6,999 research outputs found
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
A Review of Atrial Fibrillation Detection Methods as a Service
Atrial Fibrillation (AF) is a common heart arrhythmia that often goes undetected, and even if it is detected, managing the condition may be challenging. In this paper, we review how the RR interval and Electrocardiogram (ECG) signals, incorporated into a monitoring system, can be useful to track AF events. Were such an automated system to be implemented, it could be used to help manage AF and thereby reduce patient morbidity and mortality. The main impetus behind the idea of developing a service is that a greater data volume analyzed can lead to better patient outcomes. Based on the literature review, which we present herein, we introduce the methods that can be used to detect AF efficiently and automatically via the RR interval and ECG signals. A cardiovascular disease monitoring service that incorporates one or multiple of these detection methods could extend event observation to all times, and could therefore become useful to establish any AF occurrence. The development of an automated and efficient method that monitors AF in real time would likely become a key component for meeting public health goals regarding the reduction of fatalities caused by the disease. Yet, at present, significant technological and regulatory obstacles remain, which prevent the development of any proposed system. Establishment of the scientific foundation for monitoring is important to provide effective service to patients and healthcare professionals
An Efficient Algorithm for Video Super-Resolution Based On a Sequential Model
In this work, we propose a novel procedure for video super-resolution, that
is the recovery of a sequence of high-resolution images from its low-resolution
counterpart. Our approach is based on a "sequential" model (i.e., each
high-resolution frame is supposed to be a displaced version of the preceding
one) and considers the use of sparsity-enforcing priors. Both the recovery of
the high-resolution images and the motion fields relating them is tackled. This
leads to a large-dimensional, non-convex and non-smooth problem. We propose an
algorithmic framework to address the latter. Our approach relies on fast
gradient evaluation methods and modern optimization techniques for
non-differentiable/non-convex problems. Unlike some other previous works, we
show that there exists a provably-convergent method with a complexity linear in
the problem dimensions. We assess the proposed optimization method on {several
video benchmarks and emphasize its good performance with respect to the state
of the art.}Comment: 37 pages, SIAM Journal on Imaging Sciences, 201
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