Estimating Multidimensional Persistent Homology through a Finite Sampling

An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of $X$ from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it

Real-time monitoring of apples (Malus domestica var. Gala) during hot-air drying using NIR spectroscopy

Among commercial fruits, apple shows a growing trend to its worldwide consumption, where dried apple plays a major part in food industry as raw material to produce snacks, integral breakfast foods, chips, etc., which have become popular in the diet of modern consumers in parallel with the human consumption of organic products. Despite apple tissue exhibits extensive and non-homogeneous discoloration during drying, it is nowadays often dried by conventional methods which, however, are usually uncontrolled and then prone to product quality deterioration. However, because no all conventional drying treatments are allowed by the European Organic Regulation (i.e. EC No. 834/2007 and EC No. 889/2008), drying of organic apples should be carefully optimized to obtain comparable results to conventional methods. Therefore, the main objective of the proposed study was to investigate the feasibility of near-infrared (NIR) spectroscopy as smart drying technology to proactively and non-destructively detect and monitor quality change in organic apple wedges during hot-air drying

Beyond topological persistence: Starting from networks

Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to significant data types as simple graphs and quivers. We focus on categorical persistence functions that allow us to study in full generality strong kinds of connectedness such as clique communities, $k$-vertex and $k$-edge connectedness directly on simple graphs and monic coherent categories.Comment: arXiv admin note: text overlap with arXiv:1707.0967

Size Functions for the Morphological Analysis of Melanocytic Lesions

Size Functions and Support Vector Machines are used to implement a new automatic classifier of melanocytic lesions. This is mainly based on a qualitative assessment of asymmetry, performed by halving images by several lines through the center of mass, and comparing the two halves in terms of color, mass distribution, and boundary. The program is used, at clinical level, with two thresholds, so that comparison of the two outputs produces a report of low-middle-high risk. Experimental results on 977 images, with cross-validation, are reported

Multidimensional persistent homology is stable

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.Comment: 14 pages, 3 figure

Estimating multidimensional persistent homology through a finite sampling

An exact computation of the persistent Betti numbers of a submanifold X of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of X is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of X from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it

Local Triangle Choice for Impact Computation in the Tactile Exploration of a Virtual Surface

Evaluating the intersection of the trajectory of the exploring finger, with the virtual surface representing the scene, is a key problem in the VIDET project of an aid for the visually impaired. A substitute for Delaunay triangulation, which permits of local computation for that goal, is proposed

Steady and ranging sets in graph persistence

Generalised persistence functions (gp-functions) are defined on $(\mathbb{R}, \le)$-indexed diagrams in a given category. A sufficient condition for stability is also introduced. In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature. The example of steady and ranging hubs is studied in depth; their meaning is investigated in three concrete networks