Multivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures

Rieck, Bastian; Mara, Hubert; Leitte, Heike

doi:10.1109/tvcg.2012.248

Cited by 49 publications

(46 citation statements)

References 23 publications

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“…Persistent Homology [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55] is a means of topological data analysis. Now let us use an example to show how the topological data analysis methods can overcome the limitations of geometrical methods.…”

Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

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Atom probe tomography (APT) represents a revolutionary characterization tool for materials that combine atomic imaging with a time-of-flight (TOF) mass spectrometer to provide direct space three-dimensional, atomic scale resolution images of materials with the chemical identities of hundreds of millions of atoms. It involves the controlled removal of atoms from a specimen's surface by field evaporation and then sequentially analyzing them with a position sensitive detector and TOF mass spectrometer. A paradox in APT is that while on the one hand, it provides an unprecedented level of imaging resolution in three dimensions, it is very difficult to obtain an accurate perspective of morphology or shape outlined by atoms of similar chemistry and microstructure. The origins of this problem are numerous, including incomplete detection of atoms and the complexity of the evaporation fields of atoms at or near interfaces. Hence, unlike scattering techniques such as electron microscopy, interfaces appear diffused, not sharp. This, in turn, makes it challenging to visualize and quantitatively interpret the microstructure at the "meso" scale, where one is interested in the shape and form of the interfaces and their associated chemical gradients. It is here that the application of informatics at the nanoscale and statistical learning methods plays a critical role in both defining the level of uncertainty and helping to make quantitative, statistically objective interpretations where heuristics often dominate. In this chapter, we show how the tools of Topological Data Analysis provide a new and powerful tool in the field of nanoinformatics for materials characterization.

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Section: Persistent Homologymentioning

confidence: 99%

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

2018

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“…Robustness against noise is improved by simplifying the Morse- Smale complex using the concept of persistence [8]. Rieck et al [30] developed persistence-based filters and topological signatures for the multivariate analysis of data and applied the techniques to the analysis of cultural heritage artifacts. While the method performs very well on artificial data, the processing of 3D-scans requires manual segmentation of the digital object and adjustment of parameters (see Sections 6.2.1 and 6.2.2 in [30]).…”

Section: Related Workmentioning

confidence: 99%

“…Rieck et al [30] developed persistence-based filters and topological signatures for the multivariate analysis of data and applied the techniques to the analysis of cultural heritage artifacts. While the method performs very well on artificial data, the processing of 3D-scans requires manual segmentation of the digital object and adjustment of parameters (see Sections 6.2.1 and 6.2.2 in [30]). A technique to illustrate surface features and enhance the spatial impression with lines was presented by Lawonn et al [21].…”

Section: Related Workmentioning

confidence: 99%

Visualization and Extraction of Carvings for Heritage Conservation

Lawonn

Trostmann

Preim

et al. 2017

IEEE Trans. Visual. Comput. Graphics

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Abstract-We present novel techniques for visualizing, illustrating, analyzing, and generating carvings in surfaces. In particular, we consider the carvings in the plaster of the cloister of the Magdeburg cathedral, which dates to the 13th century. Due to aging and weathering, the carvings have flattened. Historians and restorers are highly interested in using digitalization techniques to analyze carvings in historic artifacts and monuments and to get impressions and illustrations of their original shape and appearance. Moreover, museums and churches are interested in such illustrations for presenting them to visitors. The techniques that we propose allow for detecting, selecting, and visualizing carving structures. In addition, we introduce an example-based method for generating carvings. The resulting tool, which integrates all techniques, was evaluated by three experienced restorers to assess the usefulness and applicability. Furthermore, we compared our approach with exaggerated shading and other state-of-the-art methods.

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“…7, 22, 23, 26, 50 Some of these persistent homology algorithms have been implemented in many software packages, namely Perseus, 50, 52 JavaPlex 71 and Dionysus. In the past few years, persistent homology has been applied to image analysis, 5, 9, 58, 67 image retrieval, 30 chaotic dynamics verification, 42, 49 sensor networks, 66 complex networks, 40, 45 data analysis, 8, 47, 53, 60, 73 computer vision, 67 shape recognition 24 and computational biology. 21, 31, 43, 86 …”

Section: Introductionmentioning

confidence: 99%

Object-oriented persistent homology

Wang

Wei

2016

Journal of Computational Physics

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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.

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Multivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures

Cited by 49 publications

References 23 publications

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

Topological Data Analysis for the Characterization of Atomic Scale Morphology from Atom Probe Tomography Images

Visualization and Extraction of Carvings for Heritage Conservation

Object-oriented persistent homology

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