481 research outputs found
Topo-Geometric Filtration Scheme for Geometric Active Contours and Level Sets: Application to Cerebrovascular Segmentation
One of the main problems of the existing methods for the
segmentation of cerebral vasculature is the appearance in the segmentation
result of wrong topological artefacts such as the kissing vessels.
In this paper, a new approach for the detection and correction of such
errors is presented. The proposed technique combines robust topological
information given by Persistent Homology with complementary geometrical
information of the vascular tree. The method was evaluated on 20
images depicting cerebral arteries. Detection and correction success rates
were 81.80% and 68.77%, respectively
Persistent Homology Analysis of Brain Artery Trees.
New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries
Tissue distribution and biochemical properties of an interspecific tumour-associated gamma foetal antigen.
A late-gestation neonatal antigen (gamma foetal antigen; gamma-FA) immunologically and biochemically unrelated to murine alpha-foetoprotein, was identified in several spontaneous and carcinogen-induced sarcomas and hepatic carcinomas of the mouse and rat. An approximate mol. wt of 35,000 for gamma-FA from both foetus and tumour was obtained by molecular-sieve chromatography and sucrose-gradient centrifugation. Radial immunodiffusion analyses of organ extracts indicated that gamma-FA could be found in several neonatal tissues, the highest concentration occurring in the spleen. In the 2-month-old mouse, only splenic tissue contained gamma-FA and at much lower levels than in the organ of the newborn mouse
Probabilistic Fréchet means for time varying persistence diagrams
© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (D<inf>p</inf>, W<inf>p</inf>), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (D<inf>p</inf>)<sup>N</sup>→ℙ(D<inf>p</inf>). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards
Data-Driven Analysis of Pareto Set Topology
When and why can evolutionary multi-objective optimization (EMO) algorithms
cover the entire Pareto set? That is a major concern for EMO researchers and
practitioners. A recent theoretical study revealed that (roughly speaking) if
the Pareto set forms a topological simplex (a curved line, a curved triangle, a
curved tetrahedron, etc.), then decomposition-based EMO algorithms can cover
the entire Pareto set. Usually, we cannot know the true Pareto set and have to
estimate its topology by using the population of EMO algorithms during or after
the runtime. This paper presents a data-driven approach to analyze the topology
of the Pareto set. We give a theory of how to recognize the topology of the
Pareto set from data and implement an algorithm to judge whether the true
Pareto set may form a topological simplex or not. Numerical experiments show
that the proposed method correctly recognizes the topology of high-dimensional
Pareto sets within reasonable population size.Comment: 8 pages, accepted at GECCO'18 as a full pape
Hierarchies and Ranks for Persistence Pairs
We develop a novel hierarchy for zero-dimensional persistence pairs, i.e.,
connected components, which is capable of capturing more fine-grained spatial
relations between persistence pairs. Our work is motivated by a lack of spatial
relationships between features in persistence diagrams, leading to a limited
expressive power. We build upon a recently-introduced hierarchy of pairs in
persistence diagrams that augments the pairing stored in persistence diagrams
with information about which components merge. Our proposed hierarchy captures
differences in branching structure. Moreover, we show how to use our hierarchy
to measure the spatial stability of a pairing and we define a rank function for
persistence pairs and demonstrate different applications.Comment: Topology-based Methods in Visualization 201
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
Persistent Intersection Homology for the Analysis of Discrete Data
Topological data analysis is becoming increasingly relevant to support the
analysis of unstructured data sets. A common assumption in data analysis is
that the data set is a sample---not necessarily a uniform one---of some
high-dimensional manifold. In such cases, persistent homology can be
successfully employed to extract features, remove noise, and compare data sets.
The underlying problems in some application domains, however, turn out to
represent multiple manifolds with different dimensions. Algebraic topology
typically analyzes such problems using intersection homology, an extension of
homology that is capable of handling configurations with singularities. In this
paper, we describe how the persistent variant of intersection homology can be
used to assist data analysis in visualization. We point out potential pitfalls
in approximating data sets with singularities and give strategies for resolving
them.Comment: Topology-based Methods in Visualization 201
Search for Top Squark Pair Production in the Dielectron Channel
This report describes the first search for top squark pair production in the
channel stop_1 stopbar_1 -> b bbar chargino_1 chargino_1 -> ee+jets+MEt using
74.9 +- 8.9 pb^-1 of data collected using the D0 detector. A 95% confidence
level upper limit on sigma*B is presented. The limit is above the theoretical
expectation for sigma*B for this process, but does show the sensitivity of the
current D0 data set to a particular topology for new physics.Comment: Five pages, including three figures, submitted to PRD Brief Report
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