Center for Stokastisk Geometri og Advanceret Bioimaging · Integral geometry and advanced stereology Eva B. Vedel Jensen / Markus Kiderlen Topological properties Andrew du Plessis

CENTRE FOR STOCHASTIC GEOMETRYAND ADVANCED BIOIMAGING

Center for Stokastisk Geometri og Advanceret Bioimaging

Research Plan

Department of Mathematical Sciences | Stereology and EM Research Laboratory

Aarhus University

Eva B. Vedel Jensen | Jens R. Nyengaard

16 June 2008 BioStoc

f x( )dx= wd-i t d-1-i I t<d K,x,u( ){ }f x+tu( )Li K;d x,u( )( )dtNor K( )

∫0

�

∫i=0

d-1

∑Rd \K

∫

f x( )dx= wd-i t d-1-i I t<d K,x,u( ){ }f x+tu( )Li K;d x,u( )( )dtNor K( )

∫0

�

∫i=0

d-1

∑Rd \K

∫

Bioimaging

Φ j,r,s K ∩ E( )µkd dE( )

A d,k( )∫

Φ j,r,s K ∩ gL( )µ dg( )G d( )∫

V1,d−10( ) K,M( ) = h K,u( )S −M,du( )

Sd−1∫

Centre for Advanced Bioimaging

and Stochastic Geometry

Hjemmeside: www.csgb.dk


CSGB - et VKR Center of Excellence

I 2010 donerede Villum Fonden 25 mio kr til et nyt VKR Center ofExcellence.

Tværvidenskabeligt samarbejde mellem

Institut for Matematik, Aarhus Universitet (AU math)Klinisk Institut, Aarhus Universitet (AU bio)Institut for Matematiske Fag, Aalborg Universitet (AAU)Datalogisk Institut, Københavns Universitet (KU)


CSGBs formål

At udvikle nye matematiske og statistiske metoder til atanalysere billeddataMange af metoderne anvender de nyeste udviklinger inden forstokastisk geometriVi anvender algebra, analyse, topologi, sandsynlighedsteori ogstatistik


CSGBs medarbejderstab

Staben består af

7 professorer14 lektorer9 postdocs/adjunkter11 PhD studenter

Den stokastiske geometri gruppe ved Karlsruhe Institute of Technology er envigtig international samarbejdspartner.


CSGBs medarbejderstab

Interne workshops to gange årligt


Center aktiviteter og forskeruddannelse

Internationale konferencer og workshops(i alt 7 møder, f.eks. Sandbjerg, juni 2011, oktober 2012,september 2014)

Internationale minisymposier og ’research kitchens’(Aarhus, juni and oktober 2010, februar og august 2011)

Internationale PhD kurser(i alt 17 kurser, f.eks. Aalborg, maj 2013)

Erasmus programmer(Frankfurt)

Udveksling af PhD studenter og postdocs(Bern, Heidelberg, Karlsruhe, Perth, Prague)


De fire deltagende forskergrupper

AU math AU bio

AAU KU


CSGBs forskningsplan

Administrative and Research support office

Advanced bioimaging Jens R. Nyengaard/

Mads Nielsen

Non-uniform samplingUte Hahn /

Jens R. Nyengaard

Fluorescence microscopy taken to the molecular level

Jens R. Nyengaard / Rasmus P. Waagepetersen

Molecular cryo-EMMonika Golas / François Lauze /

Björn Sander

Research trainingPartner universities

Integral geometry and advanced stereology

Eva B. Vedel Jensen / Markus Kiderlen

Topological propertiesAndrew du Plessis

Digital stereology Markus Kiderlen

Rotational integral geometry

Eva B. Vedel Jensen

Advisory boardHåvard Rue, Dietrich StoyanHans Hebert, Steven Zucker

Research

Statistics of stochastic geometry models

Jesper Møller / Jens Ledet Jensen

Spatial and spatio-temporal point processes

Jesper Møller / Rasmus P. Waagepetersen

Space-time lattice dataMads Nielsen /

Kristjana Jónsdóttir

Random shapes Francois Lauze / Mads Nielsen

Jens R. Nyengaard | co-directorEva B. Vedel Jensen | director


Samarbejdsprojekter


Rotations integral geometri


Rotations integral geometri

Lokal stereologi giver adgang til celle størrelser i 3DMålinger på lokale snit anvendes

T


Lokal stereologi - eksempler i 2D

Cirkel

r

Areal = πr2



Konveks mængde

rdf

Areal (skraveret område)= 1

2 · højde · grundlinie = 12 · r · rdφ = 1

2 · r2dφ



Konveks mængde

rdf

Areal =∫ 2π0

12 r2dφ = π

∫ 2π0 r2 dφ

2π = πr̄2



Ikke-konveks mængde

r0

r1

r2

Areal (skraveret område)=12(r2

0 + r22 − r2

1 )dφ

Areal= πr̄2



w

r

dr

dv=r2drdw

Volumen= 4π3 r̄3


Lokal stereologi - n dimensioner

Den generaliserede Blaschke-Petkantschin formel:

c(n − q − r , p − q − r)

∫X1

· · ·∫

Xq

g(x1, . . . , xq)

q∏i=1

dxni

=

∫Ln

p(r)

∫X1∩Lp

· · ·∫

Xq∩Lp

g(x1, . . . , xq)

×∇r+q(e1, . . . , er , x1, . . . , xq)n−pq∏

i=1

dxpi dLn

p(r)

n = 3, q = 1, r = 0, p = 1: volumen i 3D fra lokale snit!


Lokal stereologi - volumen i 3D

Isotropisk L1:23

∑x∈∂X∩L1

(−1)α(x)‖x‖3

Isotropisk L2, uniform G1 ⊆ L2 :

2A1∑

x∈∂(X∩L2)∩G1(−1)α(x)

∫ |x·ω|0

√v2 + ‖πL2L1x‖2dv

Vertikal L2, uniform G1 ⊆ L2, L1 ⊥ L1(0):π2 A1

∑x∈∂(X∩L2)∩G1

‖πL1x‖2

Isotropisk T2, uniform G1:

A2∑

x∈∂(X∩T2)∩G1(−1)α(x)

∫ |x·ω|0 F1,2

(t2

v2+‖πL⊥1

x‖2

)dv

Vertikal T2, uniform G1, L1 ⊆ L⊥1(0):

A2∑

x∈∂(X∩T2)∩G1(−1)α(x)

∫ |x·ω|0 F1,1

(t2

v2+‖π(L1(0)+L1)⊥ x‖2

)dv


Lokal stereologi - overflade areal i 3D

Isotropisk L2, isotropisk L1 ⊆ L2:2π∑

x∈X∩L1(1 + cotβx(

π2 − βx))‖x‖2

Isotropisk T2, uniform og isotropisk G1:2A2

∑x∈X∩T2∩G1

F1,2(t2/‖x‖2)−1

Vertikal T2, uniform og isotropisk G1:2A2

∑x∈X∩T2∩G1

F1,1(t2/‖πL⊥1(0)

x‖2)−1

Isotropisk T2, G1||T2:2A2

∑x∈X∩T2∩G1

F1,1(t2/‖πL⊥1

x‖2)−1


Lokal stereologi - længde i 3D

Isotropisk T2, uniform og isotropisk G2:2A1

∑x∈X∩T2∩G2

F1,2(t2/‖x‖2)−1

Vertikal T2, uniform og isotropisk G2:2A1

∑x∈X∩T2∩G2

F1,1(t2/‖πL⊥1(0)

x‖2)−1

Isotropisk T2, G2 ⊥ T2:2A1

∑x∈X∩T2∩G2

F1,1(t2/‖πL⊥1

x‖2)−1


Lokal stereologi - antal i 3D

Isotropisk T2:∑x∈X∩T2

F1,2(t2/‖x‖2)−1

Vertikal T2:∑x∈X∩T2

F1,1(t2/‖πL⊥1(0)

x‖2)−1


Digital stereologi

Projektet drejer sig om estimation af geometriske karakteristika fradiskrete binære billeder af en 2- eller 3-dimensional struktur.

Nye anden-ordens udviklinger tillader estimation af merekomplicerede geometriske karakteristika end volumen og overfladeareal. Eksempler er såkaldte indre volumina, som kan udtrykkessom krumningsintegraler.


Rumlige punkt processer

Intracellulære protein interactioner kan studeres ved hjælp af FRETmikroskopi.

Vi anvender rumlige punkt proces modeller til at studerefordelingen af proteiner og deres interaktioner.

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Non-uniform sampling

Antal kan bestemmes med større præcision, hvis synsfelter imikroskopet vælges med en sandsynlighed, der er proportional meden informativ hjælpevariabel.

Her skal antallet af granular celler i det blå lag bestemmes.Synsfelter er vist som gule firkanter.

6 Proportionator Sampling and Estimation

Figure 3. Estimating total number of granule cells in rat cerebellum. The blue granule cell layer is clearly visible

at 1.25X (upper left panel). The area of interest is delineated coarsely and partitioned into fields of view. The upper

right panel shows the fields of view with their assigned weight on a grey-scale. Middle left panel shows the distribution

of sampled fields (yellow rectangles) for the proportionator, the selected fields of view are almost surely in the granule

cell layer. As shown in the middle right panel!sampling with the traditional SURS!such fields of view may or may not

hit the blue region. The lower two panels are examples of counting at 100X magnification (oil lens).

Total number of GFP orexin neurons in mice brain

Two brains were studied from mature transgenic mice, where orexin neurons in lateral hypothalamus and

adjacent perifornical area could be visualized in situ by expression of enhanced green fluorescent protein

(Burdakov et al. 2006). Brains had been immersion fixed in 4% phosphate-buffered formaldehyde for a

few hours, cryo-protected and frozen in liquid nitrogen. The brains were cut exhaustively using a

Documents

Center for Stokastisk Geometri og Advanceret Bioimaging · Integral geometry and advanced stereology Eva B. Vedel Jensen / Markus Kiderlen Topological properties Andrew du Plessis