Gait-based carried object detection using persistent homology

There are surveillance scenarios where it is important to emit an alarm when a person carrying an object is detected. In order to detect when a person is carrying an object, we build models of naturally-walking and object-carrying persons using topological features. First, a stack of human silhouettes, extracted by background subtraction and thresholding, are glued through their gravity centers, forming a 3D digital image I. Second, different filters (i.e. orderings of the cells) are applied on ∂ K(I) (cubical complex obtained from I) which capture relations among the parts of the human body when walking. Finally, a topological signature is extracted from the persistence diagrams according to each filter. We build some clusters of persons walking naturally, without carrying object and some clusters of persons carrying bags. We obtain vector prototypes for each cluster. Simple distances to the means are calculated for detecting the presence of carrying object. The measure cosine is used to give a similarity value between topological signatures. The accuracies obtained are 95.7% and 95.9% for naturally-walking and object-carrying respectively

Finite element model for predicting the shear behavior of FRP-strengthened RC members

The shear behavior of FRP strengthened reinforced concrete (FRP strengthened RC) membrane elements can be predicted by developing logical models that satisfy the principles of mechanics of materials namely stress equilibrium, strain compatibility, and constitutive relationships of concrete, steel and, FRP reinforcements. The Softened Membrane Model (SMM), which was developed for predicting the shear behavior of reinforced concrete (RC) membrane elements, is extended to FRP strengthened RC members subjected to shear. This new analytical model, referred to as the Softened Membrane Model for FRP strengthened RC members (SMM-FRP), considers new constitutive laws for each material component of the member. Similar to the case of the SMM model for RC, this new SMM-FRP model can predict the entire stress-strain curve, including pre- and post-cracking, and the ascending and descending branches. The SMM-FRP is implemented into an OpenSees-based finite element program for a membrane 2-D element that will allow structural engineers to predict the monotonic responses of FRP strengthened RC members subjected to shear. The developed program is validated in this paper by the prediction of the monotonic responses of 10 FRP strengthened RC panels subjected to pure shear stresses. The good agreement between the experimental and analytical results confirms the validity of the analytical model in predicting the shear behavior of RC members strengthened with FRP sheets

Behavior of FRP-strengthened RC elements subjected to pure shear

The shear behavior of fiber reinforced polymer strengthened reinforced concrete (FRP-strengthened RC) has been the focus of extensive research studies. However, the mechanism of this complex phenomenon has not been fully clarified. Recent analytical models which were developed for predicting the shear capacity of FRP strengthened RC girders were based on test results of simply supported beam specimens with various shear span-to-depth (a/d) ratios. In such tests no region of the specimen is subjected to uniform stress conditions, Therefore, the results of such tests cannot predict the true pure shear behavior due to non-uniformity of stresses, the presence of flexural and other non-shear related effects such as a/d ratio that cannot be filtered out. Therefore, proper design of shear strengthening using FRP requires testing of elements that are subjected to pure shear case primary before adding other governing effects. This allows a careful investigation and full understanding of the behavior at the element level. In order to accomplish this task, panel testing of representative RC specimens strengthened with FRP sheets were needed. This paper reports the testing of 10 FRP strengthened RC panels subjected to pure shear stress field. The tests were carried out to evaluate the effects of three variables: FRP stiffness, FRP wrapping scheme, and transverse steel reinforcement ratio. The test results showed that these three variables greatly affected the shear behavior due to various types of failure modes associated with FRP strengthening. In addition, it was observed that the magnitude of increased shear capacity associated with the application of FRP sheets depends not only upon the stiffness of FRP, but also on the stiffness of internal shear reinforcement. With the increase of internal steel shear reinforcement, the effectiveness of shear gain due to externally bonded FRP decreases

Topological features for monitoring human activities at distance

In this paper, a topological approach for monitoring human activities is presented. This approach makes possible to protect the person’s privacy hiding details that are not essential for processing a security alarm. First, a stack of human silhouettes, extracted by background subtraction and thresholding, are glued through their gravity centers, forming a 3D digital binary image I. Secondly, different orders of the simplices are applied on a simplicial complex obtained from I, which capture relations among the parts of the human body when walking. Finally, a topological signature is extracted from the persistence diagrams according to each order. The measure cosine is used to give a similarity value between topological signatures. In this way, the powerful topological tool known as persistent homology is novelty adapted to deal with gender classification, person identification, carrying bag detection and simple action recognition. Four experiments show the strength of the topological feature used; three of they use the CASIA-B database, and the fourth use the KTH database to present the results in the case of simple actions recognition. In the first experiment the named topological signature is evaluated, obtaining 98.8% (lateral view) of correct classification rates for gender identification. In the second one are shown results for person identification, obtaining an average of 98.5%. In the third one the result obtained is 93.8% for carrying bag detection. And in the last experiment the results were 97.7% walking and 97.5% running, which were the actions took from the KTH database

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

Clear and Compress: Computing Persistent Homology in Chunks

We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by G\"unther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we improve the performance through parallelized computation.Comment: This result was presented at TopoInVis 2013 (http://www.sci.utah.edu/topoinvis13.html

Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational Geometr

Topological characteristics of oil and gas reservoirs and their applications

We demonstrate applications of topological characteristics of oil and gas reservoirs considered as three-dimensional bodies to geological modeling.Comment: 12 page

Mind the Gap: A Study in Global Development through Persistent Homology

The Gapminder project set out to use statistics to dispel simplistic notions about global development. In the same spirit, we use persistent homology, a technique from computational algebraic topology, to explore the relationship between country development and geography. For each country, four indicators, gross domestic product per capita; average life expectancy; infant mortality; and gross national income per capita, were used to quantify the development. Two analyses were performed. The first considers clusters of the countries based on these indicators, and the second uncovers cycles in the data when combined with geographic border structure. Our analysis is a multi-scale approach that reveals similarities and connections among countries at a variety of levels. We discover localized development patterns that are invisible in standard statistical methods

Characterising epithelial tissues using persistent entropy

In this paper, we apply persistent entropy, a novel topological statistic, for characterization of images of epithelial tissues. We have found out that persistent entropy is able to summarize topological and geometric information encoded by \alpha-complexes and persistent homology. After using some statistical tests, we can guarantee the existence of significant differences in the studied tissues.Comment: 12 pages, 7 figures, 4 table