SLIDE 1 Discrete tom ography of lattice im ages: a journey through Mathem atics
Friday, 2 Decem ber, 2 0 1 1
W orkshop on the occasion of Herm an te Riele's retirem ent from CW I Am sterdam
Centrum Wiskunde & Informatica, Amsterdam
SLIDE 2
Tom ography: acquisition
SLIDE 3
Tom ography: acquisition
SLIDE 4
Tom ography: acquisition
SLIDE 5
Tomography: reconstruction
SLIDE 6 W hat is Discrete Tom ography?
- Classical definition: Reconstruction of lattice sets
due to Larry Shepp
SLIDE 7 History of DT
- Discrete Tomography Workshops in Germany (1994),
Hungary (1997) and France (1999)
- Key application: QUANTITEM data
- ”Mapping projected potential, interfacial roughness, and
composition in general crystalline solids by quantitative transmission electron microscopy”,
- Phys. Rev. Lett. 71, 4150–4153 (1993)
SLIDE 8 Counting Atom s
J.R. Jinschek et al, Ultramicroscopy, 108(6), 589-604, 2008
SLIDE 9 Discrete Tom ography of atom s
- Atoms are discrete entities
- …
that lie on a regular grid
- Exploit this prior knowledge about
nanocrystals
SLIDE 10
Horizontal projection
SLIDE 11
Vertical projection
SLIDE 12 Reconstruction from 2 projections
S T R1 R2 R3 C1 C2 C3
2 1 2 2 2 1 1 1 1 1
SLIDE 13 S T R1 R2 R3 C1 C2 C3
2 1 2 2 2 1 1 1 1 1
Reconstruction from 2 projections
SLIDE 14
Tw o projections
SLIDE 15
More projections
SLIDE 16
More projections
SLIDE 17
Sw itching com ponents
SLIDE 18
Sw itching com ponents
SLIDE 19 Sw itching com ponents
Generating polynomial:
- L. Hajdu and R. Tijdeman, J. reine angew. Math. 534 (2001), 119-128.
SLIDE 20
Sw itching com ponents
SLIDE 21
Sw itching com ponents
SLIDE 22
Sw itching com ponents
SLIDE 23
Sw itching com ponents
SLIDE 24
Clever idea?
SLIDE 25 Atom ic resolution im aging
2010: HAADF STEM image of a silver nanocrystal Courtesy of Rolf Erni, Marta Rossell
SLIDE 26 Som e difficulties
- Typically few m easurem ents
- Difficult to keep sample stable at atomic scale
- Alignm ent m ust be extrem ely accurate
- Accurate alignment from few projections is hard
- Nonlinear im age form ation
- In particular when imaging crystalline structures
SLIDE 27 Ag nanocrystal em bedded in Al m atrix
Courtesy of Rolf Erni, Marta Rossell
SLIDE 28 Counting atom s
Courtesy of Sandra van Aert
SLIDE 29 Counting atom s
Total number of atoms: 780 Total number of atoms: 784
SLIDE 30 Algorithm
- Prior Know ledge
- Regular lattice
- One atom type
- 3D connectivity with no holes
- Slices with distance > 2 from the boundary
should contain no holes
- Minimal number of boundary voxels
- Algorithm :
- Basic simulated annealing algorithm
SLIDE 31
- S. Van Aert, K.J. Batenburg et al.,
Three-dimensional atomic imaging of crystalline nanoparticles, Nature 470, 374-377 (2011).
Atom ic resolution tom ography
SLIDE 32 How certain can w e be?
Joint work with Wagner Fortes, Robert Tijdeman and Lajos Hajdu
SLIDE 33
Model properties
SLIDE 34
Model properties
SLIDE 35
Characterizing solutions
SLIDE 36
Characterizing solutions
SLIDE 37 A distance bound
- Approach can be used to prove uniqueness
- …
but also to bound how different solutions can be
SLIDE 38 Conclusions
- Discrete Tomography relates to many fields in
Mathematics
- Combinatorics
- Graph Theory
- Algebra/ Number Theory
- Linear Algebra
- Optimization
- By effectively combining results from these
fields, a coherent framework appears
- Currently a topic of strong interest
