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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.
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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)
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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
).
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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.
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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)
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
Tensor estimation from central sections
Estimate the moment tensor of K ⊂ Rd
Ψr (K ) =1r !
∫K
x r dx
from central sections of K :
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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.
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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.
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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]
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics
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.
Eva B. Vedel Jensen · Markus Kiderlen Project 6: Image Analysis & spatial Statistics