Content of talk

1. Introduce point processes and Ripley's K-function
2. Shape valued processes
3. Currents metric and fiber K-function
4. Result on myelin fibers

Point processes

A point process $X$ on a metric space $S$ is a measurable map from some probability space $(\Omega, F, P )$ into the space of locally finite subsets of $S$.
 

Ripley’s K-function is a well-known statistical tool for

• analyzing second order moment
  structure of point processes
• measuring deviance from complete
  spatial randomness in point sets.                       

Ripley's K-function

Ripley’s K-function is a well-known statistical tool for

• analyzing second order moment
  structure of point processes
• measuring deviance from complete
  spatial randomness in point sets.                       

Ripley's K-function

$r$

Goal

Define a k-function for point processes where every point is a shape, for example fibers in $\mathbb R^3$

Space of Currents

$V_\gamma(w) = \int_\gamma w(x)^t\tau(x)d \lambda(x)$

Thus, we can think of $\gamma$ as a functional

$V_\gamma\in W^* = \{ f:W\to \mathbb R | \text{linear and cont} \}.$

This representation is called a current and formally the space of $m$-currents is the dual space of the differential $m$-forms.

By considering W as a RKHS, we get the inner product on $W^*$

$\langle V_{\gamma_1}, V_{\gamma_2} \rangle = \int_{\gamma_1}\int_{\gamma_2} \tau_{\gamma_1}(x)^tK(x,y) \tau_{\gamma_2}(x) d\lambda(x)\lambda(y)$

where $K$ is the kernel of $W$ and the induced distance measure

$d_c(V_{\gamma_1}, V_{\gamma_2}) = ||V_{\gamma_1} - V_{\gamma_2} ||$

Space of Currents

The K-function for fiber processes $X$ observed in window $W$ is

where c(.) denotes center point of curve, $\nu(S_0)$ measure on centered curves and $|W|$ size of window.

The K-function

$K(s,t) = \frac{1}{|W|\nu(S_0)}      \sum              \sum       1[||c(\gamma)-c(\gamma')|| \leq t, d_c(\gamma,\gamma')\leq s]$

$\gamma\in X:c(\gamma)\in W$

$\gamma' \neq \gamma \in X$

Uniform shape process

Uniformly

rotated lines:

Uniformly

rotated spirals:

Brownian motions:

Myelin fibers

Future Work

• Improve uniform shape process comparison 
• Review benefits from using currents metric
• Steel fibers in cement

Thank you for watching!