Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

Abstract

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fréchet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fréchet minimizer and the asymptotic distribution of the sample Fréchet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds.

A Appendix on Riemannian Manifolds

Often the manifold M in applications has a natural Riemannian metric tensor g. That is, it is given an inner product $\langle , \rangle _p$ on the tangent space $T_pM$ at p, which is smoothly defined. In local coordinates in $U_p$ given by $\psi _p(\cdot )=x= (x_1,\ldots ,x_d) \in B_p$, the functions $(g_{ij})(x) = \langle E_i,E_j\rangle _p$, with $E_i= \mathrm {d}\psi _p^{01} (\partial /\partial x_i)$ ($i,j =1,\ldots ,d$), are smooth in $B_p$. This allows one to measure the length of a smooth arc $\gamma$ joining any two points q, $q'$ in $U_p$, namely, $\int _{[a,b]}|\mathrm {d}x(t)/\mathrm {d}t| \mathrm {d}t$, $\gamma (a) = q$, $\gamma (b) = q',$ $x(t) = \psi _p\circ \gamma (t)$. Here $|\mathrm {d}x(t)/\mathrm {d}t|^2= \langle \mathrm {d}x(t)/\mathrm {d}t, \mathrm {d}x(t)/\mathrm {d}t\rangle _p$, with $\mathrm {d}x(t)/\mathrm {d}t$ expressed in the local frame $E_i$($i=1,\ldots , d$). One may also write $\mathrm {d}x(t)/\mathrm {d}t$ as $\mathrm {d}\gamma (t)/ \mathrm {d}t$. Using the compatibility condition (ii) above one now defines the length of a smooth arc joining any two points in M. The geodesic distance $\rho _g(p,q)$ between p and q is the minimum of lengths of all smooth arcs joining p and q. A standard parametrization of a curve is its arc length s: $s=\int _{[a,t]} |\mathrm {d}\gamma (u)/\mathrm {d}u| \mathrm {d}u$. In this parametrization of curves, one has $|\mathrm {d}\gamma (t)/\mathrm {d}t| =1$. We will adopt this so called unit speed parametrization unless otherwise specified. The property of local minimization of arc lengths yields a first order condition on the velocity $\mathrm {d}\gamma (t)/\mathrm {d}t$ of the minimizing curve $\gamma$ at t: the acceleration along $\gamma$ is zero at every parameter join t. If M is a submanifold ((hyper) surface) of an Euclidean space $\mathbb {R}^N$, then the second derivative $\mathrm {d}^2\gamma (t)/\mathrm {d}t^2$ is well defined, but in general does not belong to the tangent space of M at $\gamma (t)$. By ‘acceleration’ one means the orthogonal projection of the vector $\mathrm {d}^2\gamma (t)/\mathrm {d}t^2$ onto the tangent space of M at $\gamma (t)$. This projection is called the covariant derivative of the velocity and denoted $(\mathrm {D}/\mathrm {d}t) \mathrm {d}\gamma (t)/\mathrm {d}t$. The “zero acceleration” of a geodesic $\gamma$ means $(\mathrm {D}/\mathrm {d}t) \mathrm {d}\gamma (t)/\mathrm {d}t=0$. On a general differentiable manifold, which is not given explicitly as a submanifold, there is no “outside”. The proper extension of the above notion of covariant derivative by Levi–Civita, using a notion known as affine connection, for all differentiable manifolds was a milestone in the development of differential geometry (See, e.g., Do Carmo 1992 Chapter 2).

In local coordinates the equation for a geodesic is a second order ordinary differential equation. By the standard existence theorem for ordinary differential equations, a geodesic $\gamma$ is uniquely determined on a maximal interval (a, b) ($-\infty \le a<b<\infty$), given an initial point $\gamma (0) =p$ and an velocity $(\mathrm {d}\gamma (t)/\mathrm {d}t)_{t=0} = v$. According to a result of Hopf and Rinow (Do Carmo 1992, Chapter 7), the geodesics can be extended indefinitely, (i.e., $a =-\infty$ and $b= \infty$), i.e., it is geodesically complete, if and only if $(M,\rho _g)$ is a complete metric space; this in turn is equivalent to the topological condition (2). In particular, all compact Riemannian manifolds are geodesically complete. In most of the applications in this article M is compact.

On a complete Riemannian manifold, a geodesic $\gamma (t) = \gamma (t;p,v)$, $t\ge 0$, in the direction v, is completely determined by an initial point $p= \gamma (0)$, and an initial velocity $v=(\mathrm {d}\gamma (t)/\mathrm {d}t)t=0$. A cut point of p of the geodesic $\gamma$ along v is $\gamma (r(v); p,v)$, where r(v) is the supremum of all $t_0$ such that $\gamma$ is distance minimizing between $p=\gamma (0)$ and $\gamma (t_0)$. The set of all cut points (along all v) is called the cut locus of p, denoted ${\text {Cut}}(p)$. The geodesic distance $q\rightarrow \rho _g(p,q)$ may not be smooth at the cut locus ${\text {Cut}}(p)$, as Example 11 below shows. Next, define the exponential function ${\text {Exp}}_p: T_p(M) \rightarrow M: {\text {Exp}}_p (v) = \gamma (1;p,v)$ the point in M reached by the geodesic in time $t=1$, starting at p with an initial velocity v. It is known that ${\text {Exp}}_p$ is a diffeomorphism on an open ball $B(0:r_0)$ of $T_p(M)$, of radius $r_0=r_0(p)<\infty$, onto $M\backslash \,{\text {Cut}}(p)$ (Do Carmo 1992, p. 271). Here $r_0 = r_0(p)$ is the geodesic distance between p and ${\text {Cut}}(p)$). The inverse map ${\text {Exp}}_p^{-1} :M\backslash \,{\text {Cut}}(p) \rightarrow {\text {Exp}}_p (B(0: r_0)$ is called the inverse exponential, or the $\log$ map, $\log _p$, at p. The quantity ${\text {inj}}(M) = \sup \{r_0(p); p \in M)\}$ is the injectivity radius of M. The $\log _p$ map also provides the so called normal coordinates for a neighborhood of p.

Example 11

(Exponential and Log Maps on the Sphere $S^d$). Consider the unit sphere $S^d= \{x\in \mathbb {R}^{d+1}: |x|^2 \sum _{j=1}^d (x^{(j)})^2 = 1\}$. Because $|\gamma (t)| =1$ $\forall t$ for a curve on $S^d$, the tangent space at p may be identified as the set of vectors in $\mathbb {R}^{d+1}$ orthogonal to p, $T_p(S^d) = \{v \in \mathbb {R}^{d+1}: pv'=0\}$. Here we write p, v, etc. as row vectors. The geodesics are the big circles, so that the point reached at time one by the geodesic from p moving with an initial velocity v is the point on the big circle lying on the plane spanned by p and v at an arc distance |v|, i.e.,

$$\begin{aligned} {\text {Exp}}_p(v) = \cos (|v|)p + \sin (|v|)v/|v|,\; v \ne 0, {\text {Exp}}_p(0) = p\; (pv'=0). \end{aligned}$$

(A.1)

Also, the geodesic distance between p and q is the smaller of the lengths |v| of the two arcs joining p and q on the big circle,

$$\begin{aligned} \rho _g(p,q) = \arccos pq' \in [0, \pi ]. \end{aligned}$$

(A.2)

Note that the cut locus of p is ${\text {Cut}}(p) = \{-p\}$, and the distance between p and $-p$ is $\pi$, and ${\text {inj}}(S^d) = \pi$. Hence the map $\log _p(q)$ is defined on $S^d\backslash \{-p\}$ and obtained by solving for v the equation $\exp _p(v) =q$. Now $|v|= \rho _g(p,q)$. Plugging this in (A.1) (and using (A.2)), one has

$$\sqrt{[1 - (pq')^2]} v = [q - (pq')p](\arccos pq'), (p\ne q),$$

which yields

$$\begin{aligned} \log _p(q)&\nonumber = [q - (pq')p](\arccos p'q) /\sqrt{[1 - (pq')2]}\\&= [\rho _g(p,q)/ \sin \rho _g(p,q)] [q - (pq')p], \end{aligned}$$

(A.3)

for $q\ne p$, $q\ne - p$, $\log _p(p) =0$. The map $\log _p(q)$ is a diffeomorphism on $S^d\backslash \{-p\}$ onto $\{v\in T_pS^d: |v| < \pi \}$. If one uses complex coordinates for p, q then $pq'$ in the formula above are to be replaced by ${\text {Re}}(pq^*)$, etc.

Most of the manifolds we consider in this article are of the form $M=N/\mathcal G$. Here N is a complete Riemannian manifold with a metric tensor $\rho _{g,N}$ and $\mathcal G$ is a compact Lie group of isometries acting freely on N, i.e., except for the identity map, no g in $\mathcal G$ has a fixed point. This means that the orbit $O_p$ of a point p under $\mathcal G$ is in one-one correspondence with $\mathcal G$. As a subset of N, $O_p$ is a submanifold of N of dimension that of $\mathcal G$. Its tangent space $T_p O_p$ as a subspace of $T_pN$ is called the vertical subspace of $T_pN$, denoted $V_p$. The subspace $H_p$ of $T_pN$ orthogonal to $V_p$ is the horizontal subspace. M is then a Riemannian manifold with the metric tensor. The projection $\pi : N\rightarrow M$ is a Riemannian submersion. The quotient $N/\mathcal G$ is then a Riemannian manifold.

The final important notion from geometry needed in this section is that of curvature. First, consider a smooth unit speed curve $\gamma$ in $\mathbb {R}^2$: $1= |\dot{\gamma }(t) |^2 =\langle \dot{\gamma }(t), \dot{\gamma }(t) \rangle$. Differentiation shows that $\ddot{\gamma }(t) = \mathrm {d}^2\gamma (t)/\mathrm {d}t^2$ is orthogonal to $\dot{\gamma }(t) : \ddot{\gamma }(t) = \kappa (t)N(t)$, where N(t) is a unit vector orthogonal to $\dot{\gamma }(t)$ such that $(\dot{\gamma }(t), N(t))$ has the same orientation as $(\partial /\partial x_1, \partial /\partial x_2)$. Then $\kappa (t)$ is the curvature of $\gamma$ at the point $\gamma (t)$. Next, at a point p on a regular surface S in $\mathbb {R}^3$, let $N= N(p)$ denote a unit normal to S at p. A plane $\pi$ through N(p) intersects S in a smooth curve. Let $\kappa (. ; p,\pi )$ be the curvature of this curve. As $\pi$ varies by degrees of rotation, the curvature varies. Let $\kappa _1$ be the maximum and $\kappa _2$ the minimum of these curvatures, and let $\kappa = \kappa _1\kappa _2$. The Theorem Egregium of Gauss says that $\kappa = \kappa (p)$ ($p \in S$), the so-called Gaussian curvature, is intrinsic to the surface S, i.e., it is the same for all surfaces isometric to S (See, e.g., Boothby 1986, pp. 377–381). We now consider, somewhat informally, the case of a Riemannian manifold M. For $p \in M$ and $u,v \in T_p(M)$, consider the two dimensional subspace $\pi$ spanned by u,v. Consider the two-dimensional submanifold swept out by geodesics in M with initial velocities lying in this subspace. The Gaussian curvature of this submanifold, thought of locally as a surface, is called the sectional curvature of M at p for the section $\pi$.