Space Of Diffeomorphisms Research Articles

Stable isotopy connectivity of gradient-like diffeomorphisms of 2-torus

One of the most important problems in the theory of dynamical systems (mentioned in the Palis-Pugh list) is the construction of a stable arc between structural stable diffeomorphisms in the space of diffeomorphisms. The paper considers the gradient-like diffeomorphisms of 2-torus that induce an isomorphism of fundamental groups determined by a matrix (−100−1). We prove that all such diffeomorphisms are stable isotopy connected.

Construction of Smooth Source–Sink Arcs in the Space of Diffeomorphisms of a Two-Dimensional Sphere

Construction of Smooth Source–Sink Arcs in the Space of Diffeomorphisms of a Two-Dimensional Sphere

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

Space-feature measures on meshes for mapping spatial transcriptomics

Advances in the development of largely automated microscopy methods such as MERFISH for imaging cellular structures in mouse brains are providing spatial detection of micron resolution gene expression. While there has been tremendous progress made in the field of Computational Anatomy (CA) to perform diffeomorphic mapping technologies at the tissue scales for advanced neuroinformatic studies in common coordinates, integration of molecular- and cellular-scale populations through statistical averaging via common coordinates remains yet unattained. This paper describes the first set of algorithms for calculating geodesics in the space of diffeomorphisms, what we term space-feature-measure LDDMM, extending the family of large deformation diffeomorphic metric mapping (LDDMM) algorithms to accommodate a space-feature action on marked particles which extends consistently to the tissue scales. It leads to the derivation of a cross-modality alignment algorithm of transcriptomic data to common coordinate systems attached to standard atlases.We represent the brain data as geometric measures, termed as space-feature measures supported by a large number of unstructured points, each point representing a small volume in space and carrying a list of densities of features elements of a high-dimensional feature space. The shape of space-feature measure brain spaces is measured by transforming them by diffeomorphisms. The metric between these measures is obtained after embedding these objects in a linear space equipped with the norm, yielding a so-called “chordal metric”.

Equivalences between 2D dilaton gravities, their asymptotic symmetries, and their holographic duals

Dilaton gravities in two dimensions can be formulated as particular Poisson sigma models. Target space diffeomorphisms map different models to each other and establish a one-to-one correspondence between their classical solutions. We obtain a general form of such diffeomorphisms in Lorentzian and Euclidean signatures and use them to extend known holographic results, including the Schwarzian action on the asymptotic boundary, from JT to a large class of dilaton gravity models.

Target space diffeomorphisms in Poisson sigma models and asymptotic symmetries in 2D dilaton gravities

The dilaton gravity models in two dimensions, including the Jackiw--Teitelboim model and its deformations, are particular cases of Poisson sigma models. Target space diffeomorphisms map one Poisson sigma model to another. We propose to use these diffsomorphisms to identify asymptotic conditions, boundary actions, and asymptotic symmetries in distinct dilaton gravity models. As an example, we use the asymptotic conditions in Jackiw--Teitelboim gravity to construct an asymptotic problem with Virasoro symmetry in a class of asymptotically Rindler models. We show, that the method can be applied to a wide class of pairs of dilaton gravities and discuss possible generalizations.

Accelerated Optimization in the PDE Framework: Formulations for the Manifold of Diffeomorphisms

We consider the problem of optimization of cost functionals on the infinite dimensional manifold of diffeomorphisms. We present a new class of optimization methods, valid for any optimization problem setup on the space of diffeomorphisms by generalizing Nesterov accelerated optimization to the manifold of diffeomorphisms. While our framework is general for infinite dimensional manifolds, we specifically treat the case of diffeomorphisms, motivated by optical flow problems in computer vision. This is accomplished by building on a recent variational approach to a general class of accelerated optimization methods by Wibisono, Wilson, and Jordan [Proc. Natl. Acad. Sci. USA, 113 (2016), pp. E7351--E7358], which applies in finite dimensions. We generalize that approach to infinite dimensional manifolds. We derive the surprisingly simple continuum evolution equations, which are partial differential equations, for accelerated gradient descent, and relate them to simple mechanical principles from fluid mechanics. Our approach has natural connections to the optimal mass transport problem. This is because one can think of our approach as an evolution of an infinite number of particles endowed with mass (represented with a mass density) that moves in an energy landscape. The mass evolves with the optimization variable and endows the particles with dynamics. This is different from the finite dimensional case where only a single particle moves and hence the dynamics does not depend on the mass. We derive the theory, compute the PDEs for accelerated optimization, and illustrate the behavior of these new accelerated optimization schemes.

Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a surface diffeomorphism

Abstract It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy type of the closure of the invariant manifold of a saddle point of a polar diffeomorphism of a 2-torus to any prescribed homotopically nontrivial type. The arc constructed in the process is stable and does not change the topological conjugacy class of the original diffeomorphism. The ideas that are proposed here for constructing such an arc for a 2-torus can naturally be generalized to surfaces of greater genus. Bibliography: 32 titles.

Efficient momentum conservation constrained PDE-LDDMM with Gauss–Newton–Krylov optimization, Semi-Lagrangian Runge–Kutta solvers, and the band-limited parameterization

This paper proposes three efficient variants of Jacobi EPDiff PDE-LDDMM, where efficiency is achieved through Semi-Lagrangian Runge–Kutta integration and the band-limited parameterization. During Gauss–Newton–Krylov optimization, the method computes the gradient and the Hessian-vector product on the final time sample, and transports these magnitudes towards the initial time using the adjoint Jacobi equations and their incremental counterparts. Then, the optimization is performed on the initial time sample. The proposed methods have effectively achieved a considerable reduction of the computational complexity at a competitive accuracy. These variants constitute a contribution to the efficient computation of diffeomorphisms belonging to geodesics, suitable for statistical shape analysis or the construction of transversal and longitudinal models of shape variability using Principal Geodesic Analysis and Geodesic Regression in spaces of diffeomorphisms.

Divergence-Free Fitting-Based Incompressible Deformation Quantification of Liver.

Liver is an incompressible organ that maintains its volume during the respiration-induced deformation. Quantifying this deformation with the incompressible constraint is significant for liver tracking. The constraint can be accomplished with retaining the divergence-free field obtained by the deformation decomposition. However, the decomposition process is time-consuming, and the removal of non-divergence-free field weakens the deformation. In this study, a divergence-free fitting-based registration method is proposed to quantify the incompressible deformation rapidly and accurately. First, the deformation to be estimated is mapped to the velocity in a diffeomorphic space. Then, this velocity is decomposed by a fast Fourier-based Hodge-Helmholtz decomposition to obtain the divergence-free, curl-free, and harmonic fields. The curl-free field is replaced and fitted by the obtained harmonic field with a translation field to generate a new divergence-free velocity. By optimizing this velocity, the final incompressible deformation is obtained. Moreover, a deep learning framework (DLF) is constructed to accelerate the incompressible deformation quantification. An incompressible respiratory motion model is built for the DLF by using the proposed registration method and is then used to augment the training data. An encoder-decoder network is introduced to learn appearance-velocity correlation at patch scale. In the experiment, we compare the proposed registration with three state-of-the-art methods. The results show that the proposed method can accurately achieve the incompressible registration of liver with a mean liver overlap ratio of 95.33%. Moreover, the time consumed by DLF is nearly 15 times shorter than that by other methods.

Nonstationary space–time covariance functions induced by dynamical systems

AbstractThis article provides a novel approach to nonstationarity by considering a bridge between differential equations and spatial fields. We consider the dynamical transformation of a given spatial process undergoing the action of a temporal flow of space diffeomorphisms. Such dynamical deformations are shown to be connected to certain classes of ordinary and partial differential equations. The natural question arises of how such dynamical diffeomorphisms convert the original spatial covariance function, specifically if the original covariance is spatially stationary or isotropic. We first challenge this question from a general perspective, and then turn into the special cases of both d‐dimensional Euclidean spaces, and hyperspheres. Several examples of dynamical diffeomorphisms defined in these spaces are given and some emphasis has been put on the stationary reducibility problem. We provide a simple illustration to show the performance of the maximum likelihood estimation of the parameters of a family of dynamically deformed covariance functions.

Automated Superpixel-borders-guided Deformation Image Registration for Adaptive Radiotherapy

Registration between the baseline and follow-up lung computed tomography (CT) volumes plays an important role in computer-aided diagnosis and following-up care during adaptive radiotherapy. Diffeomorphic log-Demons as state of the art in Demons implementations is restricted to relatively small deformations, low accuracy, and ignoring some prior structural features. In this paper, an automated superpixel-borders-guided deformation image registration (SBG-DIR) algorithm is proposed. The proposed SBG-DIR method uses the simple linear iterative clustering (SLIC) algorithm to automated superpixels generation. Incorporation of superpixel borders into registration algorithm is implemented by a new similarity criterion based on the binary volume representation of superpixel borders. The binary volume representation enables accurate preserving motion boundaries, contributes to a faster convergence of the objective function and eliminates errors caused by manual interaction. In addition, a subtraction volume is produced by the intensity difference between the first time point CT volume and its warped follow-up CT volume. The subtraction volume can be used for detection of tumor tissue growth or shrinkage, which is an essential part of a CT-based diagnosis. Moreover, to ensure the topology preservation of biological objects, our proposed SBG-DIR method is implemented in the space of diffeomorphisms, in which meaningful biological shapes can be found. Compared with the state-of-the-art Demons, the proposed SBG-DIR method doesn’t require any additional optimization, yields a faster convergence and is more accurate and efficient in recovering large deformations. Experimental results indicate that the proposed SBG-DIR method performed better than the state-of-the-art Demons algorithms.

A scenario of the homotopy type changing of the invariant saddle manifold closure

The paper deals with surface gradient-like diffeomorphisms. The closures of the invariant manifolds of saddle points of such systems contain nodal points in their closure. In the case when there is only one such point, the closure of the invariant manifold is a closed curve that is homeomorphic to the circle. In a general case the conjugating homeomorphism changes the homotopy type of the closed curve, while the diffeomorphisms themselves may remain in the same isotopic class. This means that in the space of diffeomorphisms two such systems are connected by an arc, but every such arc necessarily undergoes bifurcations. In this paper, we describe a scenario for changing the homotopy type of the closure of the invariant saddle manifold. Moreover, the constructed arc is stable in the space of diffeomorphisms.

Fractional Sobolev metrics on spaces of immersions

We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It extends several previous results and applies to a wide range of variational partial differential equations, including the well-known Euler–Arnold equations on diffeomorphism groups as well as the geodesic equations on spaces of manifold-valued curves and surfaces.

Teichmüller space of circle diffeomorphisms with Hölder continuous derivatives

Based on the quasiconformal theory of the universal Teichmüller space, we introduce the Teichmüller space of diffeomorphisms of the unit circle with \alpha -Hölder continuous derivatives as a subspace of the universal Teichmüller space. We characterize such a diffeomorphism quantitatively in terms of the complex dilatation of its quasiconformal extension and the Schwarzian derivative given by the Bers embedding. Then, we provide a complex Banach manifold structure for it and prove that its topology coincides with the one induced by local C^{1+\alpha} -topology at the base point.

Kinematical gravitational charge algebra

When formulated in terms of connection and coframes, and in the time gauge, the phase space of general relativity consists of a pair of conjugate fields: the flux 2-form and the Ashtekar connection. On this phase-space, one has to impose the Gauss constraints, the vector, and scalar Hamiltonian constraints. These are respectively generating local SU(2) gauge transformations, spatial diffeomorphisms, and time diffeomorphisms. We write the Gauss and space diffeomorphism constraints as conservation laws for a set of boundary charges, representing spin and momenta, respectively. We prove that these kinematical charges generate a local Poincar\'e ISU(2) symmetry algebra. This gives strong support to the recent proposal of Poincar\'e charge networks as a new realm for discretized general relativity [Classical Quantum Gravity 36, 195014 (2019)].

Gravitational edge modes: from Kac–Moody charges to Poincaré networks

We revisit the canonical framework for general relativity in its connection-vierbein formulation, recasting the Gauss law, the Bianchi identity and the space diffeomorphism bulk constraints as conservation laws for boundary surface charges, respectively electric, magnetic and momentum charges. Partitioning the space manifold into 3D regions glued together through their interfaces, we focus on a single domain and its punctured 2D boundary. The punctures carry a ladder of Kac–Moody edge modes, whose 0-modes represent the electric and momentum charges while the higher modes describe the stringy vibration modes of the 1D-boundary around each puncture. In particular, this allows to identify missing observables in the discretization scheme used in loop quantum gravity (LQG) and leads to an enhanced theory upgrading spin networks to tube networks carrying Virasoro representations. In the limit where the tubes are contracted to 1D links and the string modes neglected, we do not just recover LQG but obtain a more general structure: Poincaré charge networks, which carry a representation of the 3D diffeomorphism boundary charges on top of the fluxes and gauge transformations.

Mimetic Hořava gravity

We show that the scalar field of mimetic gravity could be used to construct diffeomorphism invariant models that reduce to Hořava gravity in the synchronous gauge. The gradient of the mimetic field provides a timelike unit vector field that allows to define a projection operator of four-dimensional tensors to three-dimensional spatial tensors. Conversely, it also enables us to write quantities invariant under space diffeomorphisms in fully covariant form without the need to introduce new propagating degrees of freedom.

Geometric hydrodynamics via Madelung transform

We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important partial differential equations of hydrodynamical origin can be described in this framework in a natural way. In particular, the Madelung transform between the Schrödinger equation and Newton's equations is a symplectomorphism of the corresponding phase spaces. Furthermore, the Madelung transform turns out to be a Kähler map when the space of densities is equipped with the Fisher-Rao information metric. We describe several dynamical applications of these results.

Area-preserving diffeomorphisms of the hyperbolic plane and K-surfaces in anti-de Sitter space

We prove that any weakly acausal curve Γ in the boundary of anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike K-surfaces, one of which is past-convex and the other future-convex, for every K ∈ ( − ∞ , − 1 ) . The curve Γ is the graph of a quasisymmetric homeomorphism of the circle if and only if the K-surfaces have bounded principal curvatures. Moreover in this case a uniqueness result holds. The proofs rely on a well-known correspondence between spacelike surfaces in anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane. In fact, an important ingredient is a representation formula, which reconstructs a spacelike surface from the associated area-preserving diffeomorphism. Using this correspondence we then deduce that, for any fixed θ ∈ ( 0 , π ) , every quasisymmetric homeomorphism of the circle admits a unique extension which is a θ-landslide of the hyperbolic plane. These extensions are quasiconformal.