues a series discussing flaws in published assertions concerning fixed points in digital images.

Remarks on pseudocovering spaces in a digital topological setting: A corrigendum

After investigating some properties of the original version of a pseudo-(k0, k1)-covering space in the literature, it appears that a pseudo-(k0, k1)-covering space is equivalent to a digital (k0, k1)-covering space. Hence, as a corrigendum to [7, 9], the paper first revises one of the three conditions for a pseudo-(k0, k1)-covering space, which broadens the original version. After that, we suggest some examples for the revised version of a pseudo-(k0, k1)-covering map. Since the revised map is so related to the study of several kinds of path liftings, this new version can facilitate some studies in the field of applied topology including digital topology. We note that a weakly local (k0, k1)-isomorphic surjection is equivalent to the new version of a pseudo-(k0, k1)-covering map instead of the original version of a pseudo-(k0, k1)-covering map. The present paper only deals with k-connected digital images (X, k).

A topology constrained phase field model

This paper presents a topology constrained phase field model based on a variable mobility Cahn-Hilliard equation. Mostly in the level set framework, this issue has been addressed using concepts from digital topology. However, point based specifications can not be effective in phase fields where the interface is represented by a diffuse region. In phase field models, topology constraints are included in the energy as a penalty term. However, there are no studies which provide topological constraints in the Cahn-Hilliard model to our knowledge. In this work, we use the mobility parameter to enforce topology constraints; this allows local, explicit control of topological events, unlike the optimization based solutions. We define topologically critical regions in phase fields based on a topological analysis of phase field evolution. This analysis shows that diffuse layers formed at topological encounters behave like a Morse function and the critical points of these transitional layers give topologically critical points in phase fields. Using this property, we first detect critical points using an efficient image analysis method. Then we detect the transitional region (critical region) around each critical point using a local morphological segmentation algorithm. We define a mobility function which both has a spatial and concentration dependence. This function is designed to reverse the effect of diffuse layers formed at topological encounters and prevent the further development of topological events. Being extracted from phase field values, the mobility map naturally aligns with phase field evolution. Together with the use of an unconditionally stable semi-implicit Fourier spectral method for the variable mobility Cahn-Hilliard equation, an efficient model is provided. Local, explicit control mechanism allows the design of free energy functional for modeling different materials; this extends the use of Cahn-Hilliard equation to applications which require topology control.

Remarks on fixed point assertions in digital topology, 6

This paper continues a series discussing flaws in published assertions concerning fixed points in digital metric spaces.

Minimal graphs for contractible and dismantlable properties

The notion of a contractible transformation on a graph was introduced by Ivashchenko as a means to study molecular spaces arising from digital topology and computer image analysis, and more recently has been applied to topological data analysis. Contractible transformations involve a list of four elementary moves that can be performed on the vertices and edges of a graph, and it has been shown by Chen, Yau, and Yeh that these moves preserve the simple homotopy type of the underlying clique complex. A graph is said to be I-contractible if one can reduce it to a single isolated vertex via a sequence of contractible transformations. Inspired by the notions of collapsible and non-evasive simplicial complexes, in this paper we study certain subclasses of I-contractible graphs where one can collapse to a vertex using only a subset of these moves. Our main results involve constructions of minimal examples of graphs for which the resulting classes differ. We also relate these classes of graphs to the notion of k-dismantlable graphs and k-collapsible complexes, which also leads to a minimal counterexample to an erroneous claim of Ivashchenko from the literature. We end with some open questions.

Directional Connectivity-based Segmentation of Medical Images.

Anatomical consistency in biomarker segmentation is crucial for many medical image analysis tasks. A promising paradigm for achieving anatomically consistent segmentation via deep networks is incorporating pixel connectivity, a basic concept in digital topology, to model inter-pixel relationships. However, previous works on connectivity modeling have ignored the rich channel-wise directional information in the latent space. In this work, we demonstrate that effective disentanglement of directional sub-space from the shared latent space can significantly enhance the feature representation in the connectivity-based network. To this end, we propose a directional connectivity modeling scheme for segmentation that decouples, tracks, and utilizes the directional information across the network. Experiments on various public medical image segmentation benchmarks show the effectiveness of our model as compared to the state-of-the-art methods. Code is available at https://github.com/Zyun-Y/DconnNet.

A lumen-adapted navigation scheme with spatial awareness from monocular vision for autonomous robotic endoscopy

Lumen is a common anatomical condition in endoscopic therapy or diagnostics. The navigation of autonomous robotic endoscopy usually involves vision techniques such as detection of lumen center or anatomically specific contour features. However, these methods may fail to achieve smooth interventions and result in large conveying force without spatial awareness of the tissue state. In this paper, a novel navigation pipeline based on a spatial-aware monocular vision is proposed. The spatial awareness pipeline starts with a data-driven depth estimation technique for reconstructing real-time approximate tissue surface. The spatial shape of the lumen described by the skeleton is then extracted using digital topology. We modify the skeleton to get a smooth pathway, and design an adaptive autonomous control strategy using geometric information from the pathway. We conduct experiments on a colon phantom and ex vivo pig intestines. We test turning performance of several bending segments with different angles in phantom, as well as the overall performance of a long-range intervention task in the phantom and pig intestines. The results show our navigation scheme achieve smoother intervention with lower conveying force. The proposed navigation method with spatial awareness can effectively improve the fluency of autonomous robotic endoscopy.

Connectedness Among the Pixels at the Product Digital Topology

Abstract: The aim of this paper is to analyze at the concepts connectedness of the separated interior circles at the pixels at the product digital topology with the axioms C1, C2, C3 in the cartesian complex and also at the structure of the elements at the collections of the pixels and interior circles in that region.

Hyperspaces and function graphs in digital topology

We adapt the study of hyperspaces and function spaces from classical topology to digital topology. We define digital hyperspaces and digital function graphs, and study some of their relationships and graphical properties.

Remarks on fixed point assertions in digital topology, 5

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in “digital metric spaces” - assertions that are incorrect or incorrectly proven, or reduce to triviality.

Freezing Sets for Arbitrary Digital Dimension

The study of freezing sets is part of the theory of fixed points in digital topology. Most of the previous work on freezing sets is for digital images in the digital plane Z2. In this paper, we show how to obtain freezing sets for digital images in Zn for arbitrary n, using the c1 and cn adjacencies.

An Ultra-Broadband Polarization Beam Splitter Based on the Digital Meta-Structure at the 2 µm Waveband

The 2 μm waveband is considered to have great potential in optical communications. Driven by the demands on high-performance functional devices in this spectral band, various integrated photonic components have been demonstrated. In this work, an analog and digital topology optimization method is proposed to design an ultra-broadband polarization beam splitter at the 2 μm waveband. Within an optical bandwidth of 213 nm, the excess losses of TE and TM modes are <0.53 dB and 0.3 dB, respectively. The corresponding polarization extinction ratios are >16.5 dB and 18.1 dB. The device has a very compact footprint of only 2.52 µm × 5.4 µm. According to our best knowledge, this is a benchmark demonstration of an ultra-broadband and ultra-compact polarization beam splitter enabled by the proposed optimization method.

Semi-Jordan curve theorem on the Marcus-Wyse topological plane

<abstract><p>The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.</p></abstract>

Homotopy Theory in Digital Topology

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

≈ g(1,2)*-CLOSED AND ≈ g(1,2)*-OPEN MAPS

General topology plays vital role in many fields of applied sciences as well as in all branches of mathematics. In reality it is used in data mining, computational topology for geometric design and molecular design, computer-aided design, computer-aided geometric design, digital topology, information systems, particle physics and quantum physics etc. By researching generalizations of closed sets, some new separation axioms have been founded and they turn out to be useful in the study of digital topology. Therefore, all bi-topological sets and functions defined will have many possibilities of applications in digital topology and computer graphics.

Subsets and freezing sets in the digital plane

We continue the study of freezing sets for digital images introduced in [L. Boxer and P.C. Staecker, Fixed point sets in digital topology, 1, Applied General Topology 2020; L. Boxer, Fixed point sets in digital topology, 2, Applied General Topology 2020; L. Boxer, Convexity and Freezing Sets in Digital Topology, Applied General Topology, 2021]. We prove methods for obtaining freezing sets for digital images $(X,c_i)$ for $X \subset \mathbb{Z}^2$ and $i \in \{1,2\}$. We give examples to show how these methods can lead to the determination of minimal freezing sets.

ON SOME SETS IN DIGITAL TOPOLOGY

ON SOME SETS IN DIGITAL TOPOLOGY