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Philipp Harms

Publications and source records attributed to Philipp Harms.

35 records · Page 2Linked to original sources
arXiv2016

A Numerical Framework for Sobolev Metrics on the Space of Curves

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.

math.DG↗
arXiv2016

Consistent Recalibration of Yield Curve Models

The analytical tractability of affine (short rate) models, such as the Vasicek and the Cox-Ingersoll-Ross models, has made them a popular choice for modelling the dynamics of interest rates. However, in order to account properly for the dynamics of real data, these models need to exhibit time-dependent or even stochastic parameters. This in turn breaks their tractability, and modelling and simulating becomes an arduous task. We introduce a new class of Heath-Jarrow-Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. These CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull-White extended affine factor model with possibly different parameters. That is, we construct HJM models from "tangent" affine models. We develop a theory for a continuous path version of such models and discuss their numerical implementations within the Vasicek and Cox-Ingersoll-Ross frameworks.

q-fin.MF↗
arXiv2016

Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model

The discrete-time multifactor Vasiček model is a tractable Gaussian spot rate model. Typically, two- or three-factor versions allow one to capture the dependence structure between yields with different times to maturity in an appropriate way. In practice, re-calibration of the model to the prevailing market conditions leads to model parameters that change over time. Therefore, the model parameters should be understood as being time-dependent or even stochastic. Following the consistent re-calibration (CRC) approach, we construct models as concatenations of yield curve increments of Hull-White extended multifactor Vasiček models with different parameters. The CRC approach provides attractive tractable models that preserve the no-arbitrage premise. As a numerical example, we fit Swiss interest rates using CRC multifactor Vasiček models.

q-fin.MF↗
arXiv2015

Hörmander's condition for normal bundles on spaces of immersions

Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a selection of bundles which naturally occur in applications that they are either bracket generating or integrable.

math.DG↗
arXiv2015

Second order elastic metrics on the shape space of curves

Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. The combination of these algorithms allows to compute Karcher means in a Riemannian gradient-based optimization scheme. Our framework has the advantage that the constants determining the weights of the zero, first, and second order terms of the metric can be chosen freely. Moreover, due to its generality, it could be applied to more general spaces of mapping. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing physical objects.

math.DG↗
arXiv2015

Curve Matching with Applications in Medical Imaging

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical properties. We propose numerical solutions for the initial and boundary value problems of finding geodesics. These two methods are combined in a Riemannian gradient-based optimization scheme to compute the Karcher mean. We apply this to a study of the shape variation in HeLa cell nuclei and cycles of cardiac deformations, by computing means and principal modes of variations.

math.DG↗
arXiv2014

Metrics on Spaces of Immersions where Horizontality Equals Normality

We study metrics on shape space of immersions that have a particularly simple horizontal bundle. More specifically, we consider reparametrization invariant Sobolev metrics $G$ on the space $\operatorname{Imm}(M,N)$ of immersions of a compact manifold $M$ in a Riemannian manifold $(N,\overline{g})$. The tangent space $T_f\operatorname{Imm}(M,N)$ at each immersion $f$ has two natural splittings: one into components that are tangential/normal to the surface $f$ (with respect to $\overline{g}$) and another one into vertical/horizontal components (with respect to the projection onto the shape space $B_i(M,N)=\operatorname{Imm}(M,N)/\operatorname{Diff}(M)$ of unparametrized immersions and with respect to the metric $G$). The first splitting can be easily calculated numerically, while the second splitting is important because it mirrors the geometry of shape space and geodesics thereon. Motivated by facilitating the numerical calculation of geodesics on shape space, we characterize all metrics $G$ such that the two splittings coincide. In the special case of planar curves, we show that the regularity of curves in the metric completion can be controlled by choosing a strong enough metric within this class. We demonstrate in several examples that our approach allows us to efficiently calculate numerical solutions of the boundary value problem for geodesics on shape space.

math.DG↗
arXiv2013

Almost local metrics on shape space of hypersurfaces in n-space

This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$, or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the group of diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $$ G_f(h,k) = \int_{M} Φ(\on{Vol}(f),\operatorname{Tr}(L))\g(h, k) \operatorname{vol}(f^*\g),$$ where $\g$ is the Euclidean metric on $\mathbb R^n$, $f^*\g$ is the induced metric on $M$, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$ to the space of embeddings or immersions, where $Φ:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $\operatorname{Vol}(f) = \int_M\operatorname{vol}(f^*\g)$ is the total hypersurface volume of $f(M)$, and the trace $\operatorname{Tr}(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $Φ$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.

math.DG↗
arXiv2012

Sobolev Metrics on Shape Space, II: Weighted Sobolev Metrics and Almost Local Metrics

In continuation of [3] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^pΦ_i(\Vol(f)) \g((P_i)_fh,k) \vol(f^*\g) $$ on the space of immersions $\Imm(M,N)$ and on shape space $B_i(M,N)=\Imm(M,N)/\on{Diff}(M)$. Here $(N,\g)$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\g$ is the induced Riemannian metric on $M$, $\vol(f^*\g)$ is the induced volume density on $M$, $\Vol(f)=\int_M\vol(f^*\g)$, $Φ_i$ are positive real-valued functions, and $(P_i)_f$ are operators like some power of the Laplacian $Δ^{f^*\g}$. We derive the geodesic equations for these metrics and show that they are sometimes well-posed with the geodesic exponential mapping a local diffeomorphism. The new aspect here are the weights $\Ph_i(\Vol(f))$ which we use to construct scale invariant metrics and order 0 metrics with positive geodesic distance. We treat several concrete special cases in detail.

math.DG↗
arXiv2012

Sobolev metrics on the manifold of all Riemannian metrics

On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as described first by \cite{Ebin70}. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

math.DG↗
arXiv2012

Sobolev metrics on shape space of surfaces

Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geometry with geodesics, gradient flows and curvature. Unfortunately, the simplest such metric induces vanishing path-length distance on shape space. This discovery by Michor and Mumford was the starting point to a quest for stronger, meaningful metrics that should be able to distinguish salient features of the shapes. Sobolev metrics are a very promising approach to that. They come in two flavors: Outer metrics which are induced from metrics on the diffeomorphism group of ambient space, and inner metrics which are defined intrinsically to the shape. In this work, Sobolev inner metrics are developed and treated in a very general setting. There are no restrictions on the dimension of the immersed space or of the ambient space, and ambient space is not required to be flat. It is shown that the path-length distance induced by Sobolev inner metrics does not vanish. The geodesic equation and the conserved quantities arising from the symmetries are calculated, and well-posedness of the geodesic equation is proven. Finally examples of numerical solutions to the geodesic equation are presented.

math.DG↗
arXiv2012

The Poincaré Lemma in Subriemannian Geometry

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then the question arises: Can any two points be connected by a horizontal curve? Chow's Theorem gives an affirmative answer for bracket generating distributions. (A distribution is called bracket generating if horizontal vector fields and their iterated Lie brackets span the whole tangent space.) We present three different proofs of Chow's Theorem; each one is interesting in its own. The first proof is based on the theory of Stefan and Sussmann regarding integrability of singular distributions. The second proof is elementary and gives some insight in the shape of subriemannian balls. The third proof is based on infinite dimensional analysis of the endpoint map. Finally, the study of the endpoint map allows us to prove a regularity result for the Poincaré Lemma in a form suited to subriemannian geometry: If for some $r \geq 0$ all horizontal derivatives of a given function $f$ are known to be $r$ times continuously differentiable, then so is $f$. Sections 1 to 3 are the common work of Martin Bauer and Philipp Harms.

math.DG↗
arXiv2012

Geodesics in infinite dimensional Stiefel and Grassmann manifolds

Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We study the distance and the geodesics in these manifolds, by reducing the matter to the finite dimensional case. We then prove that any two points in those manifolds can be connected by a minimal geodesic, and characterize the cut locus.

math.DG↗
arXiv2012

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order $s\geq0$ on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on $\Diff_c(M)$ vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on $\Diff_c(M)$ is positive for $\dim(M)=1, s>\frac12$ and $\dim(M)\geq2, s\geq1$. For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$ we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$, the modified Constantin-Lax-Majda equation for $s=\frac 12$ and the Camassa-Holm equation for $s=1$.

math.DG↗
arXiv2011

Sobolev metrics on shape space of surfaces

Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ is the induced metric on $M$, $h,k \in Γ(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + AΔ^p$, where $Δ$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\bar g$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.

math.DG↗
arXiv2011

Curvature weighted metrics on shape space of hypersurfaces in $n$-space

Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from $M$ to $\mathbb R^n$. The results of \cite{Michor118}, where mean curvature weighted metrics were studied, suggest incorporating Gauß curvature weights in the definition of the metric. This leads us to study metrics on shape space that are induced by metrics on the space of immersions of the form $$ G_f(h,k) = \int_{M} Φ. \bar g(h, k) \vol(f^*\bar{g}).$$ Here $f \in \Imm(M,\R^n)$ is an immersion of $M$ into $\R^n$ and $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$. $\bar g$ is the standard metric on $\mathbb R^n$, $f^*\bar g$ is the induced metric on $M$, $\vol(f^*\bar g)$ is the induced volume density and $Φ$ is a suitable smooth function depending on the mean curvature and Gauß curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space and the conserved momenta arising from the obvious symmetries. Numerical experiments illustrate the behavior of these metrics.

math.DG↗