Computing a Link Diagram From Its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

Three-dimensional topological radiogenomics of epidermal growth factor receptor Del19 and L858R mutation subtypes on computed tomography images of lung cancer patients

OBJECTIVES

To elucidate a novel radiogenomics approach using three-dimensional (3D) topologically invariant Betti numbers (BNs) for topological characterization of epidermal growth factor receptor (EGFR) Del19 and L858R mutation subtypes.

METHODS

In total, 154 patients (wild-type EGFR, 72 patients; Del19 mutation, 45 patients; and L858R mutation, 37 patients) were retrospectively enrolled and randomly divided into 92 training and 62 test cases. Two support vector machine (SVM) models to distinguish between wild-type and mutant EGFR (mutation [M] classification) as well as between the Del19 and L858R subtypes (subtype [S] classification) were trained using 3DBN features. These features were computed from 3DBN maps by using histogram and texture analyses. The 3DBN maps were generated using computed tomography (CT) images based on the Čech complex constructed on sets of points in the images. These points were defined by coordinates of voxels with CT values higher than several threshold values. The M classification model was built using image features and demographic parameters of sex and smoking status. The SVM models were evaluated by determining their classification accuracies. The feasibility of the 3DBN model was compared with those of conventional radiomic models based on pseudo-3D BN (p3DBN), two-dimensional BN (2DBN), and CT and wavelet-decomposition (WD) images. The validation of the model was repeated with 100 times random sampling.

RESULTS

The mean test accuracies for M classification with 3DBN, p3DBN, 2DBN, CT, and WD images were 0.810, 0.733, 0.838, 0.782, and 0.799, respectively. The mean test accuracies for S classification with 3DBN, p3DBN, 2DBN, CT, and WD images were 0.773, 0.694, 0.657, 0.581, and 0.696, respectively.

CONCLUSION

3DBN features, which showed a radiogenomic association with the characteristics of the EGFR Del19/L858R mutation subtypes, yielded higher accuracy for subtype classifications in comparison with conventional features.

Topological early warning signals: Quantifying varying routes to extinction in a spatially distributed population model

Understanding and predicting critical transitions in spatially explicit ecological systems is particularly challenging due to their complex spatial and temporal dynamics and high dimensionality. Here, we explore changes in population distribution patterns during a critical transition (an extinction event) using computational topology. Computational topology allows us to quantify certain features of a population distribution pattern, such as the level of fragmentation. We create population distribution patterns via a simple coupled patch model with Ricker map growth and nearest neighbors dispersal on a two dimensional lattice. We observe two dominant paths to extinction within the explored parameter space that depend critically on the dispersal rate d and the rate of parameter drift, Δϵ. These paths to extinction are easily topologically distinguishable, so categorization can be automated. We use this population model as a theoretical proof-of-concept for the methodology, and argue that computational topology is a powerful tool for analyzing dynamical changes in systems with noisy data that are coarsely resolved in space and/or time. In addition, computational topology can provide early warning signals for chaotic dynamical systems where traditional statistical early warning signals would fail. For these reasons, we envision this work as a helpful addition to the critical transitions prediction toolbox.

Segmentation of biomedical images based on a computational topology framework

The homology groups of a topological space provide us with information about its connectivity and the number and type of holes in it. This type of information can find practical applications in describing the intrinsic structure of an image, as well as in identifying equivalence classes in collections of images. When computing homological characteristics, the existence and strength of the relationships between each pair of points in the topological space are studied. The practical use of this approach begins by building a topological space from the image, in which the computation of the homology groups can be carried out in a feasible time. Once the homological properties are obtained, what follows is the task of translating such information into operations such as image segmentation. This work presents a technique for denoising persistent diagrams and reconstructing the shape of segmented objects using the remaining classes on the diagram. A case study for the segmentation of cell nuclei in histological images is used for demonstration purposes. With this approach: a) topological denoising is achieved by aggregating trivial classes on the persistence diagram, and b) a growing seed algorithm uses the information obtained during the construction of the persistence diagram for the reconstruction of the segmented cell structures.

A topological analysis of targeted In-111 uptake in SPECT images of murine tumors

Single-photon emission computed tomography images of murine tumors are interpreted as the values of functions on a three-dimensional domain. Motivated by Morse theory, the local maxima of the tumor image functions are analyzed. This analysis captures tumor heterogeneity that cannot be identified with standard measures. Utilizing decreasing sequences of uptake values to filter the images, a modified form of the standard persistence diagrams for 0-dimensional persistent homology as well as novel childhood diagrams are constructed. Applying statistical methods to time series of persistence and childhood diagrams detects heterogeneous uptake of radioactive antibody within tumors over time and distinguishes uptake in two groups of mice injected with different labeled antibodies.

Robust detection and segmentation of cell nuclei in biomedical images based on a computational topology framework

The segmentation of cell nuclei is an important step towards the automated analysis of histological images. The presence of a large number of nuclei in whole-slide images necessitates methods that are computationally tractable in addition to being effective. In this work, a method is developed for the robust segmentation of cell nuclei in histological images based on the principles of persistent homology. More specifically, an abstract simplicial homology approach for image segmentation is established. Essentially, the approach deals with the persistence of disconnected sets in the image, thus identifying salient regions that express patterns of persistence. By introducing an image representation based on topological features, the task of segmentation is less dependent on variations of color or texture. This results in a novel approach that generalizes well and provides stable performance. The method conceptualizes regions of interest (cell nuclei) pertinent to their topological features in a successful manner. The time cost of the proposed approach is lower-bounded by an almost linear behavior and upper-bounded by O(n²) in a worst-case scenario. Time complexity matches a quasilinear behavior which is O(n^1+ɛ) for ε < 1. Images acquired from histological sections of liver tissue are used as a case study to demonstrate the effectiveness of the approach. The histological landscape consists of hepatocytes and non-parenchymal cells. The accuracy of the proposed methodology is verified against an automated workflow created by the output of a conventional filter bank (validated by experts) and the supervised training of a random forest classifier. The results are obtained on a per-object basis. The proposed workflow successfully detected both hepatocyte and non-parenchymal cell nuclei with an accuracy of 84.6%, and hepatocyte cell nuclei only with an accuracy of 86.2%. A public histological dataset with supplied ground-truth data is also used for evaluating the performance of the proposed approach (accuracy: 94.5%). Further validations are carried out with a publicly available dataset and ground-truth data from the Gland Segmentation in Colon Histology Images Challenge (GlaS) contest. The proposed method is useful for obtaining unsupervised robust initial segmentations that can be further integrated in image/data processing and management pipelines. The development of a fully automated system supporting a human expert provides tangible benefits in the context of clinical decision-making.

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f : K → R n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L ∞ distance r from f for a given r > 0 . The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1 . Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2 n - 2 , our approximation of the ( dim K - n ) th well group is exact. For the second part, we find examples of maps f , f ' : K → R n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.

Object-oriented Persistent Homology

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data classification and analysis. Indeed, persistent homology has rarely been employed for quantitative modeling and prediction. Additionally, the present persistent homology is a passive tool, rather than a proactive technique, for classification and analysis. In this work, we outline a general protocol to construct object-oriented persistent homology methods. By means of differential geometry theory of surfaces, we construct an objective functional, namely, a surface free energy defined on the data of interest. The minimization of the objective functional leads to a Laplace-Beltrami operator which generates a multiscale representation of the initial data and offers an objective oriented filtration process. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. The cubical complex based homology algorithm is employed in the present work to be compatible with the Cartesian representation of the Laplace-Beltrami flow. The proposed Laplace-Beltrami flow based persistent homology method is extensively validated. The consistence between Laplace-Beltrami flow based filtration and Euclidean distance based filtration is confirmed on the Vietoris-Rips complex for a large amount of numerical tests. The convergence and reliability of the present Laplace-Beltrami flow based cubical complex filtration approach are analyzed over various spatial and temporal mesh sizes. The Laplace-Beltrami flow based persistent homology approach is utilized to study the intrinsic topology of proteins and fullerene molecules. Based on a quantitative model which correlates the topological persistence of fullerene central cavity with the total curvature energy of the fullerene structure, the proposed method is used for the prediction of fullerene isomer stability. The efficiency and robustness of the present method are verified by more than 500 fullerene molecules. It is shown that the proposed persistent homology based quantitative model offers good predictions of total curvature energies for ten types of fullerene isomers. The present work offers the first example to design object-oriented persistent homology to enhance or preserve desirable features in the original data during the filtration process and then automatically detect or extract the corresponding topological traits from the data.