This paper presents a new approach for the visualization and analysis of the spatial variability of features of interest represented by critical points in ensemble data. Our framework, called Persistence Atlas, enables the visualization of the dominant spatial patterns of critical points, along with statistics regarding their occurrence in the ensemble. The persistence atlas represents in the geometrical domain each dominant pattern in the form of a confidence map for the appearance of critical points. As a by-product, our method also provides 2-dimensional layouts of the entire ensemble, highlighting the main trends at a global level. Our approach is based on the new notion of Persistence Map, a measure of the geometrical density in critical points which leverages the robustness to noise of topological persistence to better emphasize salient features. We show how to leverage spectral embedding to represent the ensemble members as points in a low-dimensional Euclidean space, where distances between points measure the dissimilarities between critical point layouts and where statistical tasks, such as clustering, can be easily carried out. Further, we show how the notion of mandatory critical point can be leveraged to evaluate for each cluster confidence regions for the appearance of critical points. Most of the steps of this framework can be trivially parallelized and we show how to efficiently implement them. Extensive experiments demonstrate the relevance of our approach. The accuracy of the confidence regions provided by the persistence atlas is quantitatively evaluated and compared to a baseline strategy using an off-the-shelf clustering approach. We illustrate the importance of the persistence atlas in a variety of real-life datasets, where clear trends in feature layouts are identified and analyzed.
In contrast, our approach based on persistence maps leverages the soundness of topological persistence @cite_86 while specifically focusing on the geometrical spatial layout of the critical points.
@cite_86
2056761334 2788076228 2963265143 2123207641
Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable spaces. In this paper, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence. Along the way, we extend the definition of persistence diagram to a larger setting, introduce the notions of discretization of a persistence module and associated pixelization map, define a proximity measure between persistence modules, and show how to interpolate between persistence modules, thereby lending a more analytic character to this otherwise algebraic setting. We believe these new theoretical concepts and tools shed new light on the theory of persistence, in addition to simplifying proofs and enabling new applications. Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane @math that can equivalently be seen as discrete measures in @math . When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the C ech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on @math , has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams & al., Persistence images: a stable vector representation of persistent homology] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density. Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a statistical technique, the empirical bootstrap, to separate topological signal from topological noise. In particular, we derive condence sets for persistence diagrams and condence bands for persistence landscapes. We combine topological and geometric methods to construct a multiresolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by cancelling pairs of critical points. Based on a simple notion of dependency among these cancellations, we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometry-based representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at runtime.