DOCSLIB.ORG

Diffeomorphic Shape Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Diffeomorphic Shape Analysis Diffeomorphic Shape Analysis Laurent Younes Johns Hopkins University January 2014 Partially supported by ONR, NIH, NSF Geilo, January 2014 Laurent Younes 1 Introduction • Shape analysis is the study of variation of shapes in relation with other variables. • Once a suitable “shape space” is defined, analyzing shapes involves: – Comparing them (using distances) – Finding properties of datasets in this space – Testing statistical hypotheses Geilo, January 2014 Laurent Younes 2 Some entry points to the literature • Landmarks / point sets: Kendall, Bookstein, Small, Dryden… • Active shapes: Taylor, Cootes,… • Registration: Demons, Diffeomorphic Demons (Thirion, Guimond, Ayache, Pennec, Vercauteren,…) • Harmonic analysis: SPHARM (Gerig, Steiner,…) • Medial axis and related, M-REPS (Pizer, Damon,…) • Manifolds of curves / surfaces – Theory (Mumford, Michor, Yezzi, Mennucci, Srivastava, Mio, Klassen…) • Statistics on manifolds of curves/surfaces (Pennec, Fletcher, Joshi, Marron…) Geilo, January 2014 Laurent Younes 3 On “Diffeomorphometry”… • Precursors (fluid registration): Christensen-Miller-Rabbit; Thirion… • Large Deformation Diffeomorphic Metric Mapping – LDDMM – Images: Beg, Miller, Trouvé, Y.; – Landmarks Miller, Joshi; – Measures (Glaunès, Trouvé, Y.), Currents (Glaunès, Vaillant) – Hamiltonian methods (Glaunès, Trouvé, Vialard, Y…) • Metamorphosis (Miller, Trouvé, Holm, Y…) Geilo, January 2014 Laurent Younes 4 Shape Spaces • Shape spaces are typically modeled as differential manifolds, but… • Almost all practical methods use a shape representation in a linear space of “local coordinates” • Linear statistical methods can then be employed to analyze the data. • For some of these methods (e.g., PCA), it is also important that linear combinations of the representations make sense, too. Geilo, January 2014 Laurent Younes 5 Example: Kendall Shape Space • Consider the family of collections of N distinct points in d ! d N • This forms an open subset of Q = (! ) • Identify collections that are can be deduced from each other by rotation, translation and scaling. • It is a set of subsets of Q (quotient space). • It can be structured as a differential manifold. • The Euclidean metric on Q transforms into a Riemannian metric in the quotient space. Geilo, January 2014 Laurent Younes 6 Example: Spaces of Curves 2 • The set Q of smooth functions q : [ 0 , 1 ] → ! is a Fréchet space. • Quotient out translation rotations scaling and change of parameter to obtain a shape space. • Sobolev norms on Q trickle down to Riemannian metrics on the shape space. • See papers from Michor, Mumford, Shah… Geilo, January 2014 Laurent Younes 7 Shapes Spaces and Deformable Templates • Ulf Grenander’s metric pattern theory involves transformation groups that act on shapes. • The transformations have a cost, represented by an effort functional. • Under additional assumptions, this induces a metric in the shape space. • The construction that follows will be based on these principles. • The group is the group of diffeomorphisms. • The shapes are anything that can be deformed. Geilo, January 2014 Laurent Younes 8 Tangent space representations • There is a natural way to build linear representations when the the shape space is modeled as a Riemannian manifold, i.e.: – a topological space that can be mapped locally to a vector space on which differentials can be computed (with consistency relations between local maps). – with a metric which can be used to compute lengths, and shortest paths between two points, which are called geodesics. Geilo, January 2014 Laurent Younes 9 Exponential Charts (I) • Geodesics can be characterized by a second order differential equation on the manifold. • This equation has (in general) a unique solution given its initial position and first derivative. • Fix the initial position (“template”). Denote it S . • The space of all derivatives of curves in S that start from S is the tangent space to S at S (notation T S ). S Geilo, January 2014 Laurent Younes 10 Exponential Charts (II) • The exponential map associates to each vector v T S ∈ S the solution at time t =1 of the geodesic equation starting at S in the direction v. • It is a map from T S to S , with notation S v ∈ T S exp (v) ∈ S S S • It can (generally) be restricted to a neighborhood of 0 in T S over which it is one to one, providing so-called S normal local coordinates or an exponential chart at S . Geilo, January 2014 Laurent Younes 11 SHAPES AND DIFFEOMORPHISMS Geilo, January 2014 Laurent Younes 12 General Approach • We will define shape spaces via the action of diffeomorphisms on them. • This action will induce a differential structure on the shape space. • It also allows to compare them: two shapes are similar if one can be obtained from the other via a small deformation. Geilo, January 2014 Laurent Younes 13 Notation • Diffeomorphisms of d will be denoted φ, ψ, etc. They are invertible differentiable transformations with a differentiable inverse. • Vector fields over d will be denoted v, w, etc. They are differentiable mappings from d onto itself. Geilo, January 2014 Laurent Younes 14 Ordinary Differential Equations • Diffeomorphisms can be built as flows associated to ordinary differential equations (ODEs). • Let ( t , x ) ! v ( t , x ) be a time-dependent vector field. Assume that v is differentiable in x with bounded first derivative over a time interval [ 0 , T ] . • Then there exists a unique solution of the ODE y ! = v ( t , y ) y(0) x with initial condition = 0 defined over the whole interval [ 0,T] Geilo, January 2014 Laurent Younes 15 Flows • The flow associated with the ODE y! = v ( t , y ) is the mapping (t,x) ! ϕ(t,x) such that ϕ ( t , x ) is the solution at time t of the ODE initialized with y ( 0 ) = x . • In other terms: ⎧∂tϕ = v(t,ϕ) ⎨ ϕ(0,x) = x ⎩ Geilo, January 2014 Laurent Younes 16 Flows (II) • Assuming that v ( t , ⋅ ) is continuously differentiable with bounded derivative, uniformly in time, the associated flow is such that x ! ϕ(t,x) is a diffeomorphism at all times. • If v has more (space) derivatives, they are inherited by ϕ . • We will build diffeomorphisms as flows associated to vector fields that belong in a specified reproducing kernel Hilbert space. Geilo, January 2014 Laurent Younes 17 Hilbert Spaces • A Hilbert space H is an infinite-dimensional vector space with an inner-product, such that the associated norm that makes it a complete metric space. • The inner product between two elements in H is denoted h,h' H • The set of bounded linear functionals µ : H → ! is the topological dual of H, denoted H * . • We will use the notation µ h : = µ ( h ) for µ ∈H *,h ∈H ( ) Geilo, January 2014 Laurent Younes 18 Riesz Theorem • The Riesz representation theorem states that H and H * are isometric. * • µ ∈ H if and only if there exists h ∈ H such that, for all h ' ∈ H : µh′ = h,h' ( ) H • The correspondence h will be denoted K , so that µ ! H K : H * H and its inverse A : H ! H * H → H • We therefore have µh = K µ,h and h,h' = A hh′ ( ) H H H ( H ) Geilo, January 2014 Laurent Younes 19 RKHS d • A Hilbert space H of functions defined over ! , with k values in ! is an RKHS, if H ⊂ L2 (!d ,!k ) ∩ C p (!d ,!k ) 0 ( p ≥ 0 ) with continuous inclusions in both spaces. (Here C p ( d , k ) is the space of p times differentiable 0 ! ! vector fields that converge to 0 (will all derivatives up to order p) at infinity, equipped with the supremum norm. • This means that for some constant C, and for all v ∈H max v , v ≤ C v ( 2 p,∞ ) H Geilo, January 2014 Laurent Younes 20 RKHS d k • If H is an RKHS, and x ∈ ! , a ∈ ! the linear form aδ : v ! aT v(x) x belongs to H * . • General notation: if θ is a (scalar) measure and a is a vector field defined on its support, a θ is the vector measure (aθ | w) = a(x)T w(x)dθ(x) ∫ Geilo, January 2014 Laurent Younes 21 RKHS • The reproducing kernel of V is a matrix-valued function K V d d defined on ! × ! by K H (x, y)a = K H (aδ y )(x) d with a,x ∈! K (x, y) • H is a k by k matrix. • We will denote K i the ith column of K and by K i j its i,j H H H entry. One has, in particular i j ij K H (⋅,x), K H (⋅, y) = K H (x, y) H Geilo, January 2014 Laurent Younes 22 Usual examples (scalar kernels) I • k is the d by d identity matrix. • Gaussian kernel: K (x, y) ∝ exp(− x − y 2 /2σ 2 )I H ‖ ‖ k I • Cauchy Kernel: K (x, y) ∝ k H 2 2 1+ x − y σ • Power of Laplacian: K (x, y) ∝ P ( x − y /σ )exp(− x − y /σ )I H m ‖ ‖ ‖ ‖ k with P = 1 , P = 1 + t and ∂ P − P = −tP 0 1 t m m m−1 Geilo, January 2014 Laurent Younes 23 The Interpolation Problem • Let H be a Hilbert space and µ , , µ ∈ H * . 1 … m • Solve the problem ⎡ h → min ⎢ H subject to h ,k 1, ,m ⎢ (µk ) = λk = … ⎣ m • Solution: h = a K µ where a , … , a are identified ∑ k=1 k H k 1 m using the constraints. Geilo, January 2014 Laurent Younes 24 Application • H is a scalar RKHS (k = 1). x , ,x ∈ d • 1 … m ! • Solve ⎡ h → min ⎢ H ⎢subject to h(xk ) = λk ,k = 1,…,m ⎣ m • Solution: h(⋅) = K H (⋅,xk )ak ∑ k=1 m with: K (x ,x )a = λ , k = 1,…,m ∑ l=1 H k l l k Geilo, January 2014 Laurent Younes 25 Attainable Diffeomorphisms • Fix V, an RKHS of vector fields with at least one derivative ( p ≥ 1 ). • We consider ODEs of the form y! = v ( t , y ) with v(t,⋅) ∈V • The flow evolution ( t , ) v ( t , ( t , ) ) can be interpreted ∂tϕ ⋅ = ϕ ⋅ as a control system in which the deformation, φ, is the state driven by the time-dependent vector field v (t,⋅) • The space of attainable diffeomorphisms is G = ψ : ∃T,∃v(t,⋅), v(t,⋅) bounded,ψ = ϕ v (T,⋅) V { V } where v is the flow associated to v.
Load more

Recommended publications
  • Sobolev Spaces, Theory and Applications
    Sobolev spaces, theory and applications Piotr Haj lasz1 Introduction These are the notes that I prepared for the participants of the Summer School in Mathematics in Jyv¨askyl¨a,August, 1998. I thank Pekka Koskela for his kind invitation. This is the second summer course that I delivere in Finland. Last August I delivered a similar course entitled Sobolev spaces and calculus of variations in Helsinki. The subject was similar, so it was not posible to avoid overlapping. However, the overlapping is little. I estimate it as 25%. While preparing the notes I used partially the notes that I prepared for the previous course. Moreover Lectures 9 and 10 are based on the text of my joint work with Pekka Koskela [33]. The notes probably will not cover all the material presented during the course and at the some time not all the material written here will be presented during the School. This is however, not so bad: if some of the results presented on lectures will go beyond the notes, then there will be some reasons to listen the course and at the same time if some of the results will be explained in more details in notes, then it might be worth to look at them. The notes were prepared in hurry and so there are many bugs and they are not complete. Some of the sections and theorems are unfinished. At the end of the notes I enclosed some references together with comments. This section was also prepared in hurry and so probably many of the authors who contributed to the subject were not mentioned.
    [Show full text]
  • Modular Approach on Shape Spaces, Sub-Riemannian Geometry and Computational Anatomy Barbara Gris
    Modular approach on shape spaces, Sub-Riemannian geometry and computational anatomy Barbara Gris To cite this version: Barbara Gris. Modular approach on shape spaces, Sub-Riemannian geometry and computational anatomy. General Mathematics [math.GM]. Université Paris Saclay (COmUE), 2016. English. NNT : 2016SACLN069. tel-01567466v2 HAL Id: tel-01567466 https://tel.archives-ouvertes.fr/tel-01567466v2 Submitted on 6 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 NNT : 2016SACLN069 Thèse de doctorat de l’Université Paris-Saclay préparée École Normale Supérieure de Paris-Saclay Ecole doctorale n◦574 École Doctorale de Mathématiques Hadamard (EDMH, ED 574) Spécialité de doctorat : Mathématiques appliquées par Mme. Barbara Gris Approche modulaire sur les espaces de formes, géométrie sous-riemannienne et anatomie computationnelle Thèse présentée et soutenue à École Normale Supérieure de Paris-Saclay, le 05 décembre 2016. Composition du Jury : M. Laurent YOUNES Professeur (Président du jury) Johns Hopkins University M. Tom FLETCHER Professeur (Rapporteur) University of Utah M. Simon MASNOU Professeur (Rapporteur) Université Lyon 1 Mme Julie DELON Professeure (Examinatrice) Université Paris Descartes M. Emmanuel TRÉLAT Professeur (Examinateur) Université Pierre et Marie Curie M. Stanley DURRLEMAN Chargé de recherche (Directeur de thèse) INRIA M.
    [Show full text]
  • Introduction to Sobolev Spaces
    Introduction to Sobolev Spaces Lecture Notes MM692 2018-2 Joa Weber UNICAMP December 23, 2018 Contents 1 Introduction1 1.1 Notation and conventions......................2 2 Lp-spaces5 2.1 Borel and Lebesgue measure space on Rn .............5 2.2 Definition...............................8 2.3 Basic properties............................ 11 3 Convolution 13 3.1 Convolution of functions....................... 13 3.2 Convolution of equivalence classes................. 15 3.3 Local Mollification.......................... 16 3.3.1 Locally integrable functions................. 16 3.3.2 Continuous functions..................... 17 3.4 Applications.............................. 18 4 Sobolev spaces 19 4.1 Weak derivatives of locally integrable functions.......... 19 1 4.1.1 The mother of all Sobolev spaces Lloc ........... 19 4.1.2 Examples........................... 20 4.1.3 ACL characterization.................... 21 4.1.4 Weak and partial derivatives................ 22 4.1.5 Approximation characterization............... 23 4.1.6 Bounded weakly differentiable means Lipschitz...... 24 4.1.7 Leibniz or product rule................... 24 4.1.8 Chain rule and change of coordinates............ 25 4.1.9 Equivalence classes of locally integrable functions..... 27 4.2 Definition and basic properties................... 27 4.2.1 The Sobolev spaces W k;p .................. 27 4.2.2 Difference quotient characterization of W 1;p ........ 29 k;p 4.2.3 The compact support Sobolev spaces W0 ........ 30 k;p 4.2.4 The local Sobolev spaces Wloc ............... 30 4.2.5 How the spaces relate.................... 31 4.2.6 Basic properties { products and coordinate change.... 31 i ii CONTENTS 5 Approximation and extension 33 5.1 Approximation............................ 33 5.1.1 Local approximation { any domain............. 33 5.1.2 Global approximation on bounded domains.......
    [Show full text]
  • Five Lectures on Optimal Transportation: Geometry, Regularity and Applications
    FIVE LECTURES ON OPTIMAL TRANSPORTATION: GEOMETRY, REGULARITY AND APPLICATIONS ROBERT J. MCCANN∗ AND NESTOR GUILLEN Abstract. In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x, y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Amp`ere type equations. An ap- plication to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically. c 2010 by Robert J. McCann. All rights reserved. Contents Preamble 2 1. An introduction to optimal transportation 2 1.1. Monge-Kantorovich problem: transporting ore from mines to factories 2 1.2. Wasserstein distance and geometric applications 3 1.3. Brenier’s theorem and convex gradients 4 1.4. Fully-nonlinear degenerate-elliptic Monge-Amp`eretype PDE 4 1.5. Applications 5 1.6. Euclidean isoperimetric inequality 5 1.7. Kantorovich’s reformulation of Monge’s problem 6 2. Existence, uniqueness, and characterization of optimal maps 6 2.1. Linear programming duality 8 2.2. Game theory 8 2.3. Relevance to optimal transport: Kantorovich-Koopmans duality 9 2.4. Characterizing optimality by duality 9 2.5. Existence of optimal maps and uniqueness of optimal measures 10 3. Methods for obtaining regularity of optimal mappings 11 3.1. Rectifiability: differentiability almost everywhere 12 3.2. From regularity a.e.
    [Show full text]
  • L P and Sobolev Spaces
    NOTES ON Lp AND SOBOLEV SPACES STEVE SHKOLLER 1. Lp spaces 1.1. Definitions and basic properties. Definition 1.1. Let 0 < p < 1 and let (X; M; µ) denote a measure space. If f : X ! R is a measurable function, then we define 1 Z p p kfkLp(X) := jfj dx and kfkL1(X) := ess supx2X jf(x)j : X Note that kfkLp(X) may take the value 1. Definition 1.2. The space Lp(X) is the set p L (X) = ff : X ! R j kfkLp(X) < 1g : The space Lp(X) satisfies the following vector space properties: (1) For each α 2 R, if f 2 Lp(X) then αf 2 Lp(X); (2) If f; g 2 Lp(X), then jf + gjp ≤ 2p−1(jfjp + jgjp) ; so that f + g 2 Lp(X). (3) The triangle inequality is valid if p ≥ 1. The most interesting cases are p = 1; 2; 1, while all of the Lp arise often in nonlinear estimates. Definition 1.3. The space lp, called \little Lp", will be useful when we introduce Sobolev spaces on the torus and the Fourier series. For 1 ≤ p < 1, we set ( 1 ) p 1 X p l = fxngn=1 j jxnj < 1 : n=1 1.2. Basic inequalities. Lemma 1.4. For λ 2 (0; 1), xλ ≤ (1 − λ) + λx. Proof. Set f(x) = (1 − λ) + λx − xλ; hence, f 0(x) = λ − λxλ−1 = 0 if and only if λ(1 − xλ−1) = 0 so that x = 1 is the critical point of f. In particular, the minimum occurs at x = 1 with value f(1) = 0 ≤ (1 − λ) + λx − xλ : Lemma 1.5.
    [Show full text]
  • The Logarithmic Sobolev Inequality Along the Ricci Flow
    The Logarithmic Sobolev Inequality Along The Ricci Flow (revised version) Rugang Ye Department of Mathematics University of California, Santa Barbara July 20, 2007 1. Introduction 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow 6. The κ-noncollapsing estimate Appendix A. The logarithmic Sobolev inequalities on the euclidean space Appendix B. The estimate of e−tH Appendix C. From the estimate for e−tH to the Sobolev inequality 1 Introduction Consider a compact manifold M of dimension n 3. Let g = g(t) be a smooth arXiv:0707.2424v4 [math.DG] 29 Aug 2007 solution of the Ricci flow ≥ ∂g = 2Ric (1.1) ∂t − on M [0, T ) for some (finite or infinite) T > 0 with a given initial metric g(0) = g . × 0 Theorem A For each σ > 0 and each t [0, T ) there holds ∈ R n σ u2 ln u2dvol σ ( u 2 + u2)dvol ln σ + A (t + )+ A (1.2) ≤ |∇ | 4 − 2 1 4 2 ZM ZM 1 for all u W 1,2(M) with u2dvol =1, where ∈ M R 4 A1 = 2 min Rg0 , ˜ 2 n − CS(M,g0) volg0 (M) n A = n ln C˜ (M,g )+ (ln n 1), 2 S 0 2 − and all geometric quantities are associated with the metric g(t) (e.g. the volume form dvol and the scalar curvature R), except the scalar curvature Rg0 , the modified Sobolev ˜ constant CS(M,g0) (see Section 2 for its definition) and the volume volg0 (M) which are those of the initial metric g0.
    [Show full text]
  • 27. Sobolev Inequalities 27.1
    ANALYSIS TOOLS WITH APPLICATIONS 493 27. Sobolev Inequalities 27.1. Morrey’s Inequality. d 1 d Notation 27.1. Let S − be the sphere of radius one centered at zero inside R . d 1 d For a set Γ S − ,x R , and r (0, ), let ⊂ ∈ ∈ ∞ Γx,r x + sω : ω Γ such that 0 s r . ≡ { ∈ ≤ ≤ } So Γx,r = x + Γ0,r where Γ0,r is a cone based on Γ, seeFigure49below. Γ Γ Figure 49. The cone Γ0,r. d 1 Notation 27.2. If Γ S − is a measurable set let Γ = σ(Γ) be the surface “area” of Γ. ⊂ | | Notation 27.3. If Ω Rd is a measurable set and f : Rd C is a measurable function let ⊂ → 1 fΩ := f(x)dx := f(x)dx. − m(Ω) Ω ZΩ Z By Theorem 8.35, r d 1 (27.1) f(y)dy = f(x + y)dy = dt t − f(x + tω) dσ(ω) Γx,r Γ0,r 0 Z Z Z ZΓ and letting f =1in this equation implies d (27.2) m(Γx,r)= Γ r /d. | | d 1 Lemma 27.4. Let Γ S − be a measurable set such that Γ > 0. For u 1 ⊂ | | ∈ C (Γx,r), 1 u(y) (27.3) u(y) u(x) dy |∇ d | 1 dy. − | − | ≤ Γ x y − ΓZx,r | |ΓZx,r | − | 494 BRUCE K. DRIVER† d 1 Proof. Write y = x + sω with ω S − , then by the fundamental theorem of calculus, ∈ s u(x + sω) u(x)= u(x + tω) ωdt − ∇ · Z0 and therefore, s u(x + sω) u(x) dσ(ω) u(x + tω) dσ(ω)dt | − | ≤ 0 Γ |∇ | ZΓ Z Z s d 1 u(x + tω) = t − dt |∇ d | 1 dσ(ω) 0 Γ x + tω x − Z Z | − | u(y) u(y) = |∇ d | 1 dy |∇ d | 1 dy, y x − ≤ x y − ΓZx,s | − | ΓZx,r | − | wherein the second equality we have used Eq.
    [Show full text]
  • Sobolev Inequalities in Familiar and Unfamiliar Settings
    Sobolev Inequalities in Familiar and Unfamiliar Settings Laurent Salo®-Coste Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial di®erential equations. In fact, they are extremely flexible tools and are useful in many di®erent settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1 Introduction There are few articles that have turned out to be as influential and truly im- portant as S.L. Sobolev 1938 article [93] (the American translation appeared in 1963), where he introduces his famed inequalities. It is the idea of a func- tional inequality itself that Sobolev brings to life in his paper, as well as the now so familiar notion of an a priori inequality, i.e., a functional inequality established under some strong hypothesis and that might be extended later, perhaps almost automatically, to its natural domain of de¯nition. (These ideas are also related to the theory of distributions which did not exist at the time and whose magni¯cent development by L. Schwartz was, in part, anticipated in the work of S.L. Sobolev.) The most basic and important applications of Sobolev inequalities are to the study of partial di®erential equations. Simply put, Sobolev inequalities provide some of the very basic tools in the study of the existence, regularity, and uniqueness of the solutions of all sorts of partial di®erential equations, lin- ear and nonlinear, elliptic, parabolic, and hyperbolic. I leave to others, much better quali¯ed than me, to discuss these beautiful developments.
    [Show full text]
  • Sharp Log–Sobolev Inequalities
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2903{2904 S 0002-9939(98)04406-2 SHARP LOG{SOBOLEV INEQUALITIES OSCAR S. ROTHAUS (Communicated by Palle E. T. Jorgensen) Abstract. We show existence of a wide variety of Log{Sobolev inequalities in which the constant is exactly that required by the Poincar´e inequality which may be inferred from the Log{Sobolev. We are given a smooth compact Riemannian manifold M, intrinsic gradient , and volume element dµ — we assume W.L.O.G. that µ(M)=1—andasmooth∇ positive function m, giving rise to a probability measure mdµ = dm, for which we have a log–Sobolev inequality (LSI): (1) ρ f 2dm f 2 ln f 2dm f 2dm ln f 2dm: |∇ | ≥ | | | | − | | | | ZM ZM ZM ZM It is well known that ρ 2/λ,whereλis the first non–zero eigenvalue of the Schr¨odinger operator ≥ m f ∆f + f ∇ ; → ∇ · m (∆ the usual Laplacian) attached to the Dirichlet form in (1). We call the LSI sharp if ρ =2/λ. Many examples of sharp inequalities are known, the most familiar arising from M the sphere with the usual metric scaled to give M unit volume, and m =1. We will show here that for every compact homogeneous Riemannian manifold, there are a continuum of choices of m for which sharp LSI’s exist. We follow the notation and conclusions of [1], which we now briefly review. For every M as described initially there is a least constant ρ0(M), the hypercontractive constant for M, such that (2) ρ (M) f 2dµ f 2 ln f 2dµ f 2dµ ln f 2dµ.
    [Show full text]
  • Sobolev's Inequality Under a Curvature-Dimension Condition
    Sobolev’s inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer To cite this version: Louis Dupaigne, Ivan Gentil, Simon Zugmeyer. Sobolev’s inequality under a curvature-dimension condition. 2021. hal-03006155v3 HAL Id: hal-03006155 https://hal.archives-ouvertes.fr/hal-03006155v3 Preprint submitted on 19 Jan 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sobolev's inequality under a curvature-dimension condition Louis Dupaigne, Ivan Gentil, Simon Zugmeyer January 19, 2021 R´esum´e Dans cette note, nous proposons une nouvelle preuve de l'in´egalit´ede Sobolev sur les vari´et´es`a courbure de Ricci minor´eepar une constante positive. Le r´esultatavait ´et´eobtenu en 1983 par Ilias. Nous pr´esentons une preuve tr`escourte de ce th´eor`eme, dressons l'´etatde l'art pour cette fameuse in´egalit´eet expliquons en quoi notre m´ethode, qui repose sur un flot de gradient, est simple et robuste. En particulier, nous ´elucidonsles calculs utilis´esdans des travaux pr´ec´edents, `acommencer par un c´el`ebrearticle de Bidaut-V´eron et V´eronpubli´een 1991. Abstract In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature.
    [Show full text]
  • Entorhinal and Transentorhinal Atrophy in Mild Cognitive Impairment Using Longitudinal Diffeomorphometry
    Alzheimer’s& Dementia: Diagnosis, Assessment & Disease Monitoring 9 (2017) 41-50 Neuroimaging Entorhinal and transentorhinal atrophy in mild cognitive impairment using longitudinal diffeomorphometry Daniel J. Twarda,b,*, Chelsea S. Sicata, Timothy Browna, Arnold Bakkerc, Michela Gallagherd, Marilyn Alberte, Michael Millera,b,f, for the Alzheimer’s Disease Neuroimaging Initiative aCenter for Imaging Science, Johns Hopkins University, Baltimore, MD, USA bDepartment of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA cDepartment of Psychiatry and Behavioral Sciences, Johns Hopkins University, Baltimore, MD, USA dDepartment of Psychological and Brain Sciences, Johns Hopkins School of Arts and Sciences, Baltimore, MD, USA eDepartment of Neurology, Johns Hopkins University School of Medicine, Baltimore, MD, USA fKavli Neuroscience Discovery Institute, Johns Hopkins University, Baltimore, MD, USA Abstract Introduction: Autopsy findings have shown the entorhinal cortex and transentorhinal cortex are among the earliest sites of accumulation of pathology in patients developing Alzheimer’s disease. Methods: Here, we study this region in subjects with mild cognitive impairment (n 5 36) and in control subjects (n 5 16). The cortical areas are manually segmented, and local volume and shape changes are quantified using diffeomorphometry, including a novel mapping procedure that reduces variability in anatomic definitions over time. Results: We find significant thickness and volume changes localized to the transentorhinal cortex through high field strength atlasing. Discussion: This demonstrates that in vivo neuroimaging biomarkers can detect these early changes among subjects with mild cognitive impairment. Ó 2017 The Authors. Published by Elsevier Inc. on behalf of the Alzheimer’s Association. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/ 4.0/).
    [Show full text]
  • Gagliardo-Nirenberg Inequalities on Manifolds
    GAGLIARDO-NIRENBERG INEQUALITIES ON MANIFOLDS NADINE BADR Abstract. We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs. Contents 1. Introduction 1 2. Preliminaries 5 2.1. Besov and Morrey spaces 5 2.2. Sobolev spaces on Riemannian manifolds 6 2.3. Doubling property and Poincar´einequalities 7 3. Ledoux's and Sobolev inequalities 8 3.1. The classical Sobolev inequality 9 3.2. Sobolev inequalities for Lorentz spaces 10 4. Proof of Theorem 1.1, 1.6, 1.8 and 1.9 11 References 13 1. Introduction Cohen-Meyer-Oru [5], Cohen-Devore-Petrushev-Xu [4], proved the following Gagliardo- Nirenberg type inequality n−1 1 n n (1.1) kfk1∗ ≤ Ck jrfj k1 kfk −(n−1) B1;1 1 n ∗ n for all f 2 W1 (R ) (1 = n−1 ). The proof of (1.1) is involved and based on wavelet decompositions, weak type (1,1) estimates and interpolation results. Using a simple method relying on weak type estimates and pseudo-Poincar´ein- equalities, Ledoux [14] obtained the following extension of (1.1). He proved that for 1 n 1 ≤ p < l < 1 and for every f 2 Wp (R ) θ 1−θ (1.2) kfkl ≤ Ck jrfj kpkfk θ θ−1 B1;1 p where θ = l and C > 0 only depends on l; p and n. In the same paper, he extended (1.2) to the case of Riemannian manifolds. If p = 2 he observed that (1.2) holds without any assumption on M. If p 6= 2 he assumed that the Ricci curvature is non-negative and obtained (1.2) with C > 0 only depending on l; p when 1 ≤ p ≤ 2 and on l; p and n when 2 < p < 1.
    [Show full text]