Quantum Algorithms for Topological and Geometric Analysis of Data
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Geometric Analysis of Shapes and Its Application to Medical Image Analysis
Geometric Analysis of Shapes and Its application to Medical Image Analysis by Anirban Mukhopadhyay (Under the DIRECTION of Suchendra M. Bhandarkar) Abstract Geometric analysis of shapes plays an important role in the way the visual world is per- ceived by modern computers. To this end, low-level geometric features provide most obvious and important cues towards understanding the visual scene. A novel intrinsic geometric sur- face descriptor, termed as the Geodesic Field Estimate (GFE) is proposed. Also proposed is a parallel algorithm, well suited for implementation on Graphics Processing Units, for efficient computation of the shortest geodesic paths. Another low level geometric descriptor, termed as the Biharmonic Density Estimate, is proposed to provide an intrinsic geometric scale space signature for multiscale surface feature-based representation of deformable 3D shapes. The computer vision and graphics communities rely on mid-level geometric understanding as well to analyze a scene. Symmetry detection and partial shape matching play an important role as mid-level cues. A comprehensive framework for detection and characterization of partial intrinsic symmetry over 3D shapes is proposed. To identify prominent overlapping symmetric regions, the proposed framework is decoupled into Correspondence Space Voting followed by Transformation Space Mapping procedure. Moreover, a novel multi-criteria optimization framework for matching of partially visible shapes in multiple images using joint geometric embedding is also proposed. The ultimate goal of geometric shape analysis is to resolve high level applications of modern world. This dissertation has focused on three different application scenarios. In the first scenario, a novel approach for the analysis of the non-rigid Left Ventricular (LV) endocardial surface from Multi-Detector CT images, using a generalized isometry-invariant Bag-of-Features (BoF) descriptor, is proposed and implemented. -
Topology and Data
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data. -
Karen Keskulla Uhlenbeck
2019 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck University of Texas at Austin “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” Karen Keskulla Uhlenbeck is a founder of modern by earlier work of Morse, guarantees existence of Geometric Analysis. Her perspective has permeated minimisers of geometric functionals and is successful the field and led to some of the most dramatic in the case of 1-dimensional domains, such as advances in mathematics in the last 40 years. closed geodesics. Geometric analysis is a field of mathematics where Uhlenbeck realised that the condition of Palais— techniques of analysis and differential equations are Smale fails in the case of surfaces due to topological interwoven with the study of geometrical and reasons. The papers of Uhlenbeck, co-authored with topological problems. Specifically, one studies Sacks, on the energy functional for maps of surfaces objects such as curves, surfaces, connections and into a Riemannian manifold, have been extremely fields which are critical points of functionals influential and describe in detail what happens when representing geometric quantities such as energy the Palais-Smale condition is violated. A minimising and volume. For example, minimal surfaces are sequence of mappings converges outside a finite set critical points of the area and harmonic maps are of singular points and by using rescaling arguments, critical points of the Dirichlet energy. Uhlenbeck’s they describe the behaviour near the singularities major contributions include foundational results on as bubbles or instantons, which are the standard minimal surfaces and harmonic maps, Yang-Mills solutions of the minimising map from the 2-sphere to theory, and integrable systems. -
Algebraic Topology
Algebraic Topology Vanessa Robins Department of Applied Mathematics Research School of Physics and Engineering The Australian National University Canberra ACT 0200, Australia. email: [email protected] September 11, 2013 Abstract This manuscript will be published as Chapter 5 in Wiley's textbook Mathe- matical Tools for Physicists, 2nd edition, edited by Michael Grinfeld from the University of Strathclyde. The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology, including persistent homology. arXiv:1304.7846v2 [math-ph] 10 Sep 2013 1 Contents 1 Introduction 3 2 Homotopy Theory 4 2.1 Homotopy of paths . 4 2.2 The fundamental group . 5 2.3 Homotopy of spaces . 7 2.4 Examples . 7 2.5 Covering spaces . 9 2.6 Extensions and applications . 9 3 Homology 11 3.1 Simplicial complexes . 12 3.2 Simplicial homology groups . 12 3.3 Basic properties of homology groups . 14 3.4 Homological algebra . 16 3.5 Other homology theories . 18 4 Cohomology 18 4.1 De Rham cohomology . 20 5 Morse theory 21 5.1 Basic results . 21 5.2 Extensions and applications . 23 5.3 Forman's discrete Morse theory . 24 6 Computational topology 25 6.1 The fundamental group of a simplicial complex . 26 6.2 Smith normal form for homology . 27 6.3 Persistent homology . 28 6.4 Cell complexes from data . 29 2 1 Introduction Topology is the study of those aspects of shape and structure that do not de- pend on precise knowledge of an object's geometry. -
NECESSARY CONDITIONS for DISCONTINUITIES of MULTIDIMENSIONAL SIZE FUNCTIONS Introduction Topological Persistence Is Devoted to T
NECESSARY CONDITIONS FOR DISCONTINUITIES OF MULTIDIMENSIONAL SIZE FUNCTIONS A. CERRI AND P. FROSINI Abstract. Some new results about multidimensional Topological Persistence are presented, proving that the discontinuity points of a k-dimensional size function are necessarily related to the pseudocritical or special values of the associated measuring function. Introduction Topological Persistence is devoted to the study of stable properties of sublevel sets of topological spaces and, in the course of its development, has revealed itself to be a suitable framework when dealing with applications in the field of Shape Analysis and Comparison. Since the beginning of the 1990s research on this sub- ject has been carried out under the name of Size Theory, studying the concept of size function, a mathematical tool able to describe the qualitative properties of a shape in a quantitative way. More precisely, the main idea is to model a shape by a topological space endowed with a continuous function ϕ, called measuring function. Such a functionM is chosen according to applications and can be seen as a descriptor of the features considered relevant for shape characterization. Under these assumptions, the size function ℓ( ,ϕ) associated with the pair ( , ϕ) is a de- scriptor of the topological attributes thatM persist in the sublevel setsM of induced by the variation of ϕ. According to this approach, the problem of comparingM two shapes can be reduced to the simpler comparison of the related size functions. Since their introduction, these shape descriptors have been widely studied and applied in quite a lot of concrete applications concerning Shape Comparison and Pattern Recognition (cf., e.g., [4, 8, 15, 34, 35, 36]). -
Iasinstitute for Advanced Study
IAInsti tSute for Advanced Study Faculty and Members 2012–2013 Contents Mission and History . 2 School of Historical Studies . 4 School of Mathematics . 21 School of Natural Sciences . 45 School of Social Science . 62 Program in Interdisciplinary Studies . 72 Director’s Visitors . 74 Artist-in-Residence Program . 75 Trustees and Officers of the Board and of the Corporation . 76 Administration . 78 Past Directors and Faculty . 80 Inde x . 81 Information contained herein is current as of September 24, 2012. Mission and History The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. The Institute exists to encourage and support fundamental research in the sciences and human - ities—the original, often speculative thinking that produces advances in knowledge that change the way we understand the world. It provides for the mentoring of scholars by Faculty, and it offers all who work there the freedom to undertake research that will make significant contributions in any of the broad range of fields in the sciences and humanities studied at the Institute. Y R Founded in 1930 by Louis Bamberger and his sister Caroline Bamberger O Fuld, the Institute was established through the vision of founding T S Director Abraham Flexner. Past Faculty have included Albert Einstein, I H who arrived in 1933 and remained at the Institute until his death in 1955, and other distinguished scientists and scholars such as Kurt Gödel, George F. D N Kennan, Erwin Panofsky, Homer A. Thompson, John von Neumann, and A Hermann Weyl. N O Abraham Flexner was succeeded as Director in 1939 by Frank Aydelotte, I S followed by J. -
Geometric Analysis and Integral Geometry
598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors 598 Geometric Analysis and Integral Geometry AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 22E30, 43A85, 44A12, 45Q05, 92C55; Secondary 22E46, 32L25, 35S30, 65R32. Library of Congress Cataloging-in-Publication Data AMS Special Session on Radon Transforms and Geometric Analysis (2012 : Boston, Mass.) Geometric analysis and integral geometry : AMS special session in honor of Sigurdur Helgason’s 85th birthday, radon transforms and geometric analysis, January 4-7, 2012, Boston, MA ; Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces, January 8-9, 2012, Medford, MA / Eric Todd Quinto, Fulton Gonzalez, Jens Gerlach Christensen, editors. -
Perspectives on Geometric Analysis
Surveys in Differential Geometry X Perspectives on geometric analysis Shing-Tung Yau This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I[731] worked on fundamental groups of manifolds with non-positive curva- ture. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much inter- ested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S.Y. Cheng told me about these inspiring lec- tures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [563] published his results right be- fore us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Amp`ere equations started to broaden in these years. -
25 HIGH-DIMENSIONAL TOPOLOGICAL DATA ANALYSIS Fr´Ed´Ericchazal
25 HIGH-DIMENSIONAL TOPOLOGICAL DATA ANALYSIS Fr´ed´ericChazal INTRODUCTION Modern data often come as point clouds embedded in high-dimensional Euclidean spaces, or possibly more general metric spaces. They are usually not distributed uniformly, but lie around some highly nonlinear geometric structures with nontriv- ial topology. Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical and algorithmic tools to understand the topological and geometric structure of data. This chapter provides a short introduction to this new field through a few selected topics. The focus is deliberately put on the mathe- matical foundations rather than specific applications, with a particular attention to stability results asserting the relevance of the topological information inferred from data. The chapter is organized in four sections. Section 25.1 is dedicated to distance- based approaches that establish the link between TDA and curve and surface re- construction in computational geometry. Section 25.2 considers homology inference problems and introduces the idea of interleaving of spaces and filtrations, a funda- mental notion in TDA. Section 25.3 is dedicated to the use of persistent homology and its stability properties to design robust topological estimators in TDA. Sec- tion 25.4 briefly presents a few other settings and applications of TDA, including dimensionality reduction, visualization and simplification of data. 25.1 GEOMETRIC INFERENCE AND RECONSTRUCTION Topologically correct reconstruction of geometric shapes from point clouds is a classical problem in computational geometry. The case of smooth curve and surface reconstruction in R3 has been widely studied over the last two decades and has given rise to a wide range of efficient tools and results that are specific to dimensions 2 and 3; see Chapter 35. -
Multiset Metrics on Bounded Spaces
Multiset metrics on bounded spaces∗ Stephen M. Turner Abstract We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete metrics iff it is also complete. Two of them, and the fifth func- tion, all generalise the usual Hausdorff metric on subsets. Some possible applications are also considered. 1 Introduction Metrics on subsets and multisets (subsets-with-repetition-allowed) of metric spaces have or could have numerous fields of application such as credit rating, pattern or image recognition and synthetic biology. We employ three related models (called E, F and G) for the space of multisets on the metric space (X, d). On each of E, F we define two closely-related functions. These four functions all turn out to be metrics precisely when d is bounded, and are complete iff d is also complete. Another function studied in model G has the same properties for (at least) uniformly discrete d. X is likely to be finite in many applications anyway. We show that there is an integer programming algorithm for those in model E. The three in models F and G generalise the Hausdorff metric. Beyond finiteness, no assumptions about multiset sizes are required. Various types of multiset metric on sets have been described[1.3], but the few that incorporate an underlying metric only refer to R or C. The simple and more general nature of those described here suggests that there may be other interesting possibilities. In this section, after setting out notation and required background, we men- tion briefly the existing work in this field. -
Recent Advances in Geometric Analysis
Advanced Lectures in Mathematics Volume XI Recent Advances in Geometric Analysis Editors: Yng-Ing Lee, Chang-Shou Lin, and Mao-Pei Tsui International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Yng-Ing Lee Chang-Shou Lin National Taiwan University National Chung Cheng University Mao-Pei Tsui University of Toledo Copyright © 2010 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is published and sold in China exclusively by Higher Education Press of China. No part of this work can be reproduced in any form, electronic or mechanical, recording, or by any information storage and data retrieval system, without prior approval from International Press. Requests for reproduction for scientific and/or educational purposes will normally be granted free of charge. In those cases where the author has retained copyright, requests for permission to use or reproduce any material should be addressed directly to the author. ISBN 978-1-57146-143-8 Typeset using the LaTeX system. Printed in the USA on acid-free paper. ADVANCED LECTURES IN MATHEMATICS Executive Editors Shing-Tung Yau Kefeng Liu Harvard University University of California at Los Angeles Zhejiang University Lizhen Ji Hangzhou, China University of Michigan, Ann Arbor Editorial Board Chongqing Cheng Tatsien Li Nanjing University Fudan University Nanjing, China Shanghai, China Zhong-Ci Shi Zhiying Wen Institute of Computational Mathematics Tsinghua University Chinese Academy of Sciences (CAS) Beijing, China Beijing, China Lo Yang Zhouping Xin Institute of Mathematics The Chinese University of Hong Kong Chinese Academy of Sciences (CAS) Hong Kong, China Beijing, China Weiping Zhang Xiangyu Zhou Nankai University Institute of Mathematics Tianjin, China Chinese Academy of Sciences (CAS) Beijing, China Xiping Zhu Zhongshan University Guangzhou, China Preface “2007 International Conference in Geometric Analysis” was held in Taiwan University from June 18th to 22nd, 2007. -
Geometric Analysis
IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Geometric Analysis https://doi.org/10.1090//pcms/022 IAS/PARK CITY MATHEMATICS SERIES Volume 22 Geometric Analysis Hubert L. Bray Greg Galloway Rafe Mazzeo Natasa Sesum Editors American Mathematical Society Institute for Advanced Study Hubert Lewis Bray, Gregory J. Galloway, Rafe Mazzeo, and Natasa Sesum, Volume Editors IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 53-06, 35-06, 83-06. Library of Congress Cataloging-in-Publication Data Geometric analysis / Hubert L. Bray, editor [and three others]. pages cm. — (IAS/Park City mathematics series ; volume 22) Includes bibliographical references. ISBN 978-1-4704-2313-1 (alk. paper) 1. Geometric analysis. 2. Mathematical analysis. I. Bray, Hubert L., editor. QA360.G455 2015 515.1—dc23 2015031562 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.