#
Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

#### Background

## 2. Frame Bundles, Stochastic Development, and Anisotropic Diffusions

#### 2.1. The Frame Bundle

#### 2.2. Development and Stochastic Development

#### 2.3. Adapted Coordinates

#### 2.4. Connection and Curvature

## 3. The Anisotropically Weighted Metric

#### 3.1. Sub-Riemannian Metric on the Horizontal Distribution

#### 3.2. Covariance and Nonholonomicity

#### 3.3. Riemannian Metrics on $FM$

## 4. Constrained Evolutions

**Definition**

**1.**

**Theorem**

**1**

#### 4.1. Normal Geodesics for ${g}_{FM}$

#### 4.2. Evolution in Coordinates

#### 4.3. Acceleration and Polynomials for $\mathcal{C}$

## 5. Cometric Formulation and Low-Rank Generator

## 6. Numerical Experiments

#### 6.1. Embedded Surfaces

#### 6.2. LDDMM Landmark Equations

## 7. Discussion and Concluding Remarks

#### 7.1. Statistical Estimators

#### 7.2. Priors and Low-Rank Estimation

#### 7.3. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A most probable path (MPP) for a driving Euclidean Brownian motion on an ellipsoid. The gray ellipsis over the starting point (red dot) indicates the covariance of the anisotropic diffusion. A frame ${u}_{t}$ (black/gray vectors) representing the square root covariance is parallel transported along the curve, enabling the anisotropic weighting with the precision matrix in the action functional. With isotropic covariance, normal MPPs are Riemannian geodesics. In general situations, such as the displayed anisotropic case, the family of MPPs is much larger; (

**b**) The corresponding anti-development in ${\mathbb{R}}^{2}$ (red line) of the MPP. Compare with the anti-development of a Riemannian geodesic with same initial velocity (blue dotted line). The frames ${u}_{t}\in \mathrm{GL}({\mathbb{R}}^{2},{T}_{{x}_{t}}M)$ provide local frame coordinates for each time t.

**Figure 2.**(

**a**) Normal distributions $u\mathcal{N}(0,\mathrm{Id})$ in the tangent space ${T}_{{x}_{0}}M$ with covariance $u{u}^{T}$ (blue ellipsis) can be mapped to the manifold by applying the exponential map ${\mathrm{Exp}}_{{x}_{0}}$ to sampled vectors $v\in {T}_{{x}_{0}}M$ (red vectors). This effectively linearises the geometry around ${x}_{0}$; (

**b**) The stochastic development map ${\phi}_{u}$ maps ${\mathbb{R}}^{d}$ valued paths ${w}_{t}$ to M by transporting the covariance in each step (blue ellipses) giving a covariance ${u}_{t}$ along the entire sample path. The approach does not linearise around a single point. Holonomy of the connection implies that the covariance “rotates” around closed loops—an effect which can be illustrated by continuing the transport along the loop created by the dashed path. The anisotropic metric ${g}_{FM}$ weights step lengths by the transported covariance at each time t.

**Figure 3.**Relations between the manifold, frame bundle, the horizontal distribution $HFM$, the vertical bundle $VFM$, a Riemannian metric ${g}_{R}$, and the sub-Riemannian metric ${g}_{FM}$, defined below. The connection $\mathcal{C}$ provides the splitting $TFM=HFM\oplus VFM$. The restrictions ${\pi}_{*}{|}_{{H}_{u}M}$ are invertible maps ${H}_{u}M\to {T}_{\pi \left(u\right)}M$ with inverse ${h}_{u}$, the horizontal lift. Correspondingly, the vertical bundle $VFM$ is isomorphic to the trivial bundle $FM\times \mathfrak{gl}\left(n\right)$. The metric ${g}_{FM}:{T}^{*}FM\to TFM$ has an image in the subspace $HFM$.

**Figure 4.**Curves satisfying the MPP equations (top row) and corresponding anti-development (bottom row) on three surfaces embedded in ${\mathbb{R}}^{3}$: (

**a**) An ellipsoid; (

**b**) a sphere; (

**c**) a hyperbolic surface. The family of curves is generated by rotating by $\pi /2$ radians the anisotropic covariance represented in the initial frame ${u}_{0}$ and displayed in the gray ellipse.

**Figure 5.**Minimizing normal MPPs between two fixed points (red/cyan). From isotropic covariance (top row, (

**a**)) to anisotropic (top row, (

**c**)) on ${\mathbb{S}}^{2}$. Compare with minimizing Riemannian geodesic (black curve). The MPP travels longer in the directions of high variance. Families of curves (middle row, (

**d**–

**f**)) and corresponding anti-development (bottom row, (

**g**–

**i**)) on the three surfaces in Figure 4. The family of curves is generated by rotating the covariance matrix as in Figure 4. Notice how the varying anisotropy affects the resulting minimizing curves, and how the anti-developed curves end at different points in ${\mathbb{R}}^{2}$.

**Figure 7.**(Top row) Matching of two landmarks (green) to two landmarks (red) by (

**a**) computing a minimizing Riemannian geodesic on the landmark manifold, and (

**b**–

**e**) minimizing MPPs with added covariance (arrows) in ${\mathbb{R}}^{2}$ horizontal direction (

**b**,

**c**) and vertical (

**d**,

**e**). The action of the corresponding diffeomorphisms on a regular grid is visualized by the deformed grid which is colored by the warp strain. The added covariance allows the paths to have more movement in the horizontal and vertical direction, respectively, because the anisotropically weighted metric penalizes high-covariance directions less. (bottom row, (

**f**–

**j**)) Five landmark trajectories with fixed initial velocity and anisotropic covariance but varying ${V}^{*}FM$ vertical initial momentum ${\xi}_{0}$. Changing the vertical momentum “twists” the paths.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sommer, S. Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds. *Entropy* **2016**, *18*, 425.
https://doi.org/10.3390/e18120425

**AMA Style**

Sommer S. Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds. *Entropy*. 2016; 18(12):425.
https://doi.org/10.3390/e18120425

**Chicago/Turabian Style**

Sommer, Stefan. 2016. "Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds" *Entropy* 18, no. 12: 425.
https://doi.org/10.3390/e18120425