Towards Emotion Recognition: A Persistent Entropy Application

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 (calm, happy, sad, angry, fearful, disgust and surprised)

Topological evaluation of volume reconstructions by voxel carving

Space or voxel carving [1, 4, 10, 15] is a technique for creating a three-dimensional reconstruction of an object from a series of two-dimensional images captured from cameras placed around the object at different viewing angles. However, little work has been done to date on evaluating the quality of space carving results. This paper extends the work reported in [8], where application of persistent homology was initially proposed as a tool for providing a topological analysis of the carving process along the sequence of 3D reconstructions with increasing number of cameras. We give now a more extensive treatment by: (1) developing the formal framework by which persistent homology can be applied in this context; (2) computing persistent homology of the 3D reconstructions of 66 new frames, including different poses, resolutions and camera orders; (3) studying what information about stability, topological correctness and influence of the camera orders in the carving performance can be drawn from the computed barcodes

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Study of the distribution of Malassezia species in patients with pityriasis versicolor and healthy individuals in Tehran, Iran

BACKGROUND: Pityriasis versicolor is a superficial infection of the stratum corneum which caused by a group of yeasts formerly named pityrosporium. The taxonomy of these lipophilic yeasts has recently been modified and includes seven species referred as Malassezia. The aim of this study is to compare the distribution of Malassezia species isolated from pityriasis versicolor lesions and those isolated from healthy skins. METHODS: Differentiation of all malassezia species performed using morphological features and physiological test including catalase reaction, Tween assimilation test and splitting of esculin. RESULTS: In pityriasis versicolor lesions, the most frequently isolated species was M. globosa (53.3%), followed by M. furfur (25.3%), M. sympodialis(9.3%), M. obtusa (8.1%) and M. slooffiae (4.0%). The most frequently isolated species in the skin of healthy individuals were M. globosa, M. sympodialis, M. furfur, M. sloofiae and M. restricta which respectively made up 41.7%, 25.0%, 23.3%, 6.7% and 3.3% of the isolated species. CONCLUSIONS: According to our data, M. globosa was the most prevalent species in the skin of healthy individuals which recovered only in the yeast form. However, the Mycelial form of M. globosa was isolated as the dominant species from pityriasis versicolor lesions. Therefore, the role of predisposing factors in the conversion of this yeast to mycelium and its subsequent involvement in pityriasis versicolor pathogenicity should be considered

Four Distances between Pairs of Amino Acids Provide a Precise Description of their Interaction

The three-dimensional structures of proteins are stabilized by the interactions between amino acid residues. Here we report a method where four distances are calculated between any two side chains to provide an exact spatial definition of their bonds. The data were binned into a four-dimensional grid and compared to a random model, from which the preference for specific four-distances was calculated. A clear relation between the quality of the experimental data and the tightness of the distance distribution was observed, with crystal structure data providing far tighter distance distributions than NMR data. Since the four-distance data have higher information content than classical bond descriptions, we were able to identify many unique inter-residue features not found previously in proteins. For example, we found that the side chains of Arg, Glu, Val and Leu are not symmetrical in respect to the interactions of their head groups. The described method may be developed into a function, which computationally models accurately protein structures