Heat Kernel Voting with Geometric Invariants
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Heat Kernels and Function Theory on Metric Measure Spaces
Heat kernels and function theory on metric measure spaces Alexander Grigor’yan Imperial College London SW7 2AZ United Kingdom and Institute of Control Sciences of RAS Moscow, Russia email: [email protected] Contents 1 Introduction 2 2 Definition of a heat kernel and examples 5 3 Volume of balls 8 4 Energy form 10 5 Besov spaces and energy domain 13 5.1 Besov spaces in Rn ............................ 13 5.2 Besov spaces in a metric measure space . 14 5.3 Identification of energy domains and Besov spaces . 15 5.4 Subordinated heat kernel ......................... 19 6 Intrinsic characterization of walk dimension 22 7 Inequalities for walk dimension 23 8 Embedding of Besov spaces into H¨olderspaces 29 9 Bessel potential spaces 30 1 1 Introduction The classical heat kernel in Rn is the fundamental solution to the heat equation, which is given by the following formula 1 x y 2 pt (x, y) = exp | − | . (1.1) n/2 t (4πt) − 4 ! x y 2 It is worth observing that the Gaussian term exp | − | does not depend in n, − 4t n/2 whereas the other term (4πt)− reflects the dependence of the heat kernel on the underlying space via its dimension. The notion of heat kernel extends to any Riemannian manifold M. In this case, the heat kernel pt (x, y) is the minimal positive fundamental solution to the heat ∂u equation ∂t = ∆u where ∆ is the Laplace-Beltrami operator on M, and it always exists (see [11], [13], [16]). Under certain assumptions about M, the heat kernel can be estimated similarly to (1.1). -
Heat Kernel Analysis on Graphs
Heat Kernel Analysis On Graphs Xiao Bai Submitted for the degree of Doctor of Philosophy Department of Computer Science June 7, 2007 Abstract In this thesis we aim to develop a framework for graph characterization by com- bining the methods from spectral graph theory and manifold learning theory. The algorithms are applied to graph clustering, graph matching and object recogni- tion. Spectral graph theory has been widely applied in areas such as image recog- nition, image segmentation, motion tracking, image matching and etc. The heat kernel is an important component of spectral graph theory since it can be viewed as describing the flow of information across the edges of the graph with time. Our first contribution is to investigate how to extract useful and stable invari- ants from the graph heat kernel as a means of clustering graphs. The best set of invariants are the heat kernel trace, the zeta function and its derivative at the origin. We also study heat content invariants. The polynomial co-efficients can be computed from the Laplacian eigensystem. Graph clustering is performed by applying principal components analysis to vectors constructed from the invari- ants or simply based on the unitary features extracted from the graph heat kernel. We experiment with the algorithms on the COIL and Oxford-Caltech databases. We further investigate the heat kernel as a means of graph embedding. The second contribution of the thesis is the introduction of two graph embedding methods. The first of these uses the Euclidean distance between graph nodes. To do this we equate the spectral and parametric forms of the heat kernel to com- i pute an approximate Euclidean distance between nodes. -
Proofs-6-2.Pdf
Real Analysis Chapter 6. Differentiation and Integration 6.2. Differentiability of Monotone Functions: Lebesgue’s Theorem—Proofs of Theorems December 25, 2015 () Real Analysis December 25, 2015 1 / 18 Table of contents 1 The Vitali Covering Lemma 2 Lemma 6.3 3 Lebesgue Theorem 4 Corollary 6.4 () Real Analysis December 25, 2015 2 / 18 By the countable additivity of measure and monotonicity ∞ of measure, if {Ik }k=1 ⊆ F is a collection of disjoint intervals in F the ∞ X `(Ik ) ≤ m(O) < ∞. (3) k=1 The Vitali Covering Lemma The Vitali Covering Lemma The Vitali Covering Lemma. Let E be a set of finite outer measure and F a collection of closed, bounded intervals that covers E in the sense of Vitali. Then for each n ε > 0, there is a finite disjoint subcollection {Ik }k=1 of F for which " n # ∗ [ m E \ Ik < ε. (2) k=1 Proof. Since m∗(E) < ∞, there is an open set O containing E for which m(O) < ∞ (by the definition of outer measure). Because F is a Vitali covering of E, then every x ∈ E is in some interval of length less than ε > 0 (for arbitrary ε > 0). So WLOG we can suppose each interval in F is contained in O. () Real Analysis December 25, 2015 3 / 18 The Vitali Covering Lemma The Vitali Covering Lemma The Vitali Covering Lemma. Let E be a set of finite outer measure and F a collection of closed, bounded intervals that covers E in the sense of Vitali. Then for each n ε > 0, there is a finite disjoint subcollection {Ik }k=1 of F for which " n # ∗ [ m E \ Ik < ε. -
Inverse Spectral Problems in Riemannian Geometry
Inverse Spectral Problems in Riemannian Geometry Peter A. Perry Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027, U.S.A. 1 Introduction Over twenty years ago, Marc Kac posed what is arguably one of the simplest inverse problems in pure mathematics: "Can one hear the shape of a drum?" [19]. Mathematically, the question is formulated as follows. Let /2 be a simply connected, plane domain (the drumhead) bounded by a smooth curve 7, and consider the wave equation on /2 with Dirichlet boundary condition on 7 (the drumhead is clamped at the boundary): Au(z,t) = ~utt(x,t) in/2, u(z, t) = 0 on 7. The function u(z,t) is the displacement of the drumhead, as it vibrates, at position z and time t. Looking for solutions of the form u(z, t) = Re ei~tv(z) (normal modes) leads to an eigenvalue problem for the Dirichlet Laplacian on B: v(x) = 0 on 7 (1) where A = ~2/c2. We write the infinite sequence of Dirichlet eigenvalues for this problem as {A,(/2)}n=l,c¢ or simply { A-},~=1 co it" the choice of domain/2 is clear in context. Kac's question means the following: is it possible to distinguish "drums" /21 and/22 with distinct (modulo isometrics) bounding curves 71 and 72, simply by "hearing" all of the eigenvalues of the Dirichlet Laplacian? Another way of asking the question is this. What is the geometric content of the eigenvalues of the Laplacian? Is there sufficient geometric information to determine the bounding curve 7 uniquely? In what follows we will call two domains isospectral if all of their Dirichlet eigenvalues are the same. -
Three Examples of Quantum Dynamics on the Half-Line with Smooth Bouncing
Three examples of quantum dynamics on the half-line with smooth bouncing C. R. Almeida Universidade Federal do Esp´ıritoSanto, Vit´oria,29075-910, ES, Brazil H. Bergeron ISMO, UMR 8214 CNRS, Univ Paris-Sud, France J.P. Gazeau Laboratoire APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris-Cit´e75205 Paris, France A.C. Scardua Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180 - Rio de Janeiro, RJ, Brazil Abstract This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrari- ness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bounc- ing of a charged sphere, and a smooth bouncing of \dust" sphere as a simple model of quantum Newtonian cosmology. Keywords: Integral quantization, Half-plane, Affine symmetry, Coherent states, arXiv:1708.06422v2 [quant-ph] 31 Aug 2017 Quantum smooth bouncing Email addresses: [email protected] (C. R. Almeida), [email protected] (H. Bergeron), [email protected] (J.P. -
Introduction
Traceglobfin June 3, 2011 Copyrighted Material Introduction The purpose of this book is to use the hypoelliptic Laplacian to evaluate semisimple orbital integrals, in a formalism that unifies index theory and the trace formula. 0.1 The trace formula as a Lefschetz formula Let us explain how to think formally of such a unified treatment, while allowing ourselves a temporarily unbridled use of mathematical analogies. Let X be a compact Riemannian manifold, and let ΔX be the corresponding Laplace-Beltrami operator. For t > 0, consider the trace of the heat kernel Tr pexp h ΔX 2m . If X is the Hilbert space of square-integrable functions t = L2 on X, Tr pexp htΔX =2 m is the trace of the ‘group element’ exp htΔX =2 m X acting on L2 . X X Suppose that L2 is the cohomology of an acyclic complex R on which Δ acts. Then Tr pexp htΔX =2 m can be viewed as the evaluation of a Lefschetz trace, so that cohomological methods can be applied to the evaluation of this trace. In our case, R will be the fibrewise de Rham complex of the total space Xb of a flat vector bundle over X, which contains TX as a subbundle. The Lefschetz fixed point formulas of Atiyah-Bott [ABo67, ABo68] provide a model for the evaluation of such cohomological traces. R The McKean-Singer formula [McKS67] indicates that if D is a Hodge like Laplacian operator acting on R and commuting with ΔX , for any b > 0, X L2 p hX m R p h X R 2m Tr exp tΔ =2 = Trs exp tΔ =2 − tD =2b : (0.1) In (0.1), Trs is our notation for the supertrace. -
Heat Kernels
Chapter 3 Heat Kernels In this chapter, we assume that the manifold M is compact and the general- ized Laplacian H is not necessarily symmetric. Note that if M is non-compact and H is symmetric, we can study the heat kernel following the lines of Spectral theorem and the Schwartz kernel theorem. We will not discuss it here. 3.1 heat kernels 3.1.1 What is kernel? Let H be a generalized Laplacian on a vector bundle E over a compact Riemannian oriented manifold M. Let E and F be Hermitian vector bundles over M. Let p1 and p2 be the projections from M × M onto the first and the second factor M respectively. We denote by ⊠ ∗ ⊗ ∗ E F := p1E p2F (3.1.1) over M × M. Definition 3.1.1. A continuous section P (x; y) on F ⊠E∗) is called a kernel. Using P (x; y), we could define an operator P : C 1(M; E) ! C 1(M; F ) by Z (P u)(x) = hP (x; y); u(y)iEdy: (3.1.2) y2M The kernel P (x; y) is also called the kernel of P , which is also denoted by hxjP jyi. 99 100 CHAPTER 3. HEAT KERNELS Proposition 3.1.2. If P has a kernel P (x; y), then the adjoint operator P ∗ has a kernel P ∗(x; y) = P (y; x)∗ 2 C 1(M × M; E∗ ⊠ F )1. Proof. For u 2 L2(M; E), v 2 L2(M; F ∗), we have Z ⟨Z ⟩ (P u; v)L2 = hP (x; y); u(y)iEdy; v(x) dx x2MZ y2⟨M Z F ⟩ ∗ = u(y); hP (x; y) ; v(x)iF dx dy Z ⟨ y2M Z x2M ⟩ E ∗ ∗ = u(x); hP (y; x) ; v(y)iF dy dx = (u; P v)L2 : (3.1.3) x2M y2M E So for any v 2 L2(M; F ∗), Z ∗ ∗ P v = hP (y; x) ; v(y)iF dy: (3.1.4) y2M The proof of Proposition 3.1.2 is completed. -
The Partial View Heat Kernel Descriptor for 3D Object Representation
The Partial View Heat Kernel Descriptor for 3D Object Representation Susana Brandao˜ 1, Joao˜ P. Costeira2 and Manuela Veloso3 Abstract— We introduce the Partial View Heat Kernel sensor cameras, such as the Kinect sensor, that combine (PVHK) descriptor, for the purpose of 3D object representation RGB and depth information, but are also noisy. To address and recognition from partial views, assumed to be partial these multiple challenges and goals, we introduce a new object surfaces under self occlusion. PVHK describes partial views in a geometrically meaningful way, i.e., by establishing representation for partial views, the Partial View Heat Kernel a unique relation between the shape of the view and the (PVHK) descriptor, which is: descriptor. PVHK is also stable with respect to sensor noise and 1) Informative, i.e., a single descriptor robustly describes therefore adequate for sensors, such as the current active 3D each partial view; cameras. Furthermore, PVHK takes full advantage of the dual 3D/RGB nature of current sensors and seamlessly incorporates 2) Stable, i.e., small perturbations on the surface yield appearance information onto the 3D information. We formally small perturbations on the descriptor; define the PVHK descriptor, discuss related work, provide 3) Inclusive, i.e., appearance properties, such as texture, evidence of the PVHK properties and validate them in three can be seamlessly incorporated into the geometry- purposefully diverse datasets, and demonstrate its potential for based descriptor. recognition tasks. The combination of these three characteristics results in a I. INTRODUCTION representation especially suitable for partial views captured We address the 3D representation of objects from multiple from noisy RGB+Depth sensors during robot navigation 3D partial views, where each partial view is the visible or manipulation where the object surfaces are visible with surface of the object as seen from a view angle, with no limited, if any, occlusion. -
The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO Wenhao Zhang
Claremont Colleges Scholarship @ Claremont CMC Senior Theses CMC Student Scholarship 2018 The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO Wenhao Zhang Recommended Citation Zhang, Wenhao, "The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO" (2018). CMC Senior Theses. 1907. http://scholarship.claremont.edu/cmc_theses/1907 This Open Access Senior Thesis is brought to you by Scholarship@Claremont. It has been accepted for inclusion in this collection by an authorized administrator. For more information, please contact [email protected]. CLAREMONT MCKENNA COLLEGE The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO SUBMITTED TO Professor Winston Ou AND Professor Chiu-Yen Kao BY Wenhao Zhang FOR SENIOR THESIS Spring 2018 Apr 23rd 2018 Contents Abstract 2 Chapter 1. Introduction 3 Notation 3 1.1. The space BMO 3 1.2. The Hardy-Littlewood Maximal Operator and the Strong Maximal Operator 4 1.3. Overview of the Thesis 5 1.4. Acknowledgement 6 Chapter 2. The Weak-type Estimate of Hardy-Littlewood Maximal Function 7 2.1. Vitali Covering Lemma 7 2.2. The Boundedness of Hardy-Littlewood Maximal Function on Lp 8 Chapter 3. Weighted Inequalities 10 3.1. Muckenhoupt Weights 10 3.2. Coifman-Rochberg Proposition 13 3.3. The Logarithm of Muckenhoupt weights 14 Chapter 4. The Boundedness of the Hardy-Littlewood Maximal Operator on BMO 17 4.1. Commutation Lemma and Proof of the Main Result 17 4.2. Question: The Boundedness of the Strong Maximal Function 18 Chapter 5. -
Article (458.2Kb)
British Journal of Mathematics & Computer Science 8(3): 220-228, 2015, Article no.BJMCS.2015.156 ISSN: 2231-0851 SCIENCEDOMAIN international www.sciencedomain.org On Non-metric Covering Lemmas and Extended Lebesgue-differentiation Theorem Mangatiana A. Robdera1∗ 1Department of Mathematics, University of Botswana, Private Bag 0022, Gaborone, Botswana. Article Information DOI: 10.9734/BJMCS/2015/16752 Editor(s): (1) Paul Bracken, Department of Mathematics, University of Texas-Pan American Edinburg, TX 78539, USA. Reviewers: (1) Anonymous, Turkey. (2) Danyal Soyba, Erciyes University, Turkey. (3) Anonymous, Italy. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=1032&id=6&aid=8603 Original Research Article Received: 12 February 2015 Accepted: 12 March 2015 Published: 28 March 2015 Abstract We give a non-metric version of the Besicovitch Covering Lemma and an extension of the Lebesgue-Differentiation Theorem into the setting of the integration theory of vector valued functions. Keywords: Covering lemmas; vector integration; Hardy-Littlewood maximal function; Lebesgue differ- entiation theorem. 2010 Mathematics Subject Classification: 28B05; 28B20; 28C15; 28C20; 46G12 1 Introduction The Vitali Covering Lemma that has widespread use in analysis essentially states that given a set n E ⊂ R with Lebesgue measure λ(E) < 1 and a cover of E by balls of ’arbitrary small Lebesgue measure’, one can find almost cover of E by a finite number of pairwise disjoint balls from the given cover. The more elaborate and more powerful -
Heat Kernel Smoothing in Irregular Image Domains Moo K
1 Heat Kernel Smoothing in Irregular Image Domains Moo K. Chung1, Yanli Wang2, Gurong Wu3 1University of Wisconsin, Madison, USA Email: [email protected] 2Institute of Applied Physics and Computational Mathematics, Beijing, China Email: [email protected] 3University of North Carolina, Chapel Hill, USA Email: [email protected] Abstract We present the discrete version of heat kernel smoothing on graph data structure. The method is used to smooth data in an irregularly shaped domains in 3D images. New statistical properties are derived. As an application, we show how to filter out data in the lung blood vessel trees obtained from computed tomography. The method can be further used in representing the complex vessel trees parametrically and extracting the skeleton representation of the trees. I. INTRODUCTION Heat kernel smoothing was originally introduced in the context of filtering out surface data defined on mesh vertices obtained from 3D medical images [1], [2]. The formulation uses the tangent space projection in approximating the heat kernel by iteratively applying Gaussian kernel with smaller bandwidth. Recently proposed spectral formulation to heat kernel smoothing [3] constructs the heat kernel analytically using the eigenfunctions of the Laplace-Beltrami (LB) operator, avoiding the need for the linear approximation used in [1], [4]. In this paper, we present the discrete version of heat kernel smoothing on graphs. Instead of Laplace- Beltrami operator, graph Laplacian is used to construct the discrete version of heat kernel smoothing. New statistical properties are derived for kernel smoothing that utilizes the fact heat kernel is a probability distribution. Heat kernel smoothing is used to smooth out data defined on irregularly shaped domains in 3D images. -
Lebesgue Differentiation
LEBESGUE DIFFERENTIATION ANUSH TSERUNYAN Throughout, we work in the Lebesgue measure space Rd;L(Rd);λ , where L(Rd) is the σ-algebra of Lebesgue-measurable sets and λ is the Lebesgue measure. 2 Rd For r > 0 and x , let Br(x) denote the open ball of radius r centered at x. The spaces 0 and 1 1. L Lloc The space L0 of measurable functions. By a measurable function we mean an L(Rd)- measurable function f : Rd ! R and we let L0(Rd) (or just L0) denote the vector space of all measurable functions modulo the usual equivalence relation =a:e: of a.e. equality. There are two natural notions of convergence (topologies) on L0: a.e. convergence and convergence in measure. As we know, these two notions are related, but neither implies the other. However, convergence in measure of a sequence implies a.e. convergence of a subsequence, which in many situations can be boosted to a.e. convergence of the entire sequence (recall Problem 3 of Midterm 2). Convergence in measure is captured by the following family of norm-like maps: for each α > 0 and f 2 L0, k k . · f α∗ .= α λ ∆α(f ) ; n o . 2 Rd j j k k where ∆α(f ) .= x : f (x) > α . Note that f α∗ can be infinite. 2 0 k k 6 k k Observation 1 (Chebyshev’s inequality). For any α > 0 and any f L , f α∗ f 1. 1 2 Rd The space Lloc of locally integrable functions. Differentiation at a point x is a local notion as it only depends on the values of the function in some open neighborhood of x, so, in this context, it is natural to work with a class of functions defined via a local property, as opposed to global (e.g.