Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.
Emotion recognition and classification is a very active area of research. In this paper, we present a first approach to emotion classification using persistent entropy and support vector machines. A topology-based model is applied to obtain a single real number from each raw signal. These data are used as input of a support vector machine to classify signals into 8 different emotions (neutral, calm, happy, sad, angry, fearful, disgust and surprised).
In this paper, we apply persistent entropy, a novel topological statistic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geometric information encoded by α -complexes and persistent homology. After using some statistical tests, we can guarantee the existence of significant differences in the studied tissues.
In A naturalistic open source movie for optical flow evaluation, Butler et al. create a database of ground-truth optical flow from the computer-generated video Sintel. We study the high-contrast 3× 3 patches from this video, and provide evidence that this dataset is well-modeled by a torus (a nonlinear 2-dimensional manifold). Our main tools are persistent homology and zigzag persistence, which are popular techniques from the field of computational topology. We show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, which is furthermore related to the statistics of range images.
The Jacobi set is a useful descriptor of mutual behavior of functions defined on a common domain. We introduce the piecewise linear Jacobi set for general vector fields on simplicial complexes. This definition generalizes the definition of the Jacobi set for gradients of functions introduced by Edelsbrunner and Harer.
Edge detection algorithms usually produce a discrete set of edgels (edge pixels) in a given image on a fixed pixel grid. We consider the harder problem of detecting continuous straight line segments at sub-pixel resolution. The state-of-the art Line Segment Detection Algorithm (LSDA) outputs unordered line segments whose total number cannot be easily controlled. Another motivation to improve the LSDA is to avoid intersections and small angles between line segments, hence difficulties in higher level tasks such as segmentation or contour extraction. The new Persistent Line Segment Detector (PLSD) outputs only non-intersecting line segments and ranks them by a strength, hence the user can choose a number of segments. The main novelty is an automatic selection of strongest segments along any straight line by using the persistence from Topological Data Analysis. The experiments on the Berkeley Segmentation Database of 500 real-life images show that the new algorithm outperforms the LSDA on the important measure of Boundary Recall.
We extend the notion of a gap from the square to the triangular grid, and we propose a possible classification of gaps in this grid. We give four definitions of well-composed objects in the triangular grid by translating the existing definitions of such objects in the square grid. We show that these definitions in the triangular grid are equivalent, as they are in the square grid. We give a formula relating the number of gaps of different types in an object in this grid with the number of boundary cells in the object, as well as three short intuitive proofs of this formula.
In this paper we provide a new characterization of cell decomposition (called slope complex) of a given 2-dimensional continuous surface. Each patch (cell) in the decomposition must satisfy that there exists a monotonic path for any two points in the cell. We prove that any triangulation of such surface is a slope complex and explain how to obtain new slope complexes with a smaller number of slope regions decomposing the surface. We give the minimal number of slope regions by counting certain bounding edges of a triangulation of the surface obtained from its critical points.
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.
Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of k-dimensional holes and Betti numbers. The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this study, we propose a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represent the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. We also find the most contributed HHs to the network difference based on the HH dissimilarity. We applied our proposed method to clustering the networks of 4 groups, normal controls (NC), stable and progressive mild cognitive impairment (sMCI and pMCI), and Alzheimer’s disease (AD). The results showed that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology.
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.
Inferring underlying manifold of data is one of the important issues for point cloud data analysis. This is accomplished by inferring the topological shape of the underlying manifold. This is done by estimating the number of holes in the underlying manifold in each dimension. Persistent homology is one of the means of estimating the number of holes in the underlying manifold. Calculating the persistent homology of data determines the size, number, and dimensions of holes produced from data points. However, the number of holes represented through persistent homology is far greater than that in underlying manifold. This problem is caused by noises in a result of calculating persistent homology. Therefore, reducing noises that result from calculating persistent homology is necessary to estimate the number of holes in the underlying manifold. Conventional methods cannot reduce noises adequately when data are of low density and thus cannot estimate the number of holes in the underlying manifold without manual analysis by experts. In this study, we propose a new method to estimate automatically the number of holes in the underlying manifolds. We also compare the proposed and conventional methods and show the effectiveness of the former.
In this paper, the concept of homogeneity is defined, from a topological perspective, in order to analyze how uniform is the material composition in 2D electron microscopy images. Topological multiresolution parameters are taken into account to obtain better results than classical techniques.
We propose an algorithm for persistence homology computation of orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Having as an input a real function h on the vertices of the mesh, we first compute persistent homology of filtrations obtained by adding cells incident to each vertex of the mesh, The cells to add are controlled by both the function h and a parameter δ . The parameter δ is used to control the number of cells added to each level of the filtration. Bigger δ produces less levels in the filtration and consequently more cells in each level. We then simplify each level (cluster) by merging faces of the same cluster. Our experiments demonstrate that our method allows fast computation of persistent homology of big meshes and it is persistent-homology aware in the sense that persistent homology does not change in the simplification process when fixing δ .
Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings between the associated persistent diagrams. In our paper we prove that the coherent 2D matching distance is well-defined and stable.
In this paper, a homotopy-based method is employed for the recovery of speech recordings from missing or corrupted samples taken in a noisy environment. The model for the acquisition device is a compressed sensing scenario using Gabor frames. To recover an approximation of the speech file, we used the basis pursuit denoising method with the homotopy continuation algorithm. We tested the proposed method with various speech recordings.
Persistent topology mitigates the excessive freedom of topological equivalence by studying not just a topological space but a filtration of it. This makes it a very effective class of shape descriptors, with an impressive potential for applications in the image context, in particular when it comes to images of natural origin. Research in this field is lively and follows various threads. The talk will sample some recent results without any attempt to completeness.
To prevent the release of large quantities of CO2 into the atmosphere, carbon capture and storage (CCS) represents a potential means of mitigating the contribution of fossil fuel emissions to global warming and ocean acidification. Fluvial saline aquifers are favourite targeted reservoirs for CO2 storage. These reservoirs are very heterogeneous but their heterogeneities were rarely integrated into CO2 reservoir models. Moreover, contrary to petroleum reservoirs, the available dataset is very limited and not supposed to be enriched. This leads to wide uncertainties on reservoir characteristics required for CSS management (injection location, CO2 plume migration, etc.). Stochastic simulations are classical strategies in such under-constrained context. They aim at generating a wide number of models that all fit the available dataset. The generated models serve as support for computing the required reservoir characteristics and their uncertainties. A challenge is to optimize the uncertainty computations by selecting stochastic models that should have a priori very different flow behaviours. Fluid flows depend on the connectivity of reservoir rocks (channel deposits). In this paper, it is proposed to study the variability of the Betti numbers in function of different fluvial architectures. The aim is to quantify the impact of fluvial heterogeneities and their spatial distribution on reservoir rock topology and then on CO2 storage capacities. Representative models of different scenarios of channel stacking and their internal heterogeneities are generated using geostatistical simulation approaches. The Betti numbers are computed on each generated models and statistically analysed to exhibit if fluvial architecture controls reservoir topology.
In this paper, we propose a method to compute, in parallel, the homology groups of closed meshes (i.e., orientable 2D manifolds without boundary) represented by combinatorial maps. Our experiments illustrate the interest of our approach which is really fast on big meshes and which obtains good speed-up when increasing the number of threads.
This paper discusses the connection between the texture operator LBP (local binary pattern) and an application of LBPs to persistent homology. A shape representation - the LBP scale space - is defined as a filtration based on the variation of an LBP parameter. A relation between the LBP scale space and a variation of thresholds used in the segmentation of a graylevel image is discussed. Using the LBP scale space a characterization of (parts of) shapes is demonstrated based on simple shape primitives, the observations may also be generalized for smooth curves. The LBP scale space is augmented by associating it with polar coordinates (with the origin located at the LBP center). In this way a procedure of shape reconstruction based on the LBP scale space is defined and its reconstruction accuracy is demonstrated in an experiment. Furthermore, this augmented LBP scale space representation is invariant to translation and rotation of the shape.
