Separating Topological Noise from Features Using Persistent Entropy

Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.Ministerio de Economía y Competitividad MTM2015-67072-

On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has been recently defined and successfully applied to different scenarios: characterization of the idiotypic immune network, detection of the transition between the preictal and ictal states in EEG signals, or the classification problem of real long-length noisy signals of DC electrical motors, to name a few. In this paper, we study properties of persistent entropy and prove its stability under small perturbations in the given input data. From this concept, we define three summary functions and show how to use them to detect patterns and topological features

Characterising epithelial tissues using persistent entropy

In this paper, we apply persistent entropy, a novel topological statis- tic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geomet- ric information encoded by -complexes and persistent homology. After using some statistical tests, we can guarantee the existence of signi cant di erences in the studied tissues.Ministerio de Economía y Competitividad MTM2015-67072-

Persistent entropy: a scale-invariant topological statistic for analyzing cell arrangements

In this work, we develop a method for detecting differences in the topological distribution of cells forming epithelial tissues. In particular, we extract topological information from their images using persistent homology and a summary statistic called persistent entropy. This method is scale invariant, robust to noise and sensitive to global topological features of the tissue. We have found significant differences between chick neuroepithelium and epithelium of Drosophila wing discs in both, larva and prepupal stages. Besides, we have tested our method, with good results, with images of mathematical tesselations that model biological tissues

Stable topological summaries for analyzing the organization of cells in a packed tissue

We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$

Barbed Dental Ti6Al4V Alloy Screw : Design and Bench Testing

Background context. Dental implants are designed to replace a missing tooth. Implant stability is vital to achieving osseointegration and successful implantation. Although there are many implants available on the market, there is room for improvement. Purpose. We describe a new dental implant with improved primary stability features. Study design. Lab bench test studies. Methods. We evaluated the new implant using static and flexion-compression fatigue tests with compression loads, 35 Ncm tightening torque, displacement control, 0.01 mm/s actuator movement speed, and 9-10 Hz load application frequency, obtaining a cyclic load diagram. We applied variable cyclic loadings of predetermined amplitude and recorded the number of cycles until failure. The test ended with implant failure (breakage or permanent deformation) or reaching five million cycles for each load. Results. Mean stiffness was 1151.13 ± 133.62 SD N/mm, mean elastic limit force 463.94 ± 75.03 SD N, and displacement 0.52 ± 0.04 SD mm, at failure force 663.21 ± 54.23 SD N and displacement 1.56 ± 0.18 SD mm, fatigue load limit 132.6 ± 10.4 N, and maximum bending moment 729.3 ± 69.43 mm/N. Conclusions. The implant fatigue limit is satisfactory for incisor and canine teeth and between the values for premolars and molars for healthy patients. The system exceeds five million cycles when subjected to a 132.60 N load, ensuring long-lasting life against loads below the fatigue limit

K-Factores en nubes bicromáticas

Consideramos una colección de puntos bicromática y nos preguntamos cuántos puntos adicionales son necesarios considerar para asegurar la existencia de un k {factor. Dos tipos de puntos adicionales serán tratados: puntos de Steiner y puntos blancos (con posición prefijada pero no así su color

Bionate Lumbar Disc Nucleus Prosthesis : Biomechanical Studies in Cadaveric Human Spines

Design: cadaveric spine nucleus replacement study. Objective: determining Bionate 80A nucleus replacement biomechanics in cadaveric spines. Methods: in cold preserved spines, with ligaments and discs intact, and no muscles, L3-L4, L4-L5, and L5-S1 nucleus implantation was done. Differences between customized and overdimensioned implants were compared. Flexion, extension, lateral bending, and torsion were measured in the intact spine, nucleotomy, and nucleus implantation specimens. Increasing load or bending moment was applied four times at 2, 4, 6, and 8 Nm, twice in increasing mode and twice in decreasing mode. Spine motion was recorded using stereophotogrammetry. Expulsion tests: cyclic compression of 50-550 N for 50,000 cycles, increasing the load until there was extreme flexion, implant extrusion, or anatomical structure collapse. Subsidence tests were done by increasing the compression to 6000 N load. Results: nucleotomy increased the disc mobility, which remained unchanged for the adjacent upper level but increased for the lower adjacent one, particularly in lateral bending and torsion. Nucleus implantation, compared to nucleotomy, reduced disc mobility except in flexion-extension and torsion, but intact mobility was no longer recovered, with no effect on upper or lower adjacent segments. The overdimensioned implant, compared to the customized implant, provided equal or sometimes higher mobility. Lamina, facet joint, and annulus removal during nucleotomy caused more damaged than that restored by nucleus implantation. No implant extrusion was observed under compression loads of 925-1068 N as anatomical structures collapsed before. No subsidence or vertebral body fractures were observed under compression loads of 6697.8-6812.3 N. Conclusions: nucleotomized disc and L1-S1 mobility increased moderately after cadaveric spine nucleus implantation compared to the intact status, partly due to operative anatomical damage. Our implant had shallow expulsion and subsidence risks