1. Emission Discrete Tomography and Optimization Problems Attila Kuba Department of Image Processing and Computer Graphics University of Szeged

2. OUTLINE Theoretical foundations of (Transmission) Discrete Tomography (DT)  Existence  Uniqueness  Reconstruction Theoretical foundations of Emission Discrete Tomography (EDT)  Existence  Uniqueness  Reconstruction Optimization in EDT, experiments

3. DISCRETE TOMOGRAPHY (DT) Reconstruction of functions from their projections, when the functions have known discrete range D = {d 1,...d k }

4. BINARY TOMOGRAPHY reconstruction of functions having binary range sets (characteristic functions) binary matrices 2 4 3 4 1 3 4 3 2 1 1 11 1111 111 1111 1

5. BINARY MATRICES r1r1 r2r2 rmrm s 1 s 2... s n f 11 f 12 f 1n f 21 f 22 f 2n f m1 f m2 f mn R S row sums: column sums:

6. CLASSIFICATION OF THE PROJECTIONS 3 3 1 3 2 1 331331 321321 1 1 1111 1111 1 1 1 1 inconsistent unique non-unique

7. SWITCHING COMPONENT 2 4 3 4 1 3 4 3 2 1 1 11 1111 111 1111 1 1 configuration 2 4 3 4 1 3 4 3 2 1 1 11 1111 111 1111 1 It is necessary and sufficient for non-uniqueness.

8. CONSISTENCY ∑ s j * ≥ ∑ s j ’ k = 1,…,n j=1 k k

9. CONSISTENCY ∑ s j * ≥ ∑ s j ’, k = 1,…,n j=1 k k 3 3 1 331331 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 2 3 3 1 1 1 1 1 1 1 1 1 it is a necessary and sufficient condition for the existence example: k = 2: 3+2 < 3+3 Gale, 1957, Ryser 1957

10. SETS AND THEIR PROJECTIONS F (measurable) plane set, f its characteristic function, its horizontal and vertical projections: further projections taken from arbitrary directions can be defined in a similar way after a suitable rotation of f or (F)

11. SECOND AND THIRD PROJECTIONS second projections: third projections: G F

12. UNIQUENESS AND EXISTENCE inconsistent unique non-unique G F Lorentz, 1947

13. UNIQUE SETS F

14. NON-UNIQUE SETS G F

15. SWITCHING COMPONENT

16. NON-UNIQUE SETS A set is non-unique w.r.t. its two projections if and only if it has a switching component.

17. ABSORPTION let us suppose that the space is filled with some material having known µ-absorption then points/set emitting rays with unit intensity

18. ABSORBED PROJECTIONS horizontal and vertical µ-absorption projections of G G

19. EMISSION BINARY TOMOGRAPHY reconstruction of objects emitting rays with constant intensity from absorbed projections (e.g. radioactivity detection in absorbing volume) mathematically reconstruction of a set G (equivalently, its characteristic function) from its absorbed projections P (µ) G henceforth supposed the horizontal and vertical µ-absorbed projections are given

20. RECONSTRUCTION AND CONNECTED PROBLEMS µ: given absorption coefficient, G : a sub-class of the plane sets, e.g. planar convex bodies Problem 1: RECONSTRUCTION IN CLASS G Given: p 1 and p 2 integrable functions. Task: Construct a set G in class G such that and a.e.

21. RECONSTRUCTION AND CONNECTED PROBLEMS Problem 2: CONSISTENCY IN CLASS G Given: p 1 and p 2 integrable functions. Question: Does it exist a set G in class G such that and a.e.?

22. Problem 3: UNIQUENESS IN CLASS G Given:A set G in class G. Question: Does it exist a different set G’ in class G such that its µ-absorption projections are the same as the µ-absorption projections of G a.e.? RECONSTRUCTION AND CONNECTED PROBLEMS

23. ABSORBED SECOND AND THIRD PROJECTIONS second projections: G third projections:

24. UNIQUENESS AND EXISTENCE G inconsistent unique non-unique

25. UNIQUE SETS

26. RECONSTRUCTION

27. NON-UNIQUE SETS

28. ABSORBED SWITCHING COMPONENT absorbed switching component ≈ (unabsorbed) switching component

29. NON-UNIQUE SETS A set is non-unique w.r.t. its two absorbed projections if and only if it has an absorbed switching component; it has an non-absorbed switching component; it is non-unique w.r.t. its two non-absorbed projections. the class of non-unique sets is the same as in the case of non-absorbed projections

30. ABSORBED PROJECTIONS G= {g ij } m×n : binary matrix (object) to be reconstructed µ: absorption coefficient absorbed row and column sums of G: (β=1, classical case without absorption) G G

31. AN INTERESTING SPECIAL CASE let then 1 0 0 0 1 1 or, equivalently, generally same for columns … 1 0 0 0 1 1

32. ABSORBED PROJECTIONS 1 0 0 0 1 1 1 0 0 0 1 1

33. 2D SWITCHING COMPONENT 0 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 2D elementary switching component

34. UNIQUENESS If a binary matrix has such a 2D switching component then it is non-unique w.r.t. the two projections. 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 Is the reverse statement true?

35. UNIQUENESS NO! an example: 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 they have no such 2D switching component but they have the same projections: 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 10100 = 10011 = 01111 01011 = 01100 = 10000 01111 = 10011 = 10100

36. UNIQUENESS 1 0 1 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 1 * = compositions of 2D elementary switching components = 2D switching components 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 * =

37. UNIQUENESS 1 0 0 0 1 1 0 1 1 1 1 1 0 0 another kind of composition 1 0 0 0 1 1 * 1 0 0 = 0 1 1 0 1 1 1 1 0 0

38. UNIQUENESS All 2D switching patterns can be constructed from 2D elementary switching components. The non-unique binary matrices contains the composition of 2D elementary switching components. 0 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0

39. UNIQUENESS Similar theorems seem to be applicable for other βs, like β -1 = β -2 + β -3 + β -4 β -1 = β -2 + β -3 + β -4 + β -5 … β -1 + β -2 = β -3 + β -4 + β -5 …

40. ABSORBED PROJECTIONS RECONSTRUCTION Complexity of the reconstruction problem ? P ? NP? LP-relaxation to range [0, 1] and round fractional solution (interior point methods) feasibility check

41. ABSORBED PROJECTIONS RECONSTRUCTION for certain βs the reconstruction is trivial from one projection: e.g. β ≥ 2 or, equivalently, μ ≥ log2 - c.f. numeration systems interesting (non-trivial) cases: β -1 = β -2 + β -3 β -1 + β -2 = β -3 + β -4 + β -5 …

42. ABSORBED PROJECTIONS RECONSTRUCTION G : the class of hv-convex binary matrices (the rows and columns have the consecutive 1 property) there is a reconstruction algorithm with O(n 2 ) complexity 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 The same problem in the case of unabsorbed projections is NP- hard!

43. ABSORBED PROJECTIONS RECONSTRUCTION opposite projection pairs determine the binary matrices uniquely for certain βs. 11 1111 111 1111 1 Barcucci, Frosini, Rinaldi, 2003 but not for any … e.g., if β -1 + β -4 = β -2 + β -3 1 0 0 1 0 1 1 0

44. ABSORBED PROJECTIONS RECONSTRUCTION 4 projections enough? NO! 1001 1 1 1 1 11 if β -1 + β -4 = β -2 + β -3 0 0 0 0 0 0

45. ABSORBED NOISY PROJECTIONS RECONSTRUCTION |Ag – y| < ε undetermined part

46. ABSORBED PROJECTIONS Open problems: 3D case ? # projections > 2 ?

47. ABSORBED PROJECTIONS RECONSTRUCTION Φ = ║Ag - y║ 2 + γ·║g║ 2 optimization cost function Metropolis algorithm (SA)

48. EXPERIMENTS binary object (0 – black, 1 - white) 128×128 fan-beam projections 401 detectors/proj stopping condition: there was no accepted change in the last 10.000 iterations University of Szeged Zoltán Kiss, Antal Nagy, Lajos Rodek, László Ruskó

49. NUMBER OF PROJECTIONS 32 16 8 166 s 619 s 176 s307 s 126 s90 s 10 % noise

50. DISTANCE CENTR. - DETECTOR 150 600 900 166 s619 s 682 s 494 s 509 s 425s 10 % noise #proj. = 32

51. ABSORPTION COEFFICIENT 166 s619 s 626 s652 s 0.005 0.009 0.03 10 % noise #proj. = 32

52. DIscrete REConstruction Tomography software tool for generating/reading projections reconstructing discrete objects displaying discrete objects (2D/3D) available via Internet http://www.inf.u-szeged.hu/~direct/ it is under development E-mail: direct@inf.u-szeged.hu

53. WORKSHOP ON DISCRETE TOMOGRAPHY 13-15 June, 2005 Graduate Center, City University of New York Organisers: Gabor T. Herman E-mail:gherman@gc.cuny.edu Attila Kuba E-mail:kuba@inf.u-szeged.hu