Emission Discrete Tomography and Optimization Problems

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 d1,...dk

4
BINARY TOMOGRAPHY
reconstruction of functions having binary range
sets (characteristic functions)
binary matrices
5
BINARY MATRICES
row sums
r1
f11 f12 f1n f21
f22 f2n fm1 fm2
fmn
r2
R
column sums
rm
s1 s2 . . . sn
S
6
CLASSIFICATION OF THE PROJECTIONS
3 2 1
1 1
1 1
3 3 1
1 1 1 1 1 1
1 1
1 1
1 1
1 1
3 2 1
3 3 1
inconsistent
unique
non-unique
7
SWITCHING COMPONENT
1 1
1 1
configuration
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
It is necessary and sufficient for non-uniqueness.
8
CONSISTENCY
9
CONSISTENCY
k
k
? sj ? sj, k 1,,n
j1
j1
it is a necessary and sufficient condition for
the existence
Gale, 1957, Ryser 1957
example
3 3 1
1
1
1
1
1
1
1
3 3 1
3 2 2
1
1
1
1
k 2 32 lt 33
1
1
1
1
1
1
1
1
1
1
1
3 3 1
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
12
UNIQUENESS AND EXISTENCE
inconsistent
unique
non-unique
Lorentz, 1947
13
UNIQUE SETS
F
14
NON-UNIQUE SETS
F
G
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
points/set emitting rays with unit intensity
let us suppose that the space is filled with some
material having known µ-absorption then
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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 p1 and p2 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 p1 and
p2 integrable functions. Question Does it exist
a set G in class G such that
and a.e.?
22
RECONSTRUCTION AND CONNECTED PROBLEMS
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.?
23
ABSORBED SECOND AND THIRD PROJECTIONS
G
second projections
third projections
24
UNIQUENESS AND EXISTENCE
G
inconsistent
unique
non-unique
25
UNIQUE SETS
26
UNIQUE SETS 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 gijmn binary matrix (object) to be
reconstructed µ absorption coefficient
G
G
absorbed row and column sums of G (ß1,
classical case without absorption)
31
AN INTERESTING SPECIAL CASE
let
or, equivalently,
1 0 0 0 1 1
then
generally
1 0 0 0 1 1
same for columns
32
ABSORBED PROJECTIONS
1 0 0 0 1 1 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
they have no such 2D switching component but
they have the same projections
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 1 1 0 1 1
1 0 0 0 0 1 1 1 0 1 1 1
0 1 1 1 0 0 1 0 0
0 1 1 1 1 0 0 0 1 0 0 0




compositions of 2D elementary switching
components 2D switching components
37
UNIQUENESS
another kind of composition
1 0 0 0 1 1 0 1 1 1 1 1 0 0
1 0 0
1 0 0 0 1 1 0 1 1
1 0 0 0 1 1 0 1 1 1 1 0 0
1 0 0

0 1 1 1 0 0 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 1 0 0 1 0 0
39
UNIQUENESS
Similar theorems seem to be applicable for other
ßs, like ß-1 ß-2 ß-3 ß-4 ß-1 ß-2 ß-3
ß-4 ß-5 ß-1 ß-2 ß-3 ß-4 ß-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 ßs the reconstruction is trivial from
one projection e.g. ß 2 or, equivalently, µ
log2 - c.f. numeration systems
interesting (non-trivial) cases
ß-1 ß-2 ß-3 ß-1 ß-2 ß-3 ß-4 ß-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(n2)
complexity
1 0 0 0 0 0 0 0 0 1 1 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 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 ßs.
1
1
1
1
1
1
1
1
1
1
1
1
1
Barcucci, Frosini, Rinaldi, 2003
1
but not for any e.g., if ß-1 ß-4 ß-2 ß-3
1 0 0 1 0 1 1 0
44
ABSORBED PROJECTIONS RECONSTRUCTION
4 projections enough? NO!
if ß-1 ß-4 ß-2 ß-3
1
0
0
1
1
1
0
0
1
1
0
0
1
0
0
1
45
ABSORBED NOISY PROJECTIONS RECONSTRUCTION
Ag y lt e
undetermined part
46
ABSORBED PROJECTIONS
Open problems 3D case ? projections gt 2 ?
47
ABSORBED PROJECTIONS RECONSTRUCTION
optimization
cost function
F Ag - y2 ?g2
Metropolis algorithm (SA)
48
EXPERIMENTS
binary object (0 black, 1 - white)
128128 fan-beam projections 401
detectors/proj stopping condition there was no
accepted change in the last 10.000 iterations
Zoltán Kiss, Antal Nagy, Lajos Rodek, László
Ruskó
University of Szeged
49
NUMBER OF PROJECTIONS
166 s 619 s 176 s 307 s 126 s 90 s
32
16
8
10 noise
50
DISTANCE CENTR. - DETECTOR
166 s 619 s 682 s 494 s 509 s 425s
150 600 900
proj. 32
10 noise
51
ABSORPTION COEFFICIENT
0.005 0.009 0.03
166 s 619 s 626 s 652 s 626 s 652 s
proj. 32
10 noise
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_at_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-mailgherman_at_gc.cun
y.edu Attila Kuba E-mailkuba_at_inf.u-szeged.hu