282 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
Obtaining cell complexes associated to four dimensional digital objects
In this paper, we determine a cell complex representation of a 80–adjacent doxelbased 4-dimensional object. The homological information of this polyhedral cell complex can be employed to specify topological features and characteristics of a digital object. This homological information (for example, Euler characteristic, homological classification of cycles, homology generators, relations among them...) of a discrete object can be extracted from some specific boundary operators for each cell of an object (see [3]). The different (up to isometry) polyhedral cells are 400 configurations and their local boundary information can be suitably glued for determining the global boundary of an object and consequently, its corresponding homological information. This fact allows us to implement this technique using a look-up table for the different basic configurations and its corresponding boundary operators
Homological methods in feature extraction of multidimensional images
We show that the newly developed homology algorithms are helpful in imaging problems on the example of an algorithm extracting one dimensional features from a noisy image. We indicate that in some situations the global nature of this algorithm may become advantageous when compared with the standard algorithms based on skeletonization and pruning. The algorithm works in every dimension. ©2009 IEEE.published_or_final_versionThe 2nd International Congress on Image and Signal Processing (CISP'09), Tianjin, China, 17-19 October 2009. In Proceedings of 2nd CISP, 2009, p. 1-
Point Information Gain and Multidimensional Data Analysis
We generalize the Point information gain (PIG) and derived quantities, i.e.
Point information entropy (PIE) and Point information entropy density (PIED),
for the case of R\'enyi entropy and simulate the behavior of PIG for typical
distributions. We also use these methods for the analysis of multidimensional
datasets. We demonstrate the main properties of PIE/PIED spectra for the real
data on the example of several images, and discuss possible further utilization
in other fields of data processing.Comment: 16 pages, 6 figure
Landscapes of data sets and functoriality of persistent homology
The aim of this article is to describe a new perspective on functoriality of
persistent homology and explain its intrinsic symmetry that is often
overlooked. A data set for us is a finite collection of functions, called
measurements, with a finite domain. Such a data set might contain internal
symmetries which are effectively captured by the action of a set of the domain
endomorphisms. Different choices of the set of endomorphisms encode different
symmetries of the data set. We describe various category structures on such
enriched data sets and prove some of their properties such as decompositions
and morphism formations. We also describe a data structure, based on coloured
directed graphs, which is convenient to encode the mentioned enrichment. We
show that persistent homology preserves only some aspects of these landscapes
of enriched data sets however not all. In other words persistent homology is
not a functor on the entire category of enriched data sets. Nevertheless we
show that persistent homology is functorial locally. We use the concept of
equivariant operators to capture some of the information missed by persistent
homology
Integrating topological features to enhance cardiac disease diagnosis from 3D CMR images
Treballs Finals de Grau de MatemĂ tiques, Facultat de MatemĂ tiques, Universitat de Barcelona, Any: 2023, Director: Carles Casacuberta i Polyxeni Gkontra[en] Persistent homology is a technique from the field of algebraic topology for the analysis and characterization of the shape and structure of datasets in multiple dimensions. Its use is based on the identification and quantification of topological patterns in the dataset across various scales. In this thesis, persistent homology is applied with the objective of extracting topological descriptors from three-dimensional cardiovascular magnetic resonance (CMR) imaging. Thereafter, topological descriptors are used for the detection of cardiovascular diseases by means of Machine Learning (ML) techniques.
Radiomics has been one of the recently proposed approaches for disease diagnosis. This method involves the extraction and subsequent analysis of a significant number of quantitative descriptors from medical images. These descriptors offer a characterization of the spatial distribution, texture, and intensity of the structures present in the images.
This study demonstrates that radiomics and topological descriptors achieve comparable results, providing complementary insights into the underlying structures and characteristics of anatomical tissues. Moreover, the combination of these two methods leads to a further improvement of the performance of ML models, thereby
enhancing medical diagnosis
Towards a certified computation of homology groups for digital images
International audienceIn this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on program- ming and executing inside the COQ proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in COQ from real biomedical images
- …