Geometry of diffeomorphism groups and shape matching

We introduce a differential-geometric structure for spectral density functions of discrete-time random processes. This is quite analogous to the Riemannian structure of information geometry, which is used to study perturbations of probability density functions, and which is based on the Fisher information metric. Herein, we introduce an analogous Riemannian metric, which we motivate with a problem in prediction theory. It turns out that this problem also provides a prediction theoretic interpretation to the Itakura distortion measure, which relates to our metric. Geodesies and geodesic distances are characterized in closed form and, hence, the geodesic distance between two spectral density functions provides an explicit, intrinsic (pseudo)metric on the cone of density functions. Certain other distortion measures that involve generalized means of spectral density functions are shown to lead to the same Riemannian metric. Finally, an alternative Riemannian metric is introduced, which is motivated by an analogous problem involving smoothing instead of prediction.

Nonlinear data projection based on geodesic distances requires the construction of a neighborhood graph that spans all data points so that the geodesic distance between any pair of data points could be estimated by the graph distance between the pair. This paper proposes an approach for constructing a k-edge connected neighborhood graph. The approach works by repeatedly extracting minimum spanning trees from the complete Euclidean graph of all data points. The constructed neighborhood graph has the following properties: (1) it is k-connected; (2) each point connects to its k-nearest neighbors; (3) if the graph is cut into two partitions, the cut edges contain k-shortest edges between the two partitions. Experiments show that the presented approach works well for clustered data and outperforms the nearest neighbor approaches used in Isomap for evenly distributed data.

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The geodesic flow is non-complete because geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the geodesic flow (Theorem 5.1). It immediately follows that these metrics have dense geodesics in the unit tangent bundle SM . We also prove that the closed geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the geodesic flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the geodesic flow we consider provides a good model for the WP geodesic flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP geodesic flow. See Section 7. The geometric properties of both types of flows imply that the geodesic flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these geodesic flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the geodesic flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there

Compared with 2D face recognition systems, 3D facial recognition has been a popular area of study in academia and industry, because of its ability to detect human faces in an unconstrained environment more efficiently. This study presents a new 3D face identification method that uses 3D facial images to compute geodesic distances using 3D landmarks. Feature extraction is computationally more efficient than Euclidean distance using geodesic distance. This study aims to demonstrate how to differentiate identical twins based solely on facial features by developing an efficient algorithm for calculating geodesic paths and distances. We employed the A*, Dijkstra, and fast marching (FM) algorithms to calculate the geodesic distance. Quantitative similarity metrics are obtained and then utilized as inputs in numerous classification methods: random forest (RF), extra tree classifier (ETC), light gradient boosting machine (LGBM), support vector machine (SVM), and bagging classifiers. An expression challenge dataset for 3D twins (3D-TEC), the most challenging data set for 3D face recognition research at Notre Dame University, was used to validate and test the performance of these algorithms. The recall, accuracy, precision, training time, and F1 score were compared for each classifier to determine the best performance. The results from the experiment indicate that, when compared to other algorithms and classification models, the Extra Tree Classifier is the most accurate (by 90%) in predicting face images, whereas A* is the best technique for computing geodesic distance.

Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms

We consider a [Formula: see text] free action [Formula: see text] on an [Formula: see text]-dimensional [Formula: see text] compact connected manifold [Formula: see text]. The manifold [Formula: see text] is foliated by circles if all orbits induced by [Formula: see text] are closed. We handle the periods of circle foliations, especially when the circles are closed geodesics with respect to some Riemannian metric. Then, it is known that the periods of [Formula: see text] are exactly the same if and only if there exists a Riemannian metric on [Formula: see text] with respect to which the circles are embedded as totally geodesic submanifolds of [Formula: see text] [A. W. Wadsley, Geodesic foliations by circles, J. Differ. Geom. 10 (1975) 541–549.]. It automatically follows a classical result [A. Besse, Manifolds All of Whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 93 (Springer-Verlag, New York, 1978)] that all periods are exactly the same if all the geodesic flows on the cosphere bundle [Formula: see text] are periodic (the so-called “[Formula: see text]-manifold”). Our purpose here is to adapt these results to Reeb or Hamiltonian flows on contact manifolds. Then [Formula: see text]-circles are considered as totally geodesic submanifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka–Volterra systems, and others. Lotka–Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Schrödinger operators [Formula: see text] If the spectrum of [Formula: see text] near [Formula: see text] consists of eigenvalues, we denote such a set as [Formula: see text] for small [Formula: see text]. For fixed constants [Formula: see text] and [Formula: see text], the cluster of the spectrum near [Formula: see text] (the so-called essential difference spectrum) is defined by [Formula: see text] where the notation [Formula: see text] denotes the set of cluster (accumulation) points. Under suitable assumptions, we find that [Formula: see text] In the former case, at least one of the Hamiltonian flows is not periodic on the energy surface [Formula: see text]. In the latter case, all Hamiltonian flows on [Formula: see text] are periodic with a common period [Formula: see text]. This fact represents one of the semiclassical analogies of the Helton-Guillemin theorem [V. Guillemin, Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977) 487–517; W. Helton, An operator algebra approach to partial differential equations; Propagation of singularities and spectral theory, Indiana Univ. Math. J. 26 (1977) 997–1018.].

Solving the initial value problem of discrete geodesics

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.

We show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.

The geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom whose Hamiltonian is quadratic in the momenta. Because of the homogeneity, every integral of the geodesic problem is a function of integrals that are polynomial in the momenta. The geodesic flow on a surface of genus greater than one does not admit an additional nonconstant integral at all, but on the other hand there are numerous examples of metrics on a torus whose geodesic flows are completely integrable: there are polynomial integrals of degree ≤2 that are independent of the Hamiltonian. It appears that the degree of an additional “irreducible” polynomial integral of a geodesic flow on a torus cannot exceed two. In the present paper this conjecture is proved for metrics which can arbitrarily closely approximate any metric on a two-dimensional torus. Bibliography: 12 titles.

For a given smooth compact manifold $ M $, we introduce an open class $ \mathcal G(M) $ of Riemannian metrics, which we call metrics of the gradient type. For such metrics $ g $, the geodesic flow $ v^g $ on the spherical tangent bundle $ SM \to M $ admits a Lyapunov function (so the $ v^g $-flow is traversing). It turns out, that metrics of the gradient type are exactly the non-trapping metrics.For every $ g \in \mathcal G(M) $, the geodesic scattering along the boundary $ \partial M $ can be expressed in terms of the scattering map $ C_{v^g}: \partial_1^+(SM) \to \partial_1^-(SM) $. It acts from a domain $ \partial_1^+(SM) $ in the boundary $ \partial(SM) $ to the complementary domain $ \partial_1^-(SM) $, both domains being diffeomorphic. We prove that, for a boundary generic metric $ g \in \mathcal G(M) $, the map $ C_{v^g} $ allows for a reconstruction of $ SM $ and of the geodesic foliation $ \mathcal F(v^g) $ on it, up to a homeomorphism (often a diffeomorphism).Also, for such $ g $, the knowledge of the scattering map $ C_{v^g} $ makes it possible to recover the homology of $ M $, the Gromov simplicial semi-norm on it, and the fundamental group of $ M $. Additionally, $ C_{v^g} $ allows to reconstruct the naturally stratified topological type of the space of geodesics on $ M $.We aim to understand the constraints on $ (M, g) $, under which the scattering data allow for a reconstruction of $ M $ and the metric $ g $ on it, up to a natural action of the diffeomorphism group $ \mathsf{Diff}(M, \partial M) $. In particular, we consider a closed Riemannian $ n $-manifold $ (N, g) $ which is locally symmetric and of negative sectional curvature. Let $ M $ is obtained from $ N $ by removing an $ n $-domain $ U $, such that the metric $ g|_M $ is boundary generic, of the gradient type, and the homomorphism $ \pi_1(U) \to \pi_1(N) $ of the fundamental groups is trivial. Then we prove that the scattering map $ C_{v^{g|_M}} $ makes it possible to recover $ N $ and the metric $ g $ on it.

The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action are illustrated through the Camassa-Holm equation (CH). In 1996 Misio lek observed that CH is a geodesic flow equation on the group of diffeomorphisms, preserving the H 1 metric. This example is analogous to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The momentum map and an explicit parametrization of the Virasoro group, related to recently obtained solutions for the CH equation are presented.

Detecting hands in multi-modal RGB-Depth visual data has become a challenging Computer Vision problem with several applications of interest. This task involves dealing with changes in illumination, view point variations, the articulated nature of the human body, the high flexibility of the wrist articulation, and the deformability of the hand itself. In this work, we propose an accurate and efficient automatic hand detection scheme to be applied in Human-Computer Interaction (HCI) applications in which the user is seated at the desk and, thus, only the upper body is visible. Our main hypothesis is that hand landmarks remain at a nearly constant geodesic distance from an automatically located anatomical reference point. In a given frame, the human body is segmented first in the depth image. Then, a graph representation of the body is built in which the geodesic paths are computed from the reference point. The dense optical flow vectors on the corresponding RGB image are used to reduce ambiguities of the geodesic paths' connectivity, allowing to eliminate false edges interconnecting different body parts. Finally, we are able to detect the position of both hands based on invariant geodesic distances and optical flow within the body region, without involving costly learning procedures.

Let (M, g) be a Riemannian manifold with boundary. In this paper we consider the problem of determining the metric g from its associated boundary distance function dg(x, y) = dist (x, y), x, y ∈ ∂M, that is the geodesic distance between boundary points. This problem arose in rigidity questions in Riemannian geometry [12], [6], [8]. For the case in which M is a bounded domain of Euclidean space and the metric is conformal to the Euclidean one, this problem is called the inverse kinematic problem which arose in Geophysics and has a long history starting at least in the early part of the 20th century with Herglotz [10]. He considered the case where M is a ball {x ∈ R | r = |x| ≤ R} equipped with a spherically symmetric metric ds2 = dx2/c2(r) where c(r) is a positive function depending only on the radius r = |x|. Herglotz found a formula to determine c(r) from the boundary distance function. Physically this corresponds to the case of a spherically symmetric Earth model with an index of refraction depending only on the radius. The boundary distance function corresponds to the travel times of e.g. acoustic waves going through the Earth and measured at the surface. The general problem for the case that the sound speed depends on all variables has been extensively studied (see for instance [17] and the references given there). Also, this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements, see [20].