Geometry and Physics of Spatial Random Systems...

Geometry and Physics of SpatialRandom Systems

Project 6: Image Analysis & Spatial Statistics

Eva B. Vedel Jensen, Markus Kiderlen

Norbert Henze, Daniel Hug, Klaus Mecke

December 16, 2010

CENTRE FOR STOCHASTIC GEOMETRYAND ADVANCED BIOIMAGING

Overall goal

To develop methods of extracting geometric characteristics andfeatures from physical data in a quantitative, robust andefficient manner.

Model-free inference• Minkowski functionals and tensor valuations

from digital images• Tensor valuation estimation from lower dim. sections• Shape-from-tensor problem

(−→ Project 1: Tensor valuations)

Model-based inference• non-parametric estimation of radius distribution

(−→ Project 3: Boolean models)• H.E.S.S. skymap deviations from background models

(−→ Project 4: Random fields)

Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics

Motivation: A naive digital algorithm

K KGauss−→digitization

↓ ↓

2V1(K ) = 14.5 2V1(K ) ≈ 18.0

Φ0,21 (K ) =

(48 3131 20

)Φ0,2

1 (K ) =

(8 00 10

).

Bias persists asymptotically: No multigrid convergence.No known local algorithm is multigrid convergent.


Digital algorithms for tensor valuation determination

Objectives:

Investigate asymptotic worst case errors for existing localdigital algorithms for Minkowski functionals and tensorvaluations.

Give a formal proof of the conjecture [Kenmochi, Klette(2000)] that no local algorithm is multigrid convergent.

Design and apply global or semi-local digital algorithms fortensor valuations.

→ 3 years PostDoc (financed by the Villum Foundation)→ full-time programmer position (Erlangen)


Classical Miles formulas.

Let Z be a stationaryBoolean model of random balls.

Method of moments in R3:

V 3(Z ) = 1− e−V 3

V 2(Z ) = e−V 3V 2

V 1(Z ) = e−V 3

(πV 1 −

π2

8V

22

)V 0(Z ) = e−V 3

(γ − 1

2V 1V 2 +

π

48V

32

).


Digital Miles-type formulas.

Digitized Boolean model Z ∩ tZ3

Vj = local digital algorithm.

Conjectured asymptotic digital Miles-type formula

EV3(Z ∩ tZ3) = 1− e−V 3

EV2(Z ∩ tZ3) = e−V 3V 2 + o(1)

EV1(Z ∩ tZ3) = e−V 3(

V 1 − c1V 2

)+ o(1)

EV0(Z ∩ tZ3) = e−V 3(c2

tV

22 + γ + c3V 1V 2 + c4V

32

)+ o(1),

as t → 0 + (increasing resolution).

cf. Katja Schladitz (ITWM, cooperating researcher) et al.


Digital Miles-type formulas for Boolean models

Objectives:

Develop asymptotic digital versions of Miles-type formulasfor stationary Boolean models of random balls.

Extension to medium-large but finite resolution, and totensor valuations.

Laurent expansions for large parallel volumes offinte/compact sets.

(−→ Project 3: Boolean models)


Tensor estimation from central sections

Estimate the moment tensor of K ⊂ Rd

Ψr (K ) =1r !

∫K

x r dx

from central sections of K :


Tensor estimation from central sections

Rotational integral geometry: Find a functional αr such that∫G(d ,q)

αr (K ∩ L) νq(dL) = Ψr (K ).

��Grassmannian of q-dim.

linear subspaces.

Haar measure@@I

A solution (Blaschke-Petkantschin formula):

αr (K ∩ L) = const ·∫

K∩Lx r‖x‖d−q dx .

⇒ for isotropic L, αr (K ∩ L) estimates Ψr (K ) unbiasedly.


Tensor valuation inference from planar sections

Objectives:

Construct improved unbiased estimates of moment tensorsfrom lower dimensional central sections in R3.k = 1 (line probes) k = 2 (plane probes)

use ortrip use pair of perpendic. planes.

Find (unbiased) estimates for tensor related scalars.

Extend the results to partially isotropic cases and to higherdimensions.


Examples for global estimators (surface area)

DSS/DPS estimators (digital straight/planar segment)[Dorst & Smeulders ’91], . . .

MLP estimators (minimum length polygon/polytope)2D: [Bulow & Klette ’00],3D: [Montanari ’70], [Slanski et al. ’72],. . .

Variant: RCH methods (relative convex hull), same in 2D,different in 3D.

Tangent based methods (3D: NOR methods)[Feschet & Tougne ’99], 3D: [Ellis et al. ’79]


Examples for local estimators

Marching cubes/ wrapper based algorithmsfor k < n, in particular k = n − 1(without merging or simplification)

digital geometry approaches (polygonal approach)[Bieri ’87], [K. Mecke ’94], . . .

Discretization of Crofton’s formula[Serra ’82], [Nagel, Ohser, Schladitz ’02,’03,. . .]Approximation of V0 in section planes using adjacencysystems.


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