Persistent Homology And Discrete Fourier Transform

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Persistent Homology And Discrete Fourier Transform

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Author : Victoria Callet-Feltz
language : en
Publisher: Springer Nature
Release Date : 2025-07-02

Persistent Homology And Discrete Fourier Transform written by Victoria Callet-Feltz and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-02 with Mathematics categories.

This book proposes contributions to various problems in the field of topological analysis of musical data: the objects studied are scores represented symbolically by MIDI files, and the tools used are the discrete Fourier transform and persistent homology. The manuscript is divided into three parts: the first two are devoted to the study of the aforementioned mathematical objects and the implementation of the model. More precisely, the notion of DFT introduced by Lewin is generalized to the case of dimension two, by making explicit the passage of a musical bar from a piece to a subset of Z/tZ×Z/pZ, which leads naturally to a notion of metric on the set of musical bars by their Fourier coefficients. This construction gives rise to a point cloud, to which the filtered Vietoris-Rips complex is associated, and consequently a family of barcodes given by persistent homology. This approach also makes it possible to generalize classical results such as Lewin's lemma and Babitt's Hexachord theorem. The last part of this book is devoted to musical applications of the model: the first experiment consists in extracting barcodes from artificially constructed scores, such as scales or chords. This study leads naturally to song harmonization process, which reduces a song to its melody and chord grid, thus defining the notions of graph and complexity of a piece. Persistent homology also lends itself to the problem of automatic classification of musical style, which will be treated here under the prism of symbolic descriptors given by statistics calculated directly on barcodes. Finally, the last application proposes a encoding of musical bars based on the Hausdorff distance, which leads to the study of musical textures. The book is addressed to graduate students and researchers in mathematical music theory and music information research, but also at researchers in other fields, such as applied mathematicians and topologists, who want to learn more about mathematical music theory or music information research.

Mathematics And Computation In Music

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Author : Thomas Noll
language : en
Publisher: Springer Nature
Release Date : 2024-05-22

Mathematics And Computation In Music written by Thomas Noll and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-05-22 with Language Arts & Disciplines categories.

This book constitutes the refereed proceedings of the 9th International Conference on Mathematics and Computation in Music, MCM 2024, held in Coimbra, Portugal, during June 18–21, 2024. The 30 full papers and 9 short papers included in this book were carefully reviewed and selected from 45 submissions. They were organized in topical sections as follows: mathematical scale theory and tuning; rhythm analysis and rhythm generation; categorical and algebraic approaches to music; quantum music; theory and algorithms for melodic- harmonic analysis and generation; geometric approaches to musical algorithms and microtonality; fourier analysis for music; similarity and distance measures for music; short papers; communication-performances; and tribute to Yves Hellegouarch.

Topological Signal Processing

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Author : Michael Robinson
language : en
Publisher: Springer Science & Business Media
Release Date : 2014-01-07

Topological Signal Processing written by Michael Robinson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-07 with Technology & Engineering categories.

Signal processing is the discipline of extracting information from collections of measurements. To be effective, the measurements must be organized and then filtered, detected, or transformed to expose the desired information. Distortions caused by uncertainty, noise, and clutter degrade the performance of practical signal processing systems. In aggressively uncertain situations, the full truth about an underlying signal cannot be known. This book develops the theory and practice of signal processing systems for these situations that extract useful, qualitative information using the mathematics of topology -- the study of spaces under continuous transformations. Since the collection of continuous transformations is large and varied, tools which are topologically-motivated are automatically insensitive to substantial distortion. The target audience comprises practitioners as well as researchers, but the book may also be beneficial for graduate students.

Computational Topology

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Author : Herbert Edelsbrunner
language : en
Publisher: American Mathematical Society
Release Date : 2022-01-31

Computational Topology written by Herbert Edelsbrunner and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-31 with Mathematics categories.

Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

Quantitative Tamarkin Theory

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Author : Jun Zhang
language : en
Publisher: Springer Nature
Release Date : 2020-03-09

Quantitative Tamarkin Theory written by Jun Zhang and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-09 with Mathematics categories.

This textbook offers readers a self-contained introduction to quantitative Tamarkin category theory. Functioning as a viable alternative to the standard algebraic analysis method, the categorical approach explored in this book makes microlocal sheaf theory accessible to a wide audience of readers interested in symplectic geometry. Much of this material has, until now, been scattered throughout the existing literature; this text finally collects that information into one convenient volume. After providing an overview of symplectic geometry, ranging from its background to modern developments, the author reviews the preliminaries with precision. This refresher ensures readers are prepared for the thorough exploration of the Tamarkin category that follows. A variety of applications appear throughout, such as sheaf quantization, sheaf interleaving distance, and sheaf barcodes from projectors. An appendix offers additional perspectives by highlighting further useful topics. Quantitative Tamarkin Theory is ideal for graduate students interested in symplectic geometry who seek an accessible alternative to the algebraic analysis method. A background in algebra and differential geometry is recommended. This book is part of the "Virtual Series on Symplectic Geometry" http://www.springer.com/series/16019

Topological Data Analysis For Genomics And Evolution

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Author : Raul Rabadan
language : en
Publisher: Cambridge University Press
Release Date : 2019-12-19

Topological Data Analysis For Genomics And Evolution written by Raul Rabadan and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-12-19 with Mathematics categories.

An introduction to geometric and topological methods to analyze large scale biological data; includes statistics and genomic applications.

Discrete Geometry For Computer Imagery

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Author : Walter G. Kropatsch
language : en
Publisher: Springer
Release Date : 2017-09-01

Discrete Geometry For Computer Imagery written by Walter G. Kropatsch and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-01 with Computers categories.

This book constitutes the thoroughly refereed proceedings of the 20th IAPR International Conference on Discrete Geometry for Computer Imagery, DGCI 2017, held in Vienna, Austria, in September 2017. The 28 revised full papers presented together with 3 invited talks were carefully selected from 36 submissions. The papers are organized in topical sections on geometric transforms; discrete tomography; discrete modeling and visualization; morphological analysis; discrete shape representation, recognition and analysis; discrete and combinatorial topology; discrete models and tools; models for discrete geometry.

Computational Neuroanatomy

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Author : Moo K. Chung
language : en
Publisher: World Scientific
Release Date : 2012-09-05

Computational Neuroanatomy written by Moo K. Chung and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-05 with Computers categories.

Computational neuroanatomy is an emerging field that utilizes various non-invasive brain imaging modalities, such as MRI and DTI, in quantifying the spatiotemporal dynamics of the human brain structures in both normal and clinical populations. This discipline emerged about twenty years ago and has made substantial progress in the past decade. The main goals of this book are to provide an overview of various mathematical, statistical and computational methodologies used in the field to a wide range of researchers and students, and to address important yet technically challenging topics in further detail.

Geometric And Topological Inference

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Author : Jean-Daniel Boissonnat
language : en
Publisher: Cambridge University Press
Release Date : 2018-09-27

Geometric And Topological Inference written by Jean-Daniel Boissonnat and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-27 with Computers categories.

A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.

Topological Methods In Data Analysis And Visualization Iii

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Author : Peer-Timo Bremer
language : en
Publisher: Springer Science & Business
Release Date : 2014-04-22

Topological Methods In Data Analysis And Visualization Iii written by Peer-Timo Bremer and has been published by Springer Science & Business this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-04-22 with Mathematics categories.

This collection of peer-reviewed conference papers provides comprehensive coverage of cutting-edge research in topological approaches to data analysis and visualization. It encompasses the full range of new algorithms and insights, including fast homology computation, comparative analysis of simplification techniques, and key applications in materials and medical science. The volume also features material on core research challenges such as the representation of large and complex datasets and integrating numerical methods with robust combinatorial algorithms. Reflecting the focus of the TopoInVis 2013 conference, the contributions evince the progress currently being made on finding experimental solutions to open problems in the sector. They provide an inclusive snapshot of state-of-the-art research that enables researchers to keep abreast of the latest developments and provides a foundation for future progress. With papers by some of the world’s leading experts in topological techniques, this volume is a major contribution to the literature in a field of growing importance with applications in disciplines that range from engineering to medicine.