Discrete tom ography of lattice im ages: a journey through Mathem - PowerPoint PPT Presentation

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 K. Joost Batenburg Centrum Wiskunde & Informatica, Amsterdam

Tom ography: acquisition

Tom ography: acquisition

Tom ography: acquisition

Tomography: reconstruction

W hat is Discrete Tom ography?  Classical definition: Reconstruction of lattice sets due to Larry Shepp

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)

J.R. Jinschek et al, Ultramicroscopy, 108(6), 589-604, 2008 Counting Atom s

Discrete Tom ography of atom s  Atoms are discrete entities  … that lie on a regular grid  Exploit this prior knowledge about nanocrystals

Horizontal projection

Vertical projection

Reconstruction from 2 projections S 2 1 2 R 1 R 2 R 3 1 1 1 1 C 1 C 2 C 3 2 2 1 T

Reconstruction from 2 projections S 2 1 2 R 1 R 2 R 3 1 1 1 1 C 1 C 2 C 3 2 2 1 T

Tw o projections

More projections

More projections

Sw itching com ponents

Sw itching com ponents

Sw itching com ponents L. Hajdu and R. Tijdeman, J. reine angew. Math. 534 (2001), 119-128. Generating polynomial:

Sw itching com ponents

Sw itching com ponents

Sw itching com ponents

Sw itching com ponents

Clever idea?

Atom ic resolution im aging 2010: HAADF STEM image of a silver nanocrystal Courtesy of Rolf Erni, Marta Rossell

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

em bedded in Al m atrix Ag nanocrystal Courtesy of Rolf Erni, Marta Rossell

Counting atom s Courtesy of Sandra van Aert

Total number of atoms: 784 Counting atom s Total number of atoms: 780

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

Atom ic resolution tom ography S. Van Aert, K.J. Batenburg et al., Three-dimensional atomic imaging of crystalline nanoparticles, Nature 470, 374-377 (2011).

How certain can w e be? Joint work with Wagner Fortes, Robert Tijdeman and Lajos Hajdu

Model properties

Model properties

Characterizing solutions

Characterizing solutions

A distance bound  Approach can be used to prove uniqueness  … but also to bound how different solutions can be

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