K-Functions on Fibers

Joint work with

Hans Stephensen, Jon Sporring and Stefan sommer (DIKU)

Anne-Marie Svane and Rasmus Waagepetersen (AAU)

Stine Hasselholt (AU)

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 \( S\).
 

Usually \( S= \mathbb R^n \)

Ripley's K-function for point processes \(X\) on \(\mathbb R^n\) is defined by

\(K(r) = \frac{1}{\lambda}\) E[ #events within distance r from point]

where \(\lambda \) is the density (number per unit area) of events.

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

\(r\)

\( K(r)\)

Goal

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

Issues

  1. Distance measure
  2. Well posed K-function                       
  3. Uniform shape process

Space of Currents

A piece-wise smooth curve \( \gamma \) is characterized by the path-integral over vector fields \(w\)

 

where \( \tau(x) \) is the tangent of \( \gamma \) at \(x\).

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

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

  • \( s \) shape parameter
  • \(t\) spatial parameter

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

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

\( \gamma' \neq \gamma \in X \)

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

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